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master thesis on hp discontinuous galerkin finite element method
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A. Konrad Juethner
hp-DGFEM and Transient Heat Diffusion
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q
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cc
Buv
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Dudxdydz
huvdS
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qnvdS
huvdS
Ω
Γ
Ω
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hp-DGFEM and Transient Heat Diffusion Juethner
WASHINGTON UNIVERSITY
SEVER INSTITUTE OF TECHNOLOGY
DEPARTMENT OF MECHANICAL ENGINEERING
THE hp-DISCONTINUOUS GALERKIN FEM APPLIED
TO TRANSIENT HEAT DIFFUSION PROBLEMS
by
A. Konrad Juethner
Prepared under the direction of Professor Barna A. Szabó
A thesis presented to the Sever Institute of
Washington University in partial fulfillment
of the requirements for the degree of
MASTER OF SCIENCE
December, 2001
Saint Louis, Missouri
WASHINGTON UNIVERSITY SEVER INSTITUTE OF TECHNOLOGY
DEPARTMENT OF COMPUTER SCIENCE
ABSTRACT
THE hp-DISCONTINUOUS GALERKIN FEM APPLIED TO TRANSIENT HEAT DIFFUSION PROBLEMS
by A. Konrad Juethner
ADVISOR: Professor Barna A. Szabó
December, 2001
Saint Louis, Missouri
This work addresses the application of the hp-Discontinuous Galerkin algorithm to
transient heat diffusion problems. Strong and weak formulations of the heat diffusion
equation are established first. Then, approximation spaces and convergence
characteristics are discussed. Model problems, for which exact solutions are available,
are used for investigating convergence behavior and efficiency of finite difference and
finite element methods. Separately, spatial and temporal error control methodologies
are investigated and demonstrated. For this purpose, a finite element software code was
written with h- and p-extension capabilities in one- and two- spatial dimensions and the
time dimension. It is shown that the rate of convergence of the hp-Discontinuous
Galerkin method is faster than algebraic.
copyright by
A. Konrad Juethner
2001
to my grandfather,
Dr. Ing. Konrad Jüthner
iv
Contents Tables................................................................................................................................vi
Figures .............................................................................................................................vii
Acknowledgments............................................................................................................xii
Thesis Acceptance ..........................................................................................................xiii
Defense Announcement ..................................................................................................xiv
Copyright TXu 658-006................................................................................................... xv
1. Mathematical Derivations of Heat Conduction ...................................................... 1
1.1 Strong Formulation ....................................................................................... 2
1.2 Boundary Conditions .................................................................................... 3
1.3 Generalized Formulation .............................................................................. 3
2. Numerical Approximation of the Generalized Formulation in Steady State....... 8
2.1 Approximation Spaces .................................................................................. 8
2.2 Spatial Error Control................................................................................... 10
2.3 Convergence Characteristics....................................................................... 10
2.3.1 Natural Norm ............................................................................... 11
3. Control of Spatial Errors in the Presence of Singularities................................... 12
3.1 Spatial Error in 1D by hp-Refinement ........................................................ 12
3.2 Singularity in 2D ........................................................................................ 18
4. Spatial and Temporal Error Control in 1D Diffusion Problems......................... 30
4.1 The Finite Difference Method, 1D ............................................................ 30
4.2 The Discontinuous Galerkin Method, 1D .................................................. 33
4.3 L2 Projection of Initial Conditions at t = 0+................................................ 37
4.3.1 L2 Projection of the Initial Solution f(x) = sin(πx) ...................... 40
4.3.2 L2 Projection of the Initial Solution f(x) = x(1-x)........................ 45
4.4 Temporal Error Control .............................................................................. 45
4.4.1 Adaptive Time Solvers ................................................................ 46
4.4.2 p-DGFEM and hp-DGFEM in 1D-Space and Time .................... 66
4.4.3 hp-DGFEM with Temporal Grading ........................................... 73
v
4.5 The Influence of the Spatial Grading Factor on the Temporal Error.......... 76
5. Temporal Error Control in 2D Time Dependent Problems ................................ 77
5.1 The Finite Difference Method, Two Spatial Dimensions........................... 78
5.2 The Discontinuous Galerkin Method, 2D................................................... 82
5.3 L2 Projection of Initial Conditions at t = 0+ ............................................... 82
5.4 Model Problems in 2D................................................................................ 87
5.4.1 Model Problem 4.......................................................................... 88
5.4.2 Model Problem 5.......................................................................... 88
5.5 The Initial Solution f(x,y,0) = sin(πx)sin(πy)............................................. 88
5.5.1 The Finite Difference Time Solver .............................................. 91
5.5.2 The Discontinuous Galerkin Time Solvers.................................. 91
5.6 The Initial Solution f(x,y,0) = x(1-x)y(1-y)................................................ 95
5.6.1 The Finite Difference Time Solver .............................................. 97
5.6.2 The Discontinuous Galerkin Time Solvers.................................. 98
6. Conclusions............................................................................................................. 101
Appendix A - Conventions ............................................................................................ 104
Appendix B - Shape Functions in 2D for Quadrilateral Elements with p = 8 ............... 105
Appendix C - Appropriate Spatial Discretizations for Model Problems 1, 2, and 3 ..... 106
Appendix D - MATLAB®
ODE Solver: ode15s............................................................ 113
References...................................................................................................................... 114
Vita................................................................................................................................. 116
vi
Tables 3-1. Mesh Refinement Strategy; Left Column: Overview of Solution Domain with
Increased Element Count from Top To Bottom; Right Column: Close-Up of
Singularity with Increased Refinement................................................................ 25
3-2. Convergence of p-Refinement of L-Shaped Domain, 1 Refinement,
Polynomial Orders 1 through 8............................................................................ 27
3-3. Convergence of hp Refinement of L-Shaped Domain, 8 Refinements and
Polynomial Orders 1 through 8............................................................................ 28
4-1. L2 Projection of the Polynomial Function f(x) = sin(πx)..................................... 41
4-2. Refined L2-Projection of the Polynomial Function f(x) = sin(πx)....................... 43
A-1. Physical Quantities and their Units as Used in this Paper ................................. 104
vii
Figures
3-1. Projection of Incompatible Solution onto Finite Element Solution Space: 1
Element, p=8, Error in L2 Norm = 14.9%............................................................ 14
3-2. Projection of Incompatible Solution onto Finite Element Solution Space, 1
Element, p=16, Error in L2 Norm = 8.1%............................................................ 14
3-3. Projection of Incompatible Solution onto Finite Element Solution Space, 3
Elements, Graded Mesh 10%, p=8, Error in L2 Norm = 5.0% ............................ 16
3-4. Projection of Incompatible Solution onto Finite Element Solution Space, 3
Elements, Graded Mesh 5%, p=8, Error in L2 Norm = 3.5% .............................. 16
3-5. Projection of Incompatible Solution onto Finite Element Solution Space, 3
Elements, Grading = 0.1^g, p=8, Error in L2 Can Be Made Arbitrarily Small ... 17
3-6. L-shaped Domain. Nodes are Numbered in Black; Elements are Numbered
in Red; Zero Temperature is Prescribed between Nodes 2 and 1; Zero Flux is
Prescribed between Nodes 1 and 8 ...................................................................... 18
3-7. Domain with Re-entrant Corner .......................................................................... 19
3-8. Exact Solution of uEX, i=1 ...................................................................................... 20
3-9. Exact Solution (Black Circles on Cyan Stems) Superimposed on the L-
shaped Domain; qn and ∇uΓ Displayed by Red and Blue Arrows,
Respectively......................................................................................................... 21
3-10. Finite Element Solution uFE , 3 Elements, Polynomial Degree 3 ........................ 22
3-11. Convergence of p-Refinement of L-Shaped Domain, 1 Refinement,
Polynomial Orders 1 through 8............................................................................ 27
3-12. Convergence of hp Refinement of L-Shaped Domain......................................... 28
3-13. Finite Element Solution of 8 Grading Refinements at Polynomial Order 8;
the Relative Error with Respect to the Exact Solution is 0.2682% ..................... 29
4-1. Comparison between Exact Function and its L2 Projection ................................ 41
4-2. Error Plot in the Range of ±4*10-8
....................................................................... 42
4-3. Comparison between Exact Function and its Projection ..................................... 43
4-4. Relative Error Plotted in the Range of ±2*10-15
.................................................. 44
viii
4-5. Relative Error Plotted in the Range of ±4*10-15
.................................................. 45
4-6. Time Integral of Relative Error in Energy Norm, Model Problem 1; 1
Element, p=8........................................................................................................ 48
4-7. Time Integral of Relative Error in Energy Norm, Model Problem 2, 1
Element, p=8........................................................................................................ 49
4-8. Time Integral of Relative Error in Energy Norm, Model Problem 3, 5
Geometrically Graded Elements, Spatial DOF = 8 Nodes+5 Elements*(8-1)*
p = 39 ................................................................................................................... 49
4-9. h-DGFEM with Increasing Approximation Order rm .......................................... 51
4-10. h-DGFEM Performance at Multiple Values of rm ............................................... 52
4-11. CPU Times Corresponding to Figure 4-10; Note that the Numbered Data
Points Correlate Figures 4-10 and 4-11 ............................................................... 52
4-12. Comparison of Convergence Performance: CNM and h-DGFEM at rm = 7 ....... 53
4-13. Comparison of CPU Time in the Accuracy Range of 1.94 % to 1.22*10-5
%;
The Numbered Data Points Correlate Figures 4-12 and 4-13 ............................. 53
4-14. Temporal Error Control ....................................................................................... 54
4-15. Finite Element Solution to Model Problem 1, Plotted on Uniformly Spaced
Post-Process Grid................................................................................................. 55
4-16. h-DGFEM Performance at rm = 2, Time Grading Function h(t) = t7................... 56
4-17. Convergence Rate Using Various Values rm and h(t) = t7................................... 57
4-18. CPU Time Using Various Values rm and h(t) = t7
............................................... 57
4-19. Integral of Error in Energy Norm Reduced to below 0.001% ............................. 58
4-20. Comparison of CPU Time Performance in the Accuracy Range of 0.32 % to
1.2*10-4
% Corresponding to Figure 4-19; Note that the Numbered Data
Points Correlate Figures 4-19 and 4-20 ............................................................... 58
4-21. Temporal Error Control, h-DGFEM.................................................................... 59
4-22. Solution to Model Problem 2; the h-DGFEM Mesh is Shown by the Heavy
Lines..................................................................................................................... 60
4-23. Error in Energy Norm Reduced to below 1.0%................................................... 61
4-24. Comparison of CPU Time Performance in the Accuracy Range of 0.32 % to
1.2*10-4
% Corresponding to Figure 4-23; Note that the Numbered Data
Points Correlate Figures 4-23 and 4-24 ............................................................... 61
4-25. Temporal Error Control; Time-Halving Leads to Relative Errors that are
Better than Necessary .......................................................................................... 62
ix
4-26. Convergence of Model Problem 3 Using the Integral of er(t) ............................. 63
4-27. Error in Energy Norm Reduced to below 1.0%................................................... 64
4-28. Comparison of CPU Time Performance Corresponding to Figure 4-27; Note
that the Numbered Data Points Correlate Figures 4-27 and 4-28........................ 64
4-29. Temporal Error Control ....................................................................................... 65
4-30. Solution to Model Problem 3; the h-DGFEM Mesh Is Shown by the Heavy
Lines..................................................................................................................... 66
4-31. Convergence Comparison: h- and hp-DGFEM Solving Model Problem 1........ 67
4-32. Comparison of CPU Time Performance Corresponding to Figure 4-31; Note
that the Numbered Data Points Correlate Figures 4-31 and 4-32........................ 68
4-33. Exponential Convergence of p-DGFEM ............................................................. 68
4-34. Convergence Comparison: h- and hp-DGFEM Solving Model Problem 2........ 69
4-35. Comparison of CPU Time Performance Corresponding to Figure 4-34; Note
that the Numbered Data Points Correlate with Figures 4-34 and 4-35................ 70
4-36. Initial Exponential and then Algebraic Convergence (p-DGFEM)..................... 70
4-37. Convergence Comparison: h- and hp-DGFEM Solving Model Problem 3........ 71
4-38. Comparison of CPU Time Performance Corresponding to Figure 4-37; Note
that the Numbered Data Points Correlate Figures 4-37 and 4-38........................ 72
4-39. Initial Exponential and then Algebraic Convergence (p-DGFEM)..................... 72
4-40. Model Problem 2 Converges Faster Than Algebraic Rate .................................. 74
4-41. Model Problem 2 Convergence Close to Exponential Rate ................................ 74
4-42. Model Problem 3 Converges Faster Than Algebraic Rate .................................. 75
4-43. Model Problem 3 Convergence Close to Exponential Rate ................................ 75
4-44 Optimal Spatial Grading Factor with Respect to a Specific Experiment ............ 76
5-1. Arbitrary Initial Condition ( , )f x y , 9 Elements, Refinement Level 1, p = 8 ..... 84
5-2. Result of Step 1 of L2-Projection ( , )f x y : All Nodal Values are Equal to
Zero ...................................................................................................................... 85
5-3. Result of Step 2 of L2-Projection ( , )f x y : All Element Sides are Equal to
Zero ...................................................................................................................... 85
5-4. Result of Step 3 of L2-Projection f : Error in Energy Norm = 2.59*10-5
%,
Error in Maximum Norm = 1.5*10-3
.................................................................... 86
5-5. Final Result. Assembled L2-Projection by Linear Combination of Projection
Coefficients with the Global Shape Functions .................................................... 86
5-6. Solution Domain .................................................................................................. 89
x
5-7. Initial Solution ..................................................................................................... 89
5-8. Projection Error for the Initial Condition, Error in Energy Norm: 1.57*10-7
%,
Error in Maximum Norm = 5.0*10-5
.................................................................... 90
5-9. Projected Initial Solution ..................................................................................... 90
5-10. Model Problem 4: Convergence of the Finite Difference Method ..................... 91
5-11. Convergence of the h-DGFEM for rm = 4............................................................ 92
5-12. Convergence of the h-DGFEM for rm = 8............................................................ 92
5-13. Temporal Solutions Corresponding to the Most Accurate Solution of Figure
5-12; Time Starts at Top Left (t = 0), Progresses Row by Row and Ends at
Bottom Right (t = 1.0).......................................................................................... 93
5-14. Convergence of the p-DGFEM............................................................................ 94
5-15. Convergence of the hp-DGFEM.......................................................................... 94
5-16. Temporal Solutions Corresponding to the Most Accurate Solution of Figure
5-14; Time Starts at Top Left (t = 0), Progresses Row by Row and Ends at
Bottom Right (t = 1.0).......................................................................................... 95
5-17. Initial Solution ..................................................................................................... 96
5-18. Projection Error for the Initial Condition, Error in Energy Norm = 1.3*10-28
%, Error in Maximum Norm = 8.0*10-17
............................................................. 96
5-19. Projected Initial Solution ..................................................................................... 97
5-20. Convergence of the Finite Difference Method .................................................... 97
5-21. Convergence of the h-DGFEM............................................................................ 98
5-22. Convergence of the p-DGFEM............................................................................ 99
5-23. Convergence of the hp-DGFEM.......................................................................... 99
5-24. Temporal Solutions Corresponding to the Most Accurate Solution of Figure
5-21; Time Starts at Top Left (t = 0), Progresses Row by Row and Ends at
Bottom Right (t = 1.0)........................................................................................ 100
B-1 2D Shape Functions. Top Row: Vertex Modes, Four Left Columns without
First Row: Side Modes, Columns 5 through 9: Internal Modes ........................ 105
C-1. Convergence of Model Problem 1, 1 Element in Space, p = 3.......................... 106
C-2. Convergence of Model Problem 1, 1 Element in Space, p=4............................ 107
C-3. Convergence of Model Problem 1, 1 Element in Space, p=8............................ 107
C-4. Convergence of Model Problem 1, 10 Spatial Elements, p=8........................... 108
C-5. Convergence of Model Problem 1, 10 Spatial Elements, p=16......................... 108
C-6. Convergence of Model Problem 2, 1 Spatial Element, p=3 .............................. 109
C-7. Convergence of Model Problem 2, 1 Spatial Element, p=8 .............................. 110
xi
C-8. Convergence of Model Problem 2, 10 Spatial Elements, p=8........................... 110
C-9. Convergence of Model Problem 3, 2 Spatial Elements, p=8............................. 111
C-10. Convergence of Model Problem 3, Geometrically Graded Mesh with 5
Elements, p=8, (Identical to Figure 4-8)............................................................ 112
xii
Acknowledgements
This thesis marks my most significant academic accomplishment to date. Its creation
was eye opening, enjoyable, and revealing, yet, difficult, arduous, and extremely time
consuming. Certainly not an easy feat since full-time employment at Watlow Inc
required my full attention. Although I was advised of the challenges of the dual life as
an engineer and graduate student, I saw in this task an integral part of a Master of
Science Program.
As I now reflect upon the accomplishment, I realize how much it has reduced the
interactions with my family and friends during the past two years. My wife, Salomé,
probably carried the biggest burden, dealing with my mood swings, incoherent
ramblings about mathematical tasks in delirious morning hours, the experiences of all-
time highs when things worked well, and all-time lows when programming bugs left me
searching for weeks. Many thanks for her patience and understanding during this time.
She is simply the best!
The very generous tuition reimbursement program by Watlow Inc and the supportive
mentor, Louis P. Steinhauser made this graduate work possible. Their support of both
my professional and academic careers dates back to May of 1995 and there are no words
to express my gratitude. Thank you!
I like to think of my thesis advisor, Dr. Barna Szabó, as a brilliant, picky, meticulous,
hardworking, and patient gentleman, who has driven me “bananas” during this work. I
would like to thank him for his patience in his attempt to teach me hp-FEA and DG
principles, to enhance my understanding of Mathematics, to improve my technical
writing style, and to make me a more mature engineer.
xiii
xiv
xv
xvi
1
Chapter 1
Mathematical Derivations of Heat Conduction
The derivation of the equations of linear heat conduction and the corresponding notation
are presented in fully three-dimensional setting. Mathematical models of linear heat
conduction are based on two fundamental laws of physics: Fourier’s law of heat
conduction and the conservation law. Fourier’s law of heat conduction states that
[ ] ( )q grad u= − Λ ⋅j (1-1)
where qf
is the heat flux vector with components zyx qqq ,, . The following notation will
be used:
=≡z
y
xdef
q
q
q
qqj
. (1-2)
qf
represents the heat flux per unit area (in W/m2 or equivalent units). [ ]Λ is the
symmetric positive-definite thermal conductivity matrix and u is the temperature field.
[ ]Λ can be written as
2
[ ] xx xy xz
yx yy yz
zx zy zz
λ λ λ Λ = λ λ λ λ λ λ . (1-3)
From the condition of symmetry it follows that xy yxλ = λ , etc.
1.1 Strong Formulation
The conservation law states that the heat flow rate into any volume element plus the heat
generated per unit time in the volume element equals the specific heat multiplied by the
mass density and the rate of change in temperature. In Cartesian coordinates, the
mathematical statement of the conservation law is
( ) ( ) ( )t
ucQq
zq
yq
xzyx ∂
∂=+
∂∂+∂
∂+∂∂− ρ . (1-4)
Combining (1-1), (1-2), (1-3), and (1-4) results in
xx xy xz yx yy yz
zx zy zz
u u u u u u
x x y z y x y z
u u u uQ c
z x y z t
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂λ + λ + λ + λ + λ + λ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ λ + λ + λ + = ρ ∂ ∂ ∂ ∂ ∂
(1-5)
where , , , and Q c tρ represent heat generation per unit volume per unit time, specific
heat, mass density, and time, respectively. Alternatively, equation (1-5) can be written
in the form:
([ ] ( ))u
div grad u Q ct
∂Λ ⋅ + = ρ ∂ . (1-6)
3
1.2 Boundary Conditions
Four types of boundary conditions will be considered in the following. The boundary of
the solution domain Ω will be denoted by Γ . The boundary is divided into four non-
overlapping regions, which collectively cover the entire boundary:
1. Prescribed temperature: ( , ), uu u x t x= ∈Γj j#
2. Prescribed heat flux: ( , ), n n qq q x t x= ∈Γj j j# where the flux is normal to the
boundary
3. Convection: ( ( , ) ( , )), n c c cq h u x t u x t x= − ∈Γj j j j where ch is the coefficient of
convective heat transfer measured in [W/(m2K)] and cu is the temperature of the
convective medium
4. Radiation: 44( ( , ) ( , )), r s rq f f u x t u x t xε ∞= κ ⋅ ⋅ ⋅ − ∈Γj j j j
where κ is the Stefan-
Boltzmann Constant, fε is the emissivity function with 0 1fε< ≤ , sf is the view
factor function, u is the temperature function of the radiating body, and u∞ is
the reference temperature of the environment that absorbs the radiation
1.3 Generalized Formulation
To arrive at the generalized weak form of equation (1-4), we multiply by a scalar test
function v and integrate:
yx z
qq q uv dxdydz Qv dxdydz c v dxdydz
x y z tΩ Ω Ω∂ ∂ ∂ ∂− + + + = ρ ∂ ∂ ∂ ∂ ∫∫∫ ∫∫∫ ∫∫∫ (1-7)
Using the following identities,
4
( )( )( )
xx x
y
y y
zz z
q vv dxdydz q v q dxdydz
x x x
q vv dxdydz q v q dxdydz
y y y
q vv dxdydz q v q dxdydz
z z z
Ω Ω
Ω Ω
Ω Ω
∂ ∂ ∂ = − ∂ ∂ ∂ ∂ ∂ ∂= − ∂ ∂ ∂ ∂ ∂ ∂ = − ∂ ∂ ∂
∫∫∫ ∫∫∫∫∫∫ ∫∫∫∫∫∫ ∫∫∫
(1-8)
we can rewrite (1-7) as follows:
( )
.
x y z
v v vdiv qv dxdydz q q q dxdydz
x y z
uQv dxdydz c v dxdydz
t
Ω Ω
Ω Ω
∂ ∂ ∂− ⋅ + + + + ∂ ∂ ∂ ∂+ ⋅ = ρ ⋅∂
∫∫∫ ∫∫∫∫∫∫ ∫∫∫
j
(1-9)
Applying the Gauss divergence theorem, equation (1-10) results.
( ) x y z
v v vq n v dS q q q dxdydz
x y z
uQv dxdydz c v dxdydz
t
Γ Ω
Ω Ω
∂ ∂ ∂− • ⋅ + + + + ∂ ∂ ∂ ∂+ ⋅ = ρ ⋅∂
∫∫ ∫∫∫∫∫∫ ∫∫∫
j j
(1-10)
Vector nj
is the outward positive unit normal and dS is the differential surface element.
Substituting (1-5) into (1-10), we have:
( )
.
xx xy xz
yx yy yz
zx zy zz
u u u v
x y z x
u u u vq n v dS dxdydz
x y z y
u u u v
x y z z
uQv dxdydz c v dxdydz
t
Γ Ω
Ω Ω
∂ ∂ ∂ ∂λ + λ + λ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ − • ⋅ − + λ + λ + λ + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + λ + λ + λ ∂ ∂ ∂ ∂ ∂+ ⋅ = ρ ⋅∂
∫∫ ∫∫∫
∫∫∫ ∫∫∫
j j
(1-11)
5
Defining the differential operator vector:
Tdef
Dx y z
∂ ∂ ∂= ∂ ∂ ∂ (1-12)
we can write:
( ) [ ] ( )( )
( )
.
T uD v D u dxdydz c v dxdydz
t
Qv dxdydz q n v dS
Ω Ω
Ω Γ
∂Λ + ρ ∂= − •∫∫∫ ∫∫∫∫∫∫ ∫∫ j j
(1-13)
Equation (1-13) is the generic form of the weak or generalized formulation of the
transient heat equation. The generic form is modified by the boundary conditions. The
prescribed temperature on u∂Ω is enforced by restriction. When heat flux is specified
)0( ≠Γq then the specified heat flux q~ is substituted for q in equation (1-13). When
convection is specified )0( ≠Γc then )( cc uuhq −= is substituted for q in (1-13) such
that upon rearrangement we have:
( ) ( )
( )( , ) [ ]
( ) .
c
q c
T
c
c c
B u v D v D u dxdydz h uvdS
F v Qvdxdydz q n v dS h u v dS
Ω Γ
Ω Γ Γ
= Λ += − ⋅ +∫∫∫ ∫∫∫∫∫ ∫∫ ∫∫f f (1-14)
For completeness, radiation heat transfer is mentioned as a boundary condition.
However, it is a nonlinear problem and will not be considered in the following. As a
short form for this equation, it is customary to write
)(),( vFvuB = (1-15)
6
where ( ),B u v is a bilinear form defined on ( ) ( )1 1H HΩ × Ω and ( )F v is a linear
functional defined on ( )1H Ω . The space ( )E Ω is defined by a set of functions
( ), ,u x y z that have finite energy on Ω , which is satisfied by the following condition:
∞<≤=Ω CuuBuE ),(|)( (1-16)
where C is some positive constant. We associate the norm ( )E
u Ω with the space
( )E Ω . By definition:
( )
1|| || ( , ) .
2
def
Eu B u uΩ = (1-17)
The space ( )E Ω is called the energy space. Further, we define the subsets )(~ ΩE and
( )E Ωc as follows:
( ) | ( ), ( , , ) ( , , ), ( , , ) def
uE u u E u x y z u x y z x y zΩ = ∈ Ω = ∈Γ# # (1-18)
( ) | ( ), ( , , ) 0, ( , , ) def
uE u u E u x y z x y zΩ = ∈ Ω = ∈Γc (1-19)
The generalized solution is the function ( )EXu E∈ Ω# such that
( , ) ( ) ( )EXB u v F v v E= ∀ ∈ Ωc. (1-20)
This is equivalent to finding the minimum of the functional π
1
( , ) ( )2
def
B u u F uπ = − (1-21)
7
on the space )(~ ΩE . Proof and further discussion on this subject can be found, for
example, in Finite Element Analysis [11].
8
Chapter 2
Numerical Approximation of the Generalized
Formulation in Steady State
To solve (1-13) numerically, the domain Ω is partitioned into k tetrahedral, hexahedral,
and pentahedral elements, 1,2, , ( )k M= ∆… . A particular partition is called a finite
element mesh and is denoted by ∆ . A finite dimensional subspace S of ( )E Ω is
characterized by ∆ and a polynomial degree kp is assigned to each element. A brief
description of S is presented in the following.
2.1 Approximation Spaces
We denote the subspace ( , , )pS S Q= Ω ∆f f, and define
( )
( ).
def
def
S S E
S S E
= ∩ Ω= ∩ Ωc c
# # (2-1)
S is constructed according to the partition ∆ whereby each element k is mapped from
a standard element stΩ by the mapping functions. For example, the standard
quadrilateral element stΩ is defined as:
( )
, , 1, 1, 1q
stΩ = ξ η ζ ξ ≤ η ≤ ζ ≤ (2-2)
9
The mapping between the standard quadrilateral element and the thk partition is defined
as follows:
( )
( )
( )
( , , )
( , , )
( , , ) .
k
x
k
y
k
z
x Q
y Q
z Q
= ξ η ζ= ξ η ζ= ξ η ζ
(2-3)
Let the space of polynomials of degree p defined on stΩ be pS . The finite element
space S is defined as the set of all functions ( ), ,u x y z , which lie in the energy space
( )E Ω and on the thk element ( )( )( ) ȟ,Ș,ȗ kpku Q S∈ :
( )( )( ) ( ) ( )( ) ( )( ) ( ) ( )
, ,
| ,
, , , , , , , , , 1, 2, ,k
defp
pk k k
x y z
S S Q
u u E
u Q Q Q S k M
≡ Ω ∆ =∈ Ω
ξ η ζ ξ η ζ ξ η ζ ∈ = ∆
f f
…
(2-4)
where pf
is the vector of polynomial degrees and Qf
is the vector of mapping functions
assigned to the elements
( ) ( )
1 2
1 2
, , ,
, , , .
def
M
def
M
p p p p
Q Q Q Q
∆
∆
==
f …f
… (2-5)
The continuity property of functions in S ensures that ( )S E⊂ Ω . The condition that
equation (1-13) must be satisfied for all v S∈ c results in a system of ordinary differential
equations of the form
[ ] .K a r= (2-6)
10
where [ ]K is called the stiffness matrix and r the load vector. The size of matrix
[ ]K is called the number of degrees of freedom and is denoted by N .
2.2 Spatial Error Control
The subspace S determines the finite element solution FEu and hence the error
EX FEu u− . The spatial error is controlled by proper selection of the space S . In the p-
version, a hierarchical sequence of spaces 1 2 nS S S⊂ ⊂… is constructed and
convergence is monitored. In the h-version, various adaptive methods have been
proposed in A Posteriori Error Estimates for the Finite Element Method [1] and Error
Estimates for Adaptive Finite Element Computations [2].
2.3 Convergence Characteristics
There are two fundamentally different approaches to the implementation of the finite
element method, called the h-version and the p-version. In the h-version, the solution
domain is partitioned into elements and the solution is approximated by piecewise
polynomials, defined on elements of low polynomial degree, usually 1 or 2.
Convergence is achieved by letting the size of the elements approach zero. In the p-
version, on the other hand, the partition is generally fixed and the polynomial degree of
elements is increased. Convergence is achieved by letting the lowest polynomial degree
approach infinity. Both versions can be used in combination by refining the mesh and
increasing the polynomial degree of the elements so that the finite element solution
converges to the exact solution in an optimal or nearly optimal rate. This method is
called the hp-version.
11
2.3.1 Natural Norm
The errors can be measured in various norms. The most commonly used norm is the
energy norm. The energy norm is called the natural norm because the finite element
solution satisfies the following relationship:
( ) ( )
minEX FE EXE Eu Su u u uΩ Ω∈− = − (2-7)
Referring to (1-17), the energy norm measure of error is:
( )
1( - , )
2
def
EX FE EX FE EX FEEu u B u u u uΩ− = − . (2-8)
A more useful measure is the relative error, defined by:
−=def
EX FE Er
EX E
u ue
u. (2-9)
The model problems discussed herein have been selected so as to make comparisons
with known exact solutions possible. Since exact solutions are generally not known in
engineering applications, this error can be estimated using an a posteriori estimation
procedure. Because p-extensions produce a sequence of hierarchic finite element
spaces, the convergence of the functional π with respect to the number of degrees of
freedom N given by (1-21) is monotonic.
12
Chapter 3
Control of Spatial Errors in
the Presence of Singularities
In the following, numerical examples are presented that illustrate the application of the
finite element method to heat conduction problems. A finite element code with hp
extension capabilities in one dimension (1D) and two dimensions (2D) was written in
MATLAB®
.
3.1 Spatial Error in 1D by hp-Refinement
An example showing how spatial errors are controlled is described. While complicated
to program, hp-extensions allow for very efficient treatment of any elliptic problem.
Spatial errors occur unless the exact solution happens to lie in the finite element space.
As most finite element software is based on polynomial basis functions, it is useful to
investigate how well incompatible initial solutions, such as step functions, can be
approximated. Specifically, the initial condition ( ) 1u x = with prescribed boundaries
( )0 0u = and ( )1 0u = at 0t += , defined on the solution domain ( )0,1Ω = will be
discussed.
13
Let us consider a homogeneous bar of unit length, initially at a constant temperature
0 1=u . Let us further assume that at 0t += the temperature is dropped to zero at the
ends of the bar. Thus the solution for 0>t lies in
( ) | ( ), (0) 0 (1) 0E u u E u uΩ = ∈ Ω = =c , . Since the solution at 0=t is not in ( )E Ωc , the
initial condition 0u is said to be incompatible. It is necessary to project 0u onto the
space ( )E Ωc . Using the 2L projection, this involves computing the minimum of the
integral
21
0 0
0
10
( ) , where ( ) ( )=
= − Φ Φ ∈ ⊂ Ω ∑∫ n
j j j
j
I u a x dx x S E . (3-1)
This results in a system of algebraic equations for , 1, 2, ,= …ja j n . The numerical
procedure will be discussed in detail in Section 4.3. The error between 0u and
2( )
0
1
ndefL
j j
j
u a=
= Φ∑ is measured in the least square sense that is the 2L norm. By definition:
2
21
0
10
12
0
0
n
j j
j
L
u a dx
e
u dx
= − Φ =
∑∫∫
. (3-2)
The incompatibility of the initial solution with the polynomial finite element space is
demonstrated in Figure 3-1. A single finite element is used ( )( )1M ∆ = with 8p = .
The approximation is oscillatory and the error in 2L norm is 14.9%. To reduce the error
in L2 norm, either the number of elements or their degrees of freedom must be increased.
It is shown in Figure 3-2 that doubling the polynomial order of the element leads to a
decrease in error in energy norm from 14.9% (Figure 3-1) to 8.1%.
14
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
distance x [1]
tem
pera
ture
u [
1]
uEX
and uL2
of f(x) = 1.
uEX
uL2
FIGURE 3-1. Projection of Incompatible Solution onto Finite Element Solution
Space: 1 Element, p=8, Error in L2 Norm = 14.9%
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
distance x [1]
tem
pera
ture
u [
1]
uEX
and uL2
of f(x) = 1.
uEX
uL2
FIGURE 3-2. Projection of Incompatible Solution onto Finite Element Solution
Space, 1 Element, p=16, Error in L2 Norm = 8.1%
15
However, in this particular case, it is more effective to use geometric grading toward the
ends of the domain. This is illustrated in Figure 3-3 where the results shown correspond
to a 3-element mesh. The large element is 80% of the total length, the small elements
are 10% each. It is seen that the oscillations are confined to the small elements only.
The relative error in 2L norm is 5.0 %.
By increasing the large element to 90% and reducing the small elements to 5% each, the
relative error in energy norm is further reduced to 3.5% (Figure 3-4). The obvious
question is, how well can 0u be approximated by this method.
The 2L norm error is dependent on the size of the smallest element at the end points and
can be made arbitrarily small. Considering, for example,
2( )
0
, 0
1, 1-
1, 1- 1
L
xx
u x
xx
≤ ≤ ε ε= ε < < ε − ε ≤ ≤ε (3-3)
we find:
( ) ( )2
2 212
( )
0 0
1 0 0
1 13
L xu u dx dx y dy
ε ε ε − = − = ε − = ε ∫ ∫ ∫ . (3-4)
16
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
distance x [1]
tem
pera
ture
u [
1]
uEX
and uL2
of f(x) = 1.
uEX
uL2
FIGURE 3-3. Projection of Incompatible Solution onto Finite Element Solution
Space, 3 Elements, Graded Mesh 10%, p=8, Error in L2 Norm = 5.0%
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
distance x [1]
tem
pera
ture
u [
1]
uEX
and uL2
of f(x) = 1.
uEX
uL2
FIGURE 3-4. Projection of Incompatible Solution onto Finite Element Solution
Space, 3 Elements, Graded Mesh 5%, p=8, Error in L2 Norm = 3.5%
17
Similarly:
( ) ( )2
21 1 12 2( )
0 0
1 1 0
11 1
3
L xu u dx dx y dy
−ε −ε− ε − = − = ε − = ε ∫ ∫ ∫ (3-5)
which indicates that the error in 2L norm is proportional to ε . In the numerical
example under consideration, we find a very similar dependence, which is depicted in
Figure 3-5.
100
101
10-15
10-10
10-5
100
||u(L2)
0 - u
0||
L2 / ||u
0||
L2 of f(x) = 1, f(0) = f(1) = 0, M(∆) = 3, p = 8
geometric grading factor g [1]
Err
or
in e
nerg
y n
orm
in %
FIGURE 3-5. Projection of Incompatible Solution onto FE Solution Space, 3
Elements, Grading = 0.1^g, p=8, Error in L2 Can Be Made Arbitrarily Small
Therefore, geometric grading coupled with p-extension is an effective tool for
approximating incompatible initial solutions. Although heat transfer problems are
generally "well behaved", this kind of grading is very effective for projecting
incompatible initial conditions onto the finite element space.
18
3.2 Singularity in 2D
Singularities may arise from material discontinuities, sharp corners, and abrupt changes
in boundary conditions. To demonstrate singularities in two dimensions (2D), we
consider the L-shaped domain shown in Figure 3-6. Its re-entrant corner induces a
geometric singularity. Zero flux is prescribed on the edge between nodes 1 and 8 and
zero temperature on the edge between nodes 1 and 2.
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
0
0.5
1
1.5
3
4
5
2
1
2
3
8
1
7
6
y
x
u
L-shaped domain with 3 elements.
FIGURE 3-6. L-shaped Domain. Nodes are Numbered in Black; Elements are
Numbered in Red; Zero Temperature is Prescribed between Nodes 2 and 1; Zero
Flux is Prescribed between Nodes 1 and 8
19
The exact solution of this problem in the vicinity of the re-entrant corner is of the form
i
i i
1
(cos ( 1) sin )
2 1 .
2
i
EX i
i
i
u A r
i
∞ λ=
= λ θ+ − λ θ−λ = πα
∑ (3-6)
The derivation of this is available in Finite Element Analysis [11]. The terms iA are the
expansion coefficients that depend on the other boundary conditions. α is the
complement of the angle of the reentrant corner as shown in Figure 3-7.
FIGURE 3-7. Domain with Re-entrant Corner
In this model problem 3
2
πα = , therefore 1
1
3λ = . In the following discussion we define
flux boundary conditions on the boundary segment such that the exact solution is
1
3 cos - sin3 3
EXu rθ θ = . (3-7)
A graphical illustration of EXu is shown in Figure 3-8.
20
FIGURE 3-8. Exact Solution of uEX, i=1
The boundary conditions can be handled in two ways. One is to project EXu onto the
space of the of the basis functions on the boundaries using the 2L projection method. In
this case, the boundary conditions are enforced by restriction on the space of admissible
functions. The other method is to compute the normal flux nq and enforce the boundary
conditions in the weak sense. In this work, the second method was chosen:
[ ]ˆ n EXq n uΓ= − Λ ∇ . (3-8)
Vector nΓ is the normal on the boundary Γ and [ ]Λ is the thermal conductivity tensor.
21
In this case:
2
3
2
3
1sin cos( , )
3 3 3
1 1( , ) sin cos3 3 3
EX
EX
EX
ru rr
u
u r rr
−
−
θ θ ∂ − −θ ∂∇ = = ∂ θ θ θ − + ∂θ . (3-9)
Visual representations of the boundary conditions are shown in Figure 3-9.
-0.5
0
0.5
1
1.5
-1.5
-1
-0.5
0
0.5
1
1.5
0
0.5
1
1.5
x
7
6
5
6
5
8
4
14
uEX
(cyan, black), ∇uΓq (blue), q
n (red)
1311
12
12
15
10212017
3
181922162827239
2426292235342915303336284241352136404334494841274247504056554733485457466362513961645045160495859534443515246383744453932313738322625303125201923241814131617
y
118
79
10
4
2
1
2
3
u
FIGURE 3-9. Exact Solution (Black Circles on Cyan Stems) Superimposed on the
L-shaped Domain; qn and ∇uΓ Displayed by Red and Blue Arrows, Respectively
Solving this 3-element problem using polynomial degree 3 leads to the approximate
solution shown graphically in Figure 3-10.
22
FIGURE 3-10. Finite Element Solution uFE , 3 Elements, Polynomial Degree 3
Note that the numerical solution of Figure 3-10 is substantially different from the exact
solution in Figure 3-8. The error will be estimated in energy norm.
Letting
EX FEe u u= − (3-10)
we can write:
23
[ ]2
1( ) ( ) ( , ) ( )
2
1 1( , ) ( ) ( , ) ( ) ( , )
2 2
( ) 0 .
FE EX EX EX EX
EX EX EX EX
EX E
u u e B u e u e F u e
B u u F u B u e F e B e e
u e
Π = Π − = − − − − == − + − + == Π + +
(3-11)
Therefore,
2
( ) ( )FE EXEe u u= Π −Π . (3-12)
Also, since
1
( ) ( , ) ( )2
EX EX EX EXu B u u F uΠ = − (3-13)
and
( , ) ( )EX EX EXB u u F u= (3-14)
we have
1 1
( ) ( ) ( ) .2 2
EX EX n EX EXu F u q u dsΓ
Π = − = − ∫ (3-15)
Since EXu is given by equation (3-7), ( )EXuΠ can be computed from (3-15) and
knowing ( )FEuΠ from the finite element solution:
[ ]1
2
1 2( ) ,FE n
n
r
ru a a a
r
Π = − …
B (3-16)
24
one can compute:
2
( ) ( ).FE EXEe u u= Π −Π (3-17)
The relative error is:
100 Er
EX E
ee
u= (3-18)
where
2 1 1
( ) ( ) .2 2
EX EX n EX EXEu F u q u ds
Γ= = ∫ (3-19)
For the three element mesh shown in Figure 3-6 and p = 3, we have:
( ) - 6.695974660911005 - 001
( ) - 8.471380026768022 - 001
45.8 % .
FE
EX
r
u e
u e
e
Π =Π =
= (3-20)
Errors of this magnitude are generally not acceptable for engineering purposes. The
large error is caused by the fact that EXu is not analytic, in fact strongly singular in the
reentrant corner. Hence, approximation of EXu with piecewise polynomials is not easy.
It is known that, in such cases, meshes graded in geometric progression toward the
singular point with the grading factor of ( )2
2 1− are optimal. This topic is further
discussed in Finite Element Analysis [11]. The optimal mesh refinement strategy for
controlling the error associated with the singularity at the reentrant corner is illustrated
in Table 3-1.
25
TABLE 3-1. Mesh Refinement Strategy; Left Column: Overview of Solution
Domain with Increased Element Count from Top To Bottom; Right Column:
Close-Up of Singularity with Increased Refinement
26
The optimal rate of convergence is exponential as discussed in Finite Element Analysis
[11]. To realize exponential convergence it is necessary to increase both the number of
geometrically graded layers of elements and the polynomial degree. The asymptotic rate
of convergence when p is increased on a fixed mesh is algebraic. Specifically,
provided that 1A in equation (3-6) is not zero,
1E
ke
Nλ≤ , (3-21)
see Finite Element Analysis [11].
In this model problem 1
1
3λ = , hence the theoretical rate of convergence is
1
3. It is seen
in Table 3-2 and Figure 3-11 that the numerical rate of convergence is close to 1
3.
When the number of layers of refinement is increased with p then the rate of
convergence becomes exponential. This is illustrated in Table 3-3, where it is seen that
the numerical rate is increasing as the number of degrees of freedom is increased and in
Figure 3-12, where the relative error vs. degrees of freedom curve on a log-log scale is
seen to have an increasing downward slope.
27
TABLE 3-2. Convergence of p-Refinement of L-Shaped Domain, 1 Refinement,
Polynomial Orders 1 through 8
Run# DOF Potential Energy Convergence Rate True Relative Error [%]
1
2
3
4
5
6
7
8
12
33
54
84
123
171
228
294
-7.019466711517294e-001
-7.720996474209324e-001
-7.801391374072846e-001
-7.971252036359559e-001
-8.082571556427428e-001
-8.157655245413216e-001
-8.210394799960001e-001
-8.249448993585982e-001
0.0000
0.3262
0.1151
0.3309
0.3301
0.3256
0.3199
0.3188
41.3993
29.7622
28.1227
24.2976
21.4235
19.2441
17.5522
16.1857
101
102
101.3
101.4
101.5
101.6
degrees of freedom DOF in [1]
rela
tive e
rror
e r in [
%]
Spatial Error Control by Increased Polynomial Order
||Π(uFE
)-Π(uEX
)|| / ||Π(uEX
)||E
FIGURE 3-11. Convergence of p-Refinement of L-Shaped Domain, 1 Refinement,
Polynomial Orders 1 through 8
28
TABLE 3-3. Convergence of hp Refinement of L-Shaped Domain, 8 Refinements
and Polynomial Orders 1 through 8
Run# DOF Potential Energy Convergence Rate True Relative Error [%]
1
2
3
4
5
6
7
8
12
51
114
228
411
681
1056
1554
-7.019466711517294e-001
-8.183934050633468e-001
-8.396427254420245e-001
-8.454936007622418e-001
-8.467569499625519e-001
-8.470453032705809e-001
-8.471145647099563e-001
-8.471319142504697e-001
0.0000
0.5597
0.8355
1.0942
1.2407
1.3996
1.5671
1.7439
41.3993
18.4205
9.4063
4.4058
2.1209
1.0461
0.5260
0.2682
102
101
degrees of freedom DOF in [1]
rela
tive e
rror
e r in [
%]
Spatial Error Control by Increased Polynomial Order
||Π(uFE
)-Π(uEX
)|| / ||Π(uEX
)||E
FIGURE 3-12. Convergence of hp Refinement of L-Shaped Domain
Note the substantially reduced relative error, which is 0.2682%, at 1554 degrees of
freedom. The corresponding finite element solution is illustrated in Figure 3-13.
29
FIGURE 3-13. Finite Element Solution of 8 Grading Refinements at Polynomial
Order 8; the Relative Error with Respect to the Exact Solution is 0.2682%
30
Chapter 4
Spatial and Temporal Error Control in 1D Diffusion Problems The model problem to serve as the basis of the following discussion is equation (4-1),
which is the one-dimensional equivalent of (1-13).
( ) ( )0
0 0 0
( ) (0)
L L L
x L x
u u vc A vdx A dx AQvdx Aq v L Aq v
t x x = =∂ ∂ ∂ρ + λ = − +∂ ∂ ∂∫ ∫ ∫ (4-1)
where A represents the cross section of a bar.
4.1 The Finite Difference Method, 1D
The trial functions ∈ #tu S and the test functions v S∈ c
are written in the form
1
( ) ( )n
t i i
i
u a t x=
= Φ∑ (4-2)
1
( )=
= Φ∑N i i
i
v b x (4-3)
where N is the number of degrees of freedom and ∗= +n N n , where *n represents the
number of coefficients determined by the essential boundary conditions. On substituting
(4-2) and (4-3) into (4-1), a system of N coupled ordinary differential equations results:
31
[ ] [ ] M a K a r+ =$ (4-4)
where the elements of matrices [ ] [ ] and M K are, respectively ijm and ijk :
0
0
.
Ldef
ij i j
Ldefji
ij
m cA dx
ddk A dx
dx dxλ
= Φ ΦΦΦ=
∫∫
(4-5)
The elements of vector r are:
1 1 0
0
0
0
( )
, 2,3,..., 1
( ) .
L
x
L
i i
L
n n x L
r AQ dx Aq
r AQ dx i n
r AQ dx Aq
=
=
= Φ +
= Φ = −
= Φ −
∫∫∫
$
$
$
(4-6)
The time derivative in (4-4) can be approximated by a finite difference time stepping
scheme:
( ) (1 ) .t t t t t ta a a a t+∆ +∆≈ + −θ + θ ∆$ $ (4-7)
At time steps and + ∆t t t , equation (4-4) reads:
[ ] [ ] [ ] [ ] .
t t t
t t t t t t
M a K a r
M a K a r+∆ +∆ +∆
+ =+ =
$
$ (4-8)
Multiplying the first equation of (4-8) by (1 )−θ and the second equation by θ , one can
write:
32
[ ] [ ]
[ ] [ ] (1 ) (1 ) (1 )
.
t t t
t t t t t t
M a K a r
M a K a r+∆ +∆ +∆
−θ + −θ = −θθ + θ = θ
$
$ (4-9)
Adding the equations in (4-9), we get:
[ ] ( ) [ ] [ ] (1 ) (1 ) (1 ) .t t t t t t t t t
M a a K a K a r r+∆ +∆ +∆−θ + θ + −θ + θ = −θ + θ$ $ (4-10)
Recognizing that the term multiplying [ ]M is represented in equation (4-7), we can
write:
[ ] [ ] [ ] [ ]
1 1
(1 )
(1 ) .
t t t t t t
t t t
M a M a K a K at t
r r
+∆ +∆
+∆
− + −θ + θ =∆ ∆= −θ + θ
(4-11)
Combining:
[ ] [ ] [ ] [ ] 1 1(1 ) (1 ) .
t t t t t tM K a M K a r r
t t+∆ +∆ + θ = − −θ + −θ + θ ∆ ∆ (4-12)
This equation is well suited for electronic computations. If 0.5θ ≥ then the method is
known to be unconditionally stable through many references, such as Solution of
Diffusion Problems by the Finite Element Method [10]. If ∆t is not changed, then the
coefficient matrix on the left needs to be reduced only once. The computation of time
steps involves successive substitutions.
An algorithm was developed and programmed in MATLAB®
. A linear and constant
time stepping scheme is outlined in the following:
1. Define geometry, boundary conditions, and material properties, specify
constant time step t∆
33
2. Compute stiffness matrix [ ]K and mass matrices [ ]M
3. Perform 2L projection of the initial conditions onto the finite element
subspace
4. Build load vector r incorporating the boundary conditions and modify
[ ] [ ] and K M accordingly
5. Prepare time integration by reducing to upper triangular form and solving
by Gaussian elimination (see “\” operator in MATLAB®
):
a. term1 = (M+θ*∆t *K)\(M-(1-θ)*∆t *K);
b. term2 = (M+θ*∆t *K)\( ∆t *r);
6. Solve at every time step:
a. For I = 2 to number of time steps
i. Compute at+∆t
= term1* at + term2;
ii. Check time step and reduce if necessary
b. End
7. Assemble solution at every time step
8. Perform post-processing operations
4.2 The Discontinuous Galerkin Method, 1D
The fundamental idea of this method is to use p-extension in both space and time
dimensions. To do this, we define the time domain ( )0,J T= and partition it into time
steps mI .
11, where ( , ), 1
M
m m m mmI I t t m M−= = ≤ ≤ (4-13)
34
Each time step mI is associated with a temporal approximation order 0mr ≥ and a
similar or same finite element space as the spatial problem introduced in Chapters 1 and
2. With this approach, we seek fully discrete solutions mU on each time interval mI .
, ,
0
( , ) ( , ) ( ) ( ), , m
m
r
m j m j m mI xj
U t x U t x u x t t I xΩ == = ϕ ∈ ∈Ω∑ (4-14)
where ,j mu , respectively ,j mϕ , is a spatial basis function, respectively a polynomial time
function. As used in Chapter 3 to represent geometric basis functions, scaled Legendre
polynomials are also employed here as time basis functions ,j mϕ . More information on
these functions can be found in Finite Element Analysis [11]. mU is called a trial
function. The test functions are written in the form:
, ,0( , ) ( ) ( ), , .
mr
m i m i m miV t x v x t t I x== ϕ ∈ ∈Ω∑ (4-15)
Using the procedure described in The Discontinuous Galerkin Time-Stepping Algorithm
in hp-Version Context [6], the discontinuous Galerkin time stepping method is to find a
solution ( , )U t x by solving successively on each time step mI the problem: Find
( ),mU t x of the form (4-14) such that
1 1
1 1
m
m
mm m m m m
I
m m m
I
UV U V dxdt U V dx
t
gV dxdt U V dx
+ +− −Ω Ω
− +− −Ω Ω
∂ +∇ ∇ + ∂ = +∫ ∫ ∫∫ ∫ ∫ (4-16)
for all test functions ( ),mV t x of the form (4-15). Problem (4-16) is a discrete variational
formulation of (4-1) with the special case 1c Aρ = and 1kA = . )(1 xU m
−− and 1( )mU x+− are
the left and right handed limits of the function mU at the beginning of the time step mI .
35
)(1 xU m
−− is also the initial data for computing mU on the time step mI . 1( )mU x+−
represents the discontinuous step from )(1 xU m
−− at 1mt − .
Combining (4-14) through (4-16), implementing the scaled Legendre polynomials as
temporal shape functions, considering the generic time step ( )0 1,I t t= , and omitting the
time step index m to avoid cumbersome notation, the following is true:
0 0
, 0
0
0
[ ( ) ( )]( , ) [ ] ( , )
( , ) ( , ) ( )
m
m m
m
m
r
j i j i j i H j i j i
i j I I
r
i i H o i H i
i I
dt t t u v dt a u v
g dt v U v t
+=
− +=
′ϕ ϕ + ϕ ϕ + ϕ ϕ
= ϕ + ϕ∑ ∫ ∫∑ ∫
(4-17)
where the scalar inner product ( , )j i Hu v and the energy inner product ( , )j ia u v are the
mass and stiffness matrices, [ ] [ ] and M K , respectively. The time step I can be
mapped from the reference interval (-1,1) by using the following domain mapping:
0 1 1 0
1 1ˆ ˆ( ) ( ) , .
2 2t F t t t tk k t t= = + + = − (4-18)
For clarity, the following abbreviations are introduced:
1
1
ˆ ˆˆ ˆ ˆ ˆ( 1) ( 1)ij j i j iA dt + +−
′= ϕ ϕ +ϕ − ϕ −∫ (4-19)
1
1
ˆ ˆˆ ˆ:ij j iB dt−
= ϕ ϕ∫ (4-20)
Hii vlvf ),ˆ(:)(ˆ 11 = (4-21)
1
1
1
ˆ ˆˆ: ( )i il g F dt−
= ϕ∫ c (4-22)
36
Hii vlvf ),ˆ(:)(ˆ 22 = (4-23)
2
0ˆ ˆ: ( 1)i il U −= ϕ − . (4-24)
With (4-19) through (4-24), (4-17) can be compactly expressed as:
1 2
, 0 0
ˆ ˆˆ ˆ( , ) ( , ) ( ) ( )2 2
m mr r
ij j i H ij j i i i i i
i j i
k kA u v B a u v f v f v
= =+ = +∑ ∑ . (4-25)
The matrices A and B are hierarchical. This is exploited computationally by
calculating the largest temporal approximation order once and writing the matrices to a
storage medium for all future references.
The spatial problems are discretized as in Chapter 2. By using a finite dimensional
subspace pSf, the test and trial functions are expressed as linear combinations of the
basis functions ks . Mass and stiffness matrices are obtained which lead to the complete
matrix expression of the elliptic problem (4-26).
00 01 2
0 0 0
1 2
0
ˆ ˆ2
2
ˆ ˆ2
r
r r r
r rr
kA M K A M
u f fk
u f fkA M A M K
+ = + +
A f ff
B D B B B Bf ff
A
(4-26)
An algorithm was developed and programmed in MATLAB®
. Its adaptive time
stepping scheme is outlined:
1. Define geometry, boundary conditions, and material properties, specify
constant time step t∆
2. Compute stiffness matrix [ ]K and mass matrix [ ]M
37
3. Perform 2L projection of the initial conditions onto the finite element
subspace
4. Build load vectors 1 2and f ff f
incorporating the boundary conditions,
modify K and M accordingly, and build block matrix BM by Kronecker
tensor product
5. Solve each time interval by reducing to upper triangular form and solving
by Gaussian elimination (see “\” operator in MATLAB®
): at every time
step I : ( ) 1 2\2
k ia BM f f
= + f f
;
6. Assemble solution for each time interval
7. Perform post-processing operations
The solution of the fully discrete system (4-26) for large systems is demanding
computationally. For efficient implementation of engineering problems, this system
should be decoupled. It has been shown that these systems are diagonalizable in the
complex number space at least up to a temporal approximation order of 100 (hp
Discontinuous Galerkin Time Stepping for Parabolic Problems [13].) Since the size of
matrices considered herein is small, diagonalization was not implemented.
4.3 L2 Projection of Initial Conditions at t = 0+
The initial conditions ( ),0f x + need to be expressed in terms of the coefficient vector
0ta+= . Therefore, ( ),0f x must be projected onto the finite element space ( , , )pS QΩ ∆f ff
defined in (2-4). The solution domain is partitioned into k elements of polynomial
order p . Each element is mapped from the standard element 1 1− ≤ ξ ≤ via the mapping
function (4-27),
38
1
1 1.
2 2k kx x x +
−ξ + ξ= + (4-27)
We need to express ( ),0f x + as a piecewise polynomial function. On 1 1− ≤ ξ ≤ , it can
be approximated with linear combinations of the basis functions in p
iN Sf:
1 1 2 2 3 3 1 1( ,0 ) .p pf x a N a N a N a N+ + += + + + +… (4-28)
It should be noted that ( )1 ,0ka f x= and ( )2 1,0ka f x += because
( ) ( ) ( ) ( )1 1 2 21 1, 1 0, 1 1, 1 0, and 0 at 1 for 3iN N N N N i− = = = − = = ξ = ± ≥ . Therefore
we can rewrite (4-28) and define the zero-bounded function ( )g ξ and the difference
function ( )iF a .
( )1 3 3 4 4 1 1
21
3 3 4 4 1 1
1
1 1( ) ( ) ( )
2 2
( ) ( )2 k k
def
k k p p
defk
i p p
g f f x f x a N a N a N
lF a g a N a N a N d
+ + ++
+ +−
−ξ + ξξ = − − = + + += ξ − − − − ξ∫
…
… (4-29)
where ( 1) ( 1) 0g g− = + = . The coefficients ( )3, 4, , 1i ka i p= +… are determined by
minimizing ( )iF a with respect to ia :
0=∂∂
ia
F. (4-30)
The minimization condition (4-30) is necessary and sufficient for the determination of
( ) 3, 4, , 1ia i p= −… because F is a convex quadratic function, the minimum of which
is necessarily global. This results in a system of simultaneous equations:
39
( )( )( )
( )
1
3 3 4 4 1 1 3
1
1
3 3 4 4 1 1 4
1
1
3 3 4 4 1 1
1
1
3 3 4 4 1 1 1
1
( ) 0
( ) 0
( ) 0
( ) 0 .
p p
p p
p p i
p p p
g a N a N a N N d
g a N a N a N N d
g a N a N a N N d
g a N a N a N N d
++ +
−+
+ +−
++ +
−
++ + +
−
ξ − − − − ξ =
ξ − − − − ξ =
ξ − − − − ξ =
ξ − − − − ξ =
∫∫∫∫
…
…
B
…
B
…
(4-31)
Defining
1
2 2
1
, , 1, 2, , 1ij i j kc N N d i j p+ +−
= ξ = −∫ … (4-32)
and
1
1
1
( ) , 1, 2, , 1i i kr g N d i p+−
= ξ ξ = −∫ … (4-33)
we have a system of simultaneous equations from which ( )3, 4, , 1i ka i p= +… can be
computed:
11 3 12 4 1, 1 1 1
21 3 22 4 2, 1 1 2
1 3 2 4 , 1 1
1,1 3 1,2 4 1, 1 1 1 .
p p
p p
i i i p p i
p p p p p p
c a c a c a r
c a c a c a r
c a c a c a r
c a c a c a r
− +− +
− +
− − − − + −
+ + + =+ + + =+ + + =
+ + + =
……B
…B
…
(4-34)
40
The terms ijc can be determined once and for all, however, the terms ir generally
require numerical integration. Since all elements are mapped onto the standard element,
the Gauss-Legendre quadrature was employed.
4.3.1 L2 Projection of the Initial Solution f(x) = sin(πx)
The initial solution ( ) ( )sinf x x= π cannot be represented exactly because the sine
function is not in the finite element space pSf. However, when ( )f x is a smooth
function as in this case, then the error in maximum norm can be made arbitrarily small
by choosing p sufficiently high. For example, considering the finite element space in
Table 4-1.
Four positions of greater point density in Figure 4-1 coincide with the element
transitions. This is because the location of points corresponds to the quadrature points.
The approximation error corresponding to 3 elements and 6p = is shown in Figure 4-2.
The error at every element boundary is zero because the function values are assigned by
collocation.
In addition, it should be noted that the center element produces a much smaller error,
which can be explained by the fact that for the center element all of the asymmetric
basis functions are zero and the symmetric basis functions are good approximations of
the sinusoidal exact solution. Figure 4-2 also indicates that the error, corresponding to 3
elements and 6p = , is on the order of 10-8
.
41
TABLE 4-1. L2 Projection of the Polynomial Function f(x) = sin(πx)
Initial Condition ( )f x
sin( )xπ
Number of Gauss Pts. GN 20
Solution Domain Ω 0 1x x≤ ≤
Number of Elements ∆ 3
Polynomial Order p 6
Left BC 0u ( )0 0f =
Right BC Lu ( ) 0f L =
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
6 6 6
1D mesh with 3 elements.
length x [1]th
ickness y
[1]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
distance x [1]
tem
pe
ratu
re u
[1
]
u0 and u(L
2)
0 of f(x) = sin(pi*x), f(0) = f(1) = 0, M(∆) = 3, p = 6
u0
u(L2)
0
FIGURE 4-1. Comparison between Exact Function and its L2 Projection
42
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-4
-3
-2
-1
0
1
2
3
4
x 10-8
distance x [1]
err
or
e [
1]
u(L2)
0 - u
0, f(x) = sin(pi*x), f(0) = f(1) = 0, M(∆) = 3, p = 6
spline
true data
FIGURE 4-2. Error Plot in the Range of ±4*10-8
The corresponding error in 2L norm amounts to 62.7 10 %−⋅ . Spatial error control can be
exercised by increasing the number of degrees of freedom. One can either increase the
number of elements or raise their polynomial order or both. In this particular case, we
found by trial and error that 7 elements of polynomial order 10 are sufficient to reach the
error, which is of the order of the machine ε . Of course, other 2L projections can be
used also. Table 4-2 lists the respective element configuration. The following Figures
4-3 and 4-4 present the comparison between exact and projected functions as well as the
error, which is on the order of the machine ε .
43
TABLE 4-2. Refined L2-Projection of the Polynomial Function f(x) = sin(πx)
Initial Condition ( )f x
sin( )xπ
Number of Gauss Pts. GN 20
Solution Domain Ω 0 1x x≤ ≤
Number of Elements ∆ 7
Polynomial Order p 10
Left BC 0u ( )0 0f =
Right BC Lu ( ) 0f L =
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
10 10 10 10 10 10 10
1D mesh with 7 elements.
length x [1]
thic
kness y
[1]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
distance x [1]
tem
pe
ratu
re u
[1
]
u0 and u(L
2)
0 of f(x) = sin(pi*x), f(0) = f(1) = 0, M(∆) = 7, p = 10
u0
u(L2)
0
FIGURE 4-3. Comparison between the Exact Function and its Projection
44
The error in 2L norm is 131.6 10−⋅ . In summary, we have seen three methods that will
improve projection error and, as we will later see, increase the quality of the numerical
solution. The first is to increase in the number of elements, the second is to increase the
polynomial order of the elements, and the third is to apply geometric grading toward the
end points of the solution domain. The third method is effective for unsmooth problems.
The first two methods were presented herein. An application of the third method was
demonstrated in Chapter 3. Equivalently, fewer elements and higher polynomial order
will work as well. In Figure 4-5, 5 elements are used at polynomial order 11 to achieve
similar results as in Figure 4-4.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
x 10-15
distance x [1]
err
or
e [
1]
u(L2)
0 - u
0, f(x) = sin(pi*x), f(0) = f(1) = 0, M(∆) = 7, p = 10
spline
true data
FIGURE 4-4. Relative Error Plotted in the Range of ±2*10-15
45
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-5
-4
-3
-2
-1
0
1
2
3
x 10-15
distance x [1]
err
or
e [
1]
u(L2)
0 - u
0, f(x) = sin(pi*x), f(0) = f(1) = 0, M(∆) = 5, p = 11
spline
true data
FIGURE 4-5. Relative Error Plotted in the Range of ±4*10-15
4.3.2 L2 Projection of the Initial Solution f(x) = x(1-x)
In the special case when ( )f x is a polynomial or piecewise polynomial function, the
initial condition can be represented exactly. For example, if ( ) (1 )f x x x= − then a
single element with 2p ≥ is sufficient for the description of the initial condition. This
would not be sufficient to represent the solution in time, however.
4.4 Temporal Error Control
The Crank-Nicolson method (CNM) is second order accurate in time and is
unconditionally stable. For this reason, it is a widely used method for integrating
ordinary differential equations. Higher order time methods can be used for integrating
smooth solutions more accurately, however, they are not applicable when the solution
changes substantially over small time intervals. Lower order methods have greater
generality in this regard.
46
Spatial errors largely depend upon the smoothness of the initial conditions. Since spatial
solution gradients diminish with time, the solution becomes smoother and smoother.
Therefore spatial errors decrease with time. Both the spatial and temporal errors will be
investigated with respect to specific cases.
4.4.1 Adaptive Time Solvers
The finite difference and discontinuous Galerkin algorithms described in Sections 4.1
and 4.2 were implemented in MATLAB®
. These solvers enabled performance tests with
respect to convergence and CPU time. The tests were conducted using the three initial
conditions, representing two very smooth and compatible initial conditions and one
incompatible initial condition, which were 2L projected in Sections 4.3 and 3.1. The
boundary conditions were enforced by restriction in the boundary points. The three
problem statements are expressed in (4-35), (4-36), and (4-37) and will be referred to as
Model Problems 1, 2, and 3, respectively.
( )1
( ,0) sin( ), (0, ) 0, ( , ) 0u x x u t u L t= π = = (4-35)
( )2
( ,0) (1 ), (0, ) 0, ( , ) 0u x x x u t u L t= − = = (4-36)
( )3
( ,0) 1, (0, ) 0, ( , ) 0u x u t u L t= = = (4-37)
hp-DGFEM for Parabolic Evolution Problems [5] gives the exact solutions for the three
model problems:
( ) 21
( , ) sin( ) t
EXu x t x e−π= π ⋅ (4-38)
( ) 2 22
3 31
1 cos( )( , ) 4 sin( )l t
EX
l
lu x t e l x
l
∞ − π=
− π= ππ∑ (4-39)
( ) 2 23
1
1 cos( )( , ) 2 sin( )l t
EX
l
lu x t e l x
l
∞ − π=
− π= ππ∑ . (4-40)
47
The computer program, which was written in support of this work, computes the relative
error in energy norm on the basis of (2-9). This value is compared with the maximum
acceptable relative error specified by the user. Upon starting the h-DGFEM option, the
solver starts with one finite time increment, computes a numerical solution for the entire
time domain, and determines the maximum relative error in energy norm. As long as
the maximum acceptable error in energy norm is smaller than the computed value, the
solver automatically creates a new mesh in the time domain, using a temporal grading
function, and starts over. In hp-DFGEM mode, the solver increases the degrees of
freedom in time by both doubling the number of elements in time and their polynomial
order. There is also a program option that provides a choice between monitoring the
relative spatial error in energy norm with respect to time ( )re t or its time integral
( )0
1T
r
t
e t dtT =∫ , where T is the total time of transient solution. Since 1T = in this work,
the term 1
T is omitted in the following.
The numerical experiments were performed with both error options. The first and more
stringent error option strongly enforces the error limit, which might lead to unreasonably
fine and computationally expensive discretizations. In these cases, the overall solution
quality is driven well below the error limit only to accommodate a localized error. For
example, this is true for incompatible initial conditions. The second and less stringent
error option is used throughout this work.
In the following, convergence is investigated by monitoring the maximum relative error
using (2-9) versus the degrees of freedom of the increasingly refined mesh. To assess
solver performance, the CPU time of all solver runs inclusive of assembly time is
recorded within the program. Since the computation of the exact solutions is very time-
intensive for Model Problems 2 and 3 and is not indicative of solver performance, its
clock time is subtracted from the total CPU time.
48
When computing with finite difference methods, most of the computer time is spent on
computing the continuously changing coefficient vector of the spatial solution through
time. It should be pointed out that, in the interest of numerical efficiency, all matrix
reductions were performed before this loop was entered.
The Performance of the h-DGFEM with rm = 0 is Similar to that of a Finite
Difference Solver.
Certain similarities exist between the implementations of finite difference and
discontinuous Galerkin solvers. Both implementations employ spatial finite element
solvers with hp-extension capabilities. From the multitude of available finite difference
methods, the Backward-Euler method and the CNM were chosen. Since they are both
first order time approximations, they are comparable to temporal approximation order 1
of the h-DGFEM. This is illustrated in Figures 4-6, 4-7, and 4-8, where the time integral
of the relative error is computed for Model Problems 1 and 2 while solving with the
Backward-Euler, Crank-Nicolson, and h-DGFEM ( )0mr = . These errors plotted against
the number of degrees of freedom on a log-log scale.
100
101
102
103
104
10-2
10-1
100
101
102
103
Comparison of Convergence Performance, DOF in Space = 7
tim
e inte
gra
l of
the r
ela
tive e
rror
e r fro
m t
=0 t
o T
DOF in time [1]
1
2
34
5
6
7
8
9
10
11
12
13
14
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
Backward Euler, p=8
CNM, p=8h-DGFEM, p=8, r
m=0
FIGURE 4-6. Time Integral of Relative Error in Energy Norm, Model Problem 1;
1 Element, p=8
49
100
101
102
103
104
10-2
10-1
100
101
102
103
Comparison of Convergence Performance, DOF in Space = 7
tim
e inte
gra
l of
the r
ela
tive e
rror
e r fro
m t
=0 t
o T
DOF in time [1]
1
2
34
5
6
7
8
9
10
11
12
13
14
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
Backward Euler, p=8
CNM, p=8h-DGFEM, p=8, r
m=0
FIGURE 4-7. Time Integral of Relative Error in Energy Norm, Model Problem 2,
1 Element, p=8
100
101
102
103
104
10-1
100
101
102
103
Comparison of Convergence Performance, DOF in Space = 39
tim
e inte
gra
l of
the r
ela
tive e
rror
e r fro
m t
=0 t
o T
DOF in time [1]
1
2
3 45
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
CNM, p=8h-DGFEM, p=8, r
m=0
FIGURE 4-8. Time Integral of Relative Error in Energy Norm, Model Problem 3,
5 Geometrically Graded Elements, Spatial DOF = 8 Nodes+5 Elements*(8-1)*p =
39
50
The convergence paths of Model Problems 1 and 2 have almost identical appearance.
However, close inspection reveals that they are slightly different from one another. The
rates of convergence of the CNM and the h-DGFEM are approximately equal. While
the CNM has a smaller error in the time integral of the relative error in energy norm in
the first iteration, the h-DGFEM takes over in all following iterations. The Backward
Euler Method has lower errors than the other methods, however, it generates larger
errors beginning at iteration 3. In addition, its rate of convergence is smaller than those
displayed by the other two methods.
Due to extremely slow convergence of Model Problem 3, the Backward Euler algorithm
was not used. However, to reach an error in energy norm less than 1.0%, geometric
grading had to be used in this case to control the spatial error. h-DGFEM outperforms
CNM after the first iteration, although CNM settles at a higher rate of convergence
beginning with iteration 10.
Of course, Figures 4-6, 4-7, and 4-8 are only meaningful if the spatial error is negligible
in relation to the temporal error. By observing temporal convergence characteristics for
increasingly refined spatial discretizations, it was determined that one element at
polynomial order 8 is sufficient for Model Problems 1 and 2. Model Problem 3 requires
a geometrically graded mesh of 5 elements. See Appendix C for more detail.
The Initial Solution f(x) = sin(πx).
By increasing the temporal approximation order of the h-DGFEM, it is demonstrated in
Figure 4-9 that the rate of convergence increases linearly. Slopes of -1, -2, -3, and -4 are
obtained for approximation orders 0, 1, 2, and 3, respectively. When comparing the h-
DGFEM with finite difference solvers through experimental computations, the finite-
difference solvers were adjusted for best performance. By trial and error, it was
determined that one element with a spatial polynomial order of 8 resulted in the best
convergence at minimal solution times.
51
100
101
102
103
10-6
10-4
10-2
100
102
12
3
4
5
6
7
8
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
1
2
3
4
5
6
Comparison of Convergence Performance, DOF in Space = 49
tim
e inte
gra
l of
the r
ela
tive e
rror
e r fro
m t
=0 t
o T
DOF in time [1]
h-DGFEM, rm
=0
h-DGFEM, rm
=1
h-DGFEM, rm
=2
h-DGFEM, rm
=3
FIGURE 4-9. h-DGFEM with Increasing Approximation Order rm
In this exploratory procedure, it was also determined that the finite difference solvers
can reach a relative error in energy norm of less than 10-4
% within reasonable CPU
time. Accordingly, one spatial element was used for the h-DGFEM method, whereby an
optimal polynomial approximation order mr was selected from Figures 4-10 and 4-11,
which illustrate convergence paths and CPU times, respectively.
From the results, it was concluded that the temporal approximation order 7mr = was the
most effective temporal discretization of the h-DGFEM to reach an error in energy norm
of less than 10-4
% and to meet the requirement of converging slightly better than the
CNM. Using these solver parameters, approximate solutions of the heat equation for
Model Problem 1 over the time period of one second were computed. Performance
comparisons of temporal approximation orders 1 through 9 with respect to convergence
and CPU time are given in Figures 4-12 and 4-13.
52
100
101
102
103
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
1
2
3
4
5
6
1
2
3
4
5
1
2
3
4
1
2
3
1
2
3
1
2
3
1
2
Comparison of Convergence Performance, DOF in Space = 7
tim
e inte
gra
l of
the r
ela
tive e
rror
e r fro
m t
=0 t
o T
DOF in time [1]
h-DGFEM, rm
=1
h-DGFEM, rm
=2
h-DGFEM, rm
=3
h-DGFEM, rm
=4
h-DGFEM, rm
=5
h-DGFEM, rm
=6
h-DGFEM, rm
=7
h-DGFEM, rm
=8
h-DGFEM, rm
=9
FIGURE 4-10. h-DGFEM Performance at Multiple Values of rm
100
101
102
103
10-2
10-1
100
101
102
1
2
3
4
5
6
7
8
9
1 2
3
4
5
6
7
2 3
4
5
6
1
2
3
4
5
2
3
4
1
2
3
1
2
3
1
2
3
1 2
Comparison of Time Performance
CP
U t
ime t
[sec]
DOF in time [1]
h-DGFEM, rm
=1
h-DGFEM, rm
=2
h-DGFEM, rm
=3
h-DGFEM, rm
=4
h-DGFEM, rm
=5
h-DGFEM, rm
=6
h-DGFEM, rm
=7
h-DGFEM, rm
=8
h-DGFEM, rm
=9
FIGURE 4-11. CPU Times Corresponding to Figure 4-10; Note that the Numbered
Data Points Correlate Figures 4-10 and 4-11
53
100
101
102
103
104
10-6
10-4
10-2
100
102
Comparison of Convergence Performance, DOF in Space = 7
tim
e inte
gra
l of
rela
tive e
rror
e r fro
m t
=0 t
o T
DOF in time [1]
12
3
4
5
6
7
8
9
10
11
12
13
1
2
3
CNMh-DGFEM, r
m=7
FIGURE 4-12. Comparison of Convergence Performance: CNM and h-DGFEM at
rm = 7
100
101
102
103
104
10-2
10-1
100
101
102
1
7
8
9
10
11
12
13
1 2
3
Comparison of Time Performance
CP
U t
ime t
[sec]
DOF in time [1]
CNMh-DGFEM, r
m=7
FIGURE 4-13. Comparison of CPU Time in the Accuracy Range of 1.94 % to
1.22*10-5
%; The Numbered Data Points Correlate Figures 4-12 and 4-13
54
When the mesh is coarse then the CPU time is very short and therefore it is not very
repeatable. Hence, a comparison is difficult. However, for coarse meshes the relative
errors are so large that they are not important for engineering purposes. For this reason,
relative errors greater than 2.0% were disregarded. Based on Figure 4-12, the discussion
will focus on iterations 6 through 13 of the CNM and 1 through 3 for the h-DGFEM.
Figure 4-13 illustrates the comparison of CPU time needed to perform the computations
of Figure 4-12. After 0.032 seconds of solver time, CNM and h-DGFEM reach iteration
8 and iteration 2. Their respective errors in energy norm are 24.5 10 %−⋅ and 31.1 10 %−⋅ .
As iterations are increased from this point, the h-DGFEM outperforms the CNM with an
increasing margin. At iteration 13, the CNM reaches an error of 54.9 10 %−⋅ in 14.9
seconds and, after 0.047 seconds, the h-DGFEM reaches iteration 3 reporting an error of
67.5 10 %−⋅ . In this particular case, the h-DGFEM outperforms the CNM by a factor of
315 on a time basis at a 6.5 fold smaller integral error. Figure 4-14 illustrates the error
in energy norm over the entire time domain for the CNM and the h-DGFEM. Figure 4-
15 illustrates the finite element solution FEu .
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110
-6
10-5
10-4
10-3
Temporal Error History, IC: u(x,t = 0) = sin(pi*x), 100*||uFE
-uEX
||L2
/ ||uEX
||L2
time t [sec]
rela
tive e
rror
e r [%
]
h-DGFEM, p=8, rm
=8
CNM, p=8
FIGURE 4-14. Temporal Error Control
55
FIGURE 4-15. Finite Element Solution to Model Problem 1, Plotted on Uniformly
Spaced Post-Process Grid
The Initial Solution f(x) = x(1-x).
Analogously, an approximate solution of the heat equation was computed for this model
problem. This solution is “not arbitrarily smooth in time and therefore convergence
rates are dominated by the temporal regularity” of the solution, as stated in hp-DGFEM
for Parabolic Evolution Problems [5]. However, the same reference indicates that “the
optimal convergence rates can be recovered by the use of graded time meshes.” In
accordance with the cited reference, the temporal grading function 2 3
( ) mrh t t+= was used
to obtain Figure 4-16.
56
100
101
102
103
10-4
10-3
10-2
10-1
100
101
102
Comparison of Convergence Performance, DOF in Space = 63
tim
e inte
gra
l of
rela
tive e
rror
e r fro
m t
=0 t
o T
DOF in time [1]
1
2
3
4
5
6
7
8
1 2
3
4
5
6
7
uniform h-DGFEM, rm
=2
graded h-DGFEM, rm
=2
FIGURE 4-16. h-DGFEM Performance at rm = 2, Time Grading Function h(t) = t7
This figure indicates that for 2mr = , the optimal convergence slope of –3 is obtained by
use of the temporal grading ( )h t . In preparation to a comparison between h-DGFEM
and finite difference algorithms for this model problem, Figures 4-17 and 4-18 were
prepared, which, at various values for mr , illustrate the rate of convergence and CPU
time, respectively. Inspection of these figures led to the conclusion that from the range
4 9mr≤ ≤ , the temporal approximation order 9mr = resulted in the best performance.
In conjunction with the temporal grading function 2 3
( ) mrh t t+= , the h-DGFEM
converged to an error in energy norm of less than 10-3
%, which was slightly better than
the CNM. In addition, one geometrically graded layer of elements had to be used at the
boundary points to achieve errors below 10-1
%. This demonstrates the fact that the
solution to Model Problem 2 is not arbitrarily smooth although its initial solution lies in
2L . Using these solver parameters, approximate solutions of the heat equation for
Model Problem 2 over the time period of one second were computed. Performance
57
comparisons with respect to convergence and CPU time are illustrated in Figures 4-19
and 4-20.
100
101
102
103
10-5
10-4
10-3
10-2
10-1
100
101
1 23
4
5
6
1 23
4
5
6
1 23
4
5
6
1 23
4
5
1 23
4
5
1 2 3
4
5
Comparison of Convergence Performance, DOF in Space = 23
tim
e inte
gra
l of
rela
tive e
rror
e r fro
m t
=0 t
o T
DOF in time [1]
h-DGFEM, rm
=4
h-DGFEM, rm
=5
h-DGFEM, rm
=6
h-DGFEM, rm
=7
h-DGFEM, rm
=8
h-DGFEM, rm
=9
FIGURE 4-17. Convergence Rate Using Various Values rm and h(t) = t7
100
101
102
103
100
101
102
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
1
2
3
4
5
1
2
3
4
5
Comparison of Time Performance
CP
U t
ime t
[sec]
DOF in time [1]
h-DGFEM, rm
=4
h-DGFEM, rm
=5
h-DGFEM, rm
=6
h-DGFEM, rm
=7
h-DGFEM, rm
=8
h-DGFEM, rm
=9
FIGURE 4-18. CPU Time Using Various Values rm and h(t) = t7
58
100
101
102
103
104
10-4
10-3
10-2
10-1
100
101
102
103
Comparison of Convergence Performance, DOF in Space = 23
tim
e inte
gra
l of
rela
tive e
rror
e r fro
m t
=0 t
o T
DOF in time [1]
12
3
4
5
6
7
8
9
10
11
12
1 2 3
4
5
CNMh-DGFEM, r
m=9
FIGURE 4-19. Integral of Error in Energy Norm Reduced to below 0.001%
100
101
102
103
104
10-2
10-1
100
101
102
1
3
5
6
7
8
9
10
11
12
1
2
3
4
5
Comparison of Time Performance
CP
U t
ime t
[sec]
DOF in time [1]
CNMh-DGFEM, r
m=9
FIGURE 4-20. Comparison of CPU Time Performance in the Accuracy Range of
0.32 % to 1.2*10-4
% Corresponding to Figure 4-19; Note that the Numbered Data
Points Correlate Figures 4-19 and 4-20
59
Considering the magnitudes of error in energy norm in Figure 4-19, only CNM
iterations 6 through 12 and all h-DGFEM are of interest from an engineering
perspective. Cross-referencing Figure 4-20, the CNM converges in 7 iterations and
0.063 seconds to an error of 0.32% and the h-DGFEM converges in 1 iteration and
0.078 seconds to an error of 0.44%. As iterations are increased from this point, the h-
DGFEM again outperforms the CNM with increasing margin. In 12 iterations, the CNM
reaches 3.3*10-4
% in 17.1 seconds; the h-DGFEM converges to 1.2*10-4
% in 5
iterations and 2.48 seconds. In this particular case, the h-DGFEM outperforms the
CNM by a factor of 6.8 on a time basis at approximately equal integral error in energy
norm. Figure 4-21 demonstrates the algorithmic control over the error in energy norm
over the entire time domain. Figure 4-22 shows the finite element solution for Model
Problem 2 and the geometrically graded 3-element mesh used in connection with the h-
DGFEM solution. In the time domain, the adaptive time solver used 4 temporal
elements of increasing length. The increasing length is based on the temporal grading
function2 3
( ) mrh t t+= . The superimposed grid indicates the location of 20 Gauss points,
which were specified along both spatial and temporal elements.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110
-7
10-6
10-5
10-4
10-3
10-2
10-1
Temporal Error History, IC: u(x,t = 0) = x.*(1-x)
time t [sec]
rela
tive e
rror
e r [%
]
h-DGFEM, p=8, rm
=9
CNM, p=8
FIGURE 4-21. Temporal Error Control, h-DGFEM
60
FIGURE 4-22. Solution to Model Problem 2; the h-DGFEM Mesh is Shown by the
Heavy Lines
The Initial Solution f(x) = 1, f(0) = f(1) = 0.
Approximate solutions of the heat equation were computed for this model problem. In
this case, the initial solution has strongly singular characteristics at the boundary points
of the domain. Furthermore, the projected initial solution exhibits oscillations at the
boundary points. A comparison of solver performance between CNM and h-DGFEM is
illustrated in Figure 4-23 and the corresponding solution times are shown in Figure 4-24.
61
100
101
102
103
104
10-1
100
101
102
103
Comparison of Convergence Performance, DOF in Space = 55
tim
e inte
gra
l of
rela
tive e
rror
e r fro
m t
=0 t
o T
DOF in time [1]
1
2
3 45
6
7
8
9
10
11
12
13
14
1
2
3
4
CNM, p=8h-DGFEM, p=8, r
m=8
FIGURE 4-23. Error in Energy Norm Reduced to below 1.0%
100
101
102
103
104
10-2
10-1
100
101
102
103
1
3 5
6
7
8
9
10
11
12
13
14
1
2
3
4
Comparison of Time Performance
CP
U t
ime t
[sec]
DOF in time [1]
CNM, p-8h-DGFEM, p=8, r
m=8
FIGURE 4-24. Comparison of CPU Time Performance in the Accuracy Range of
0.32 % to 1.2*10-4
% Corresponding to Figure 4-23; Note that the Numbered Data
Points Correlate Figures 4-23 and 4-24
62
Considering the magnitudes of error in energy norm in Figure 4-23, only CNM
iterations 13 and 14 and h-DGFEM iterations 3 and 4 are of interest from an engineering
perspective. To reach an error in energy norm of less than 1.0%, the CNM takes 415.69
seconds (0.74%) while the h-DGFEM converges in 1.27 seconds (0.43%).
Figure 4-25 illustrates the relative error in energy norm for the CNM and the h-DGFEM.
The figure reveals that the simple time-halving algorithm converges to the desired
accuracy with respect to both methods, however, produces inefficient time
discretization. As the solvers address the start-up singularity with repeated refinements
along the entire time domain, the error toward the end of the time domain becomes
smaller than necessary. More sophisticated adaptive algorithms are likely to improve
this inefficiency.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110
-5
10-4
10-3
10-2
10-1
100
101
102
Temporal Error History, IC: u(x,t = 0) = 1, 100*||uFE
-uEX
||L2
/ ||uEX
||L2
time t [sec]
rela
tive e
rror
e r [%
]
h-DGFEM, p=8, rm
=8
CNM
FIGURE 4-25. Temporal Error Control; Time-Halving Leads to Relative Errors
that are Better than Necessary
Better results are expected with graded meshes. To improve the h-DGFEM, the
temporal grading function 2 3
( ) mrh t t+= was used again. Since the CNM was
63
significantly outperformed in the previous calculation, the MATLAB®
-internal ODE
Solver Suite was substituted. Trial and error experimentation indicated that from
various solver options, the option ode15s, described in Appendix E, performed best with
respect to this model problem. Since substantial local errors at time zero were expected,
integral error monitoring was chosen. A 5-element geometrically graded mesh with
8p = was employed.
Using the grading function ( )h t , the convergence paths of several temporal
approximation orders were computed. The results are shown in Figure 4-26.
100
101
102
103
10-3
10-2
10-1
100
101
102
Convergence for Increasing Temporal Polynomial Order, DOF in Space = 55
tim
e inte
gra
l of
rela
tive e
rror
e r fro
m t
=0 t
o T
DOF in time [1]
1 2
3
4
5
1 2
3
4
5
1 2
3
4
5
1 2
3
4
5
1 2
3
4
1 23
4
h-DGFEM, rm
=3
h-DGFEM, rm
=4
h-DGFEM, rm
=5
h-DGFEM, rm
=6
h-DGFEM, rm
=7
h-DGFEM, rm
=8
FIGURE 4-26. Convergence of Model Problem 3 Using the Integral of er(t)
Selecting 8mr = , the h-DGFEM is compared with ode15s. The convergence paths and
corresponding CPU times are shown in Figures 4-27 and 4-28. In the interest of keeping
the spatial error small, a geometrically graded mesh with 7 elements at 8p = was used.
64
100
101
102
103
10-2
10-1
100
101
Comparison of Convergence Performance, DOF in Space = 55
tim
e inte
gra
l of
rela
tive e
rror
e r fro
m t
=0 t
o T
DOF in time [1]
1
12
3
4
ode15sh-DGFEM, p=8, r
m=8
FIGURE 4-27. Error in Energy Norm Reduced to below 1.0%
100
101
102
103
10-1
100
101
1
1
2
3
4
Comparison of Time Performance
CP
U t
ime t
[sec]
DOF in time [1]
ode15sh-DGFEM, p=8, r
m=8
FIGURE 4-28. Comparison of CPU Time Performance Corresponding to Figure 4-
27; Note that the Numbered Data Points Correlate Figures 4-27 and 4-28
65
Comparing the ode15s iteration 1 and h-DGFEM iteration 4, it is found that these
algorithms converge to errors in energy norm of 0.02% in 0.69 seconds and to 0.04% in
1.42 seconds, respectively. Although ode15s is slightly better than h-DGFEM, it is
interesting to note this research version of h-DGFEM is quite effective and competitive
with this professionally developed solver. Therefore the development of a sophisticated
adaptive algorithm for h-DGFEM appears very promising.
Figure 4-29 illustrates the relative error in energy norm for the ode15s and the h-
DGFEM. The solvers behave in very similar fashion. Since they have been developed
from one another independently, it is reassuring of the validity of this work.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110
-4
10-3
10-2
10-1
100
101
102
Temporal Error History, IC: u(x,t = 0) = 1, 100*||uFE
-uEX
||L2
/ ||uEX
||L2
time t [sec]
rela
tive e
rror
e r [%
]
h-DGFEM, p=8, rm
=8
ode15s
FIGURE 4-29. Temporal Error Control
Finally, Figure 4-30 illustrates the h-DGFEM solution.
66
FIGURE 4-30. Solution to Model Problem 3; the h-DGFEM Mesh Is Shown by the
Heavy Lines
4.4.2 p-DGFEM and hp-DGFEM in 1D-Space and Time
The p-DGFEM and hp-DGFEM introduced in Chapter 4.2 were also programmed in
MATLAB®
. To produce higher degrees of freedom systematically, the p-DGFEM
increments the temporal approximation order to achieve higher degrees of freedom.
Analogously, the hp-DGFEM increases the temporal approximation order and the
number of elements in the time domain simultaneously. With these methods, greater
speeds of convergence than that of h-DGFEM are expected for Model Problems 1, 2,
and 3.
67
The Initial Solution f(x) = sin(πx).
Using 4 spatial elements with temporal approximation order 7mr = , it is illustrated in
Figure 4-31 that the hp-DGFEM converges faster than the h-DGFEM. The
corresponding CPU times are plotted in Figure 4-32. Since the exact solution of Model
Problem 1 can be computed directly, very small errors in energy norm can be achieved
within relatively short CPU times. In Figure 4-33, it is demonstrated that the p-DGFEM
converges exponentially when the exact solutions are analytic in time.
100
101
102
103
10-12
10-10
10-8
10-6
10-4
10-2
100
102
Comparison of Convergence Performance, DOF in Space = 39
inte
gra
l of
rela
tive e
rror
e r(t)
DOF in time [1]
1
2
3
4
5
2
3
4
5
6
7
h-DGFEM, rm
=7
hp-DGFEM, rm
=7
FIGURE 4-31. Convergence Comparison: h- and hp-DGFEM Solving Model
Problem 1
68
100
101
102
103
10-2
10-1
100
101
1
2
3
4
5
2
3
4
5
6
7
Comparison of Time Performance
CP
U t
ime t
[sec]
DOF in time [1]
h-DGFEM, rm
=7
hp-DGFEM, rm
=7
FIGURE 4-32. Comparison of CPU Time Performance Corresponding to Figure 4-
31; Note that the Numbered Data Points Correlate Figures 4-31 and 4-32
0 2 4 6 8 10 12 14 1610
-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
102
103
Time Convergence, Spatial DOF = 39
inte
gra
l of
e r(t)
[%]
DOF in time [1]
100* ∫ er(t)dt
FIGURE 4-33. Exponential Convergence of p-DGFEM
69
The Initial Solution f(x) = x(1-x).
Figures 4-34 and 4-35 illustrate that the hp-DGFEM does not improve performance
when using a 4-element mesh with 8p = and grading toward the endpoints. h and hp
methods exhibit approximately the same performance. In Section 4.4.3, it is shown that
the hp-DGFEM will outperform the h-DGFEM when an appropriate time grading is
used.
The p-DGFEM, however accomplishes faster than algebraic convergence initially and
then algebraic convergence once the spatial error becomes dominant (Figure 4-36.) To
continue faster than algebraic convergence as degrees of freedom in time are increased,
the spatial discretization would need to be refined. Appendix C provides examples to
verify this point.
100
101
102
103
10-4
10-3
10-2
10-1
100
101
102
103
Comparison of Convergence Performance, DOF in Space = 63
inte
gra
l of
rela
tive e
rror
e r(t)
DOF in time [1]
1
2
3
4
5
2
3
4
5
6
7
h-DGFEM, rm
=9
hp-DGFEM, rm
=9
FIGURE 4-34. Convergence Comparison: h- and hp-DGFEM Solving Model
Problem 2
70
100
101
102
103
10-2
10-1
100
101
1
2
3
4
5
2
3
4
5
6
7
Comparison of Time Performance
CP
U t
ime t
[sec]
DOF in time [1]
h-DGFEM, rm
=9
hp-DGFEM, rm
=9
FIGURE 4-35. Comparison of CPU Time Performance Corresponding to Figure 4-
34; Note that the Numbered Data Points Correlate Figures 4-34 and 4-35
0 2 4 6 8 10 1210
-1
100
101
102
103
Time Convergence, Spatial DOF = 63
inte
gra
l of
e r(t)
[%]
DOF in time [1]
100* ∫ er(t)dt
FIGURE 4-36. Initial Exponential and then Algebraic Convergence (p-DGFEM)
71
The Initial Solution f(x) = 1, f(0) = f(1) = 0.
Figures 4-37 and 4-38 illustrate that the hp-DGFEM does not improve performance
when using a 4-element mesh with 8p = and grading order 2 at the endpoints of the
domain.
h-DGFEM and hp-DGFEM methods display approximately equal performance. As will
be seen in the next section, the hp-DGFEM will outperform the h-DGFEM and p-
DGFEM when an appropriate time grading is used. The p-DGFEM achieves close to
exponential convergence initially and then algebraic convergence once the spatial error
becomes dominant (Figure 4-39.)
100
101
102
103
10-2
10-1
100
101
102
103
Comparison of Convergence Performance, DOF in Space = 63
inte
gra
l of
rela
tive e
rror
e r(t)
DOF in time [1]
1
2
3
4
5
6
2
3
4
5
6
7
h-DGFEM, rm
=9
hp-DGFEM, rm
=9
FIGURE 4-37. Convergence Comparison: h- and hp-DGFEM Solving Model
Problem 3
72
100
101
102
103
10-2
10-1
100
101
1
2
3
4
5
6
2
3
4
5
6
7
Comparison of Time Performance
CP
U t
ime t
[sec]
DOF in time [1]
h-DGFEM, rm
=9
hp-DGFEM, rm
=9
FIGURE 4-38. Comparison of CPU Time Performance Corresponding to Figure 4-
37; Note that the Numbered Data Points Correlate Figures 4-37 and 4-38
0 2 4 6 8 10 12 1410
0
101
102
103
Time Convergence, Spatial DOF = 63
inte
gra
l of
e r(t)
[%]
DOF in time [1]
100* ∫ er(t)dt
FIGURE 4-39. Initial Exponential and then Algebraic Convergence (p-DGFEM)
73
4.4.3 hp-DGFEM with Temporal Grading
Finally, it is shown that with the appropriate time grading functions ( )h t , faster
convergence than that achieved in Section 4.4.2 can be achieved with the hp-DGFEM.
In this algorithm, both spatial and temporal polynomial degrees are incremented to
achieve greater rates of convergence. An example of such a time grading function is:
( ) nh t t= (4-41)
where n is a positive real number.
In addition, a temporal approximation order function can be defined and used in the
search for optimal convergence. An example of such a function is:
mr m= µ (4-42)
where µ is a positive real number.
By experimenting with the temporal grading function (4-41) using 2 3mn r= + and the
temporal approximation order function (4-42) using 0.5, 0.75, 1.0, and 1.5µ = , very
high rates of convergence can be achieved. Figures 4-40 and 4-42 illustrate faster than
algebraic convergence on a semi-logarithmic scale. Representing the same data on a
logarithmic scale in Figures 4-41 and 4-43, close to exponential convergence is
observed.
74
The Initial Solution f(x) = x(1-x).
100
101
102
10-2
10-1
100
101
102
103
Comparison of Convergence Performance, DOF in Space = 59
inte
gra
l of
e r(t)
DOF in time [1]
1
2
3
4
5
1
2
3
4
1
2
3
4
1
2
3
4
µ = 0.5
µ =0.75
µ =1.0
µ = 1.5
FIGURE 4-40. Model Problem 2 Converges Faster Than Algebraic Rate
0 5 10 15 20 25 30 35 40 45 5010
-2
10-1
100
101
102
103
Comparison of Convergence Performance, DOF in Space = 59
inte
gra
l of
e r(t)
DOF in time [1]
1
2
3
4
5
1
2
3
4
1
2
3
4
1
2
3
4
µ = 0.5
µ =0.75
µ =1.0
µ = 1.5
FIGURE 4-41. Model Problem 2 Convergence Close to Exponential Rate
75
The Initial Solution f(x) = 1, f(0) = f(1) = 0.
100
101
102
10-2
10-1
100
101
102
103
Comparison of Convergence Performance, DOF in Space = 59
inte
gra
l of
e r(t)
DOF in time [1]
1
2
3
4
5
1
2
3
4
5
1
2
3
4
1
2
3
4
µ = 0.5
µ =0.75
µ =1.0
µ = 1.5
FIGURE 4-42. Model Problem 3 Converges Faster Than Algebraic Rate
0 10 20 30 40 50 60 7010
-2
10-1
100
101
102
103
Comparison of Convergence Performance, DOF in Space = 59
inte
gra
l of
e r(t)
DOF in time [1]
1
2
3
4
5
1
2
3
4
5
1
2
3
4
1
2
3
4
µ = 0.5
µ =0.75
µ =1.0
µ = 1.5
FIGURE 4-43. Model Problem 3 Convergence Close to Exponential Rate
76
4.5 The Influence of the Spatial Grading Factor on the
Temporal Error
In the balance between temporal and spatial errors, it is of interest which kind of spatial
grading will be optimal for controlling the error of approximations in time. At this
point, it is not known what kind of grading is optimal and whether it is a constant or a
function. An experiment performed with 8 time elements using 8mr = and 2 3
( ) mrh t t+=
verifies the existence of the optimal spatial grading factor, which amounts to 0.14 in this
particular case. This is captured in Figure 4-44.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.28.929
8.9295
8.93
8.9305
8.931
8.9315
grading factor [1]
err
or
in e
nerg
y n
orm
of
tem
pora
l solu
tion [
%]
m(∆) = 2, grading level = 2
FIGURE 4-44. Optimal Spatial Grading Factor with Respect to a Specific
Experiment
77
Chapter 5
Temporal Error Control in 2D Time
Dependent Problems
The transient solver feature described in Chapter 4 was combined with the 2D spatial
solver described in Chapter 3 and integrated with the MATLAB®
program structure.
The result of this effort was a two dimensional (2D) spatial solver with h, p, and hp
extension capabilities in both the spatial and time domains. The model problem to serve
as the basis of discussion is equation (5-1), which is the two-dimensional equivalent of
equation (1-13).
( ) [ ] ( ) =
c
q c
T
z c z z
z n z c c z
uD v D u t dxdy h uvt ds c vt dxdy
t
Qvt dxdy q vt ds h u vt ds
Ω Γ Ω
Ω Γ Γ
∂Λ + + ρ =∂− +
∫∫ ∫ ∫∫∫∫ ∫ ∫ (5-1)
where zt is the element thickness, and
Tdef
Dx y
∂ ∂= ∂ ∂ . (5-2)
The remaining variables are defined in Chapter 1.
78
5.1 The Finite Difference Method, Two Spatial Dimensions
Equation (4-12) is applicable to two-dimensional problems without any changes.
However, the spatial approximation requires a new set of shape functions and
computation of the stiffness matrix [ ]K , the mass matrix [ ]M , and the load vector r
in two dimensions. The hierarchic shape functions described in Finite Element Analysis
[11] and illustrated in Appendix B were implemented.
The coefficients ijk of the stiffness matrix [ ]K are computed element by element and the
global stiffness matrix is assembled by summing the element-level stiffness matrices
( )
[ ] [ ]( )
e
ij ij
e
e
e
k k
K K
==∑∑ (5-3)
where e is the element number, the subscripts and i j represent the serial numbers of
the basis functions defined on the solution domain Ω , [ ]( )eK are the inflated elemental
stiffness matrices, and
( ) [ ] ( ) [ ][ ] ( )
e
Te
ij i j zk D D t dxdyΩ
= Φ Λ Φ∫∫ . (5-4)
It should be noted that the local stiffness matrices [ ]( )eK are inflated to enable the
simple notation of equation (5-3). A one to one relationship between the global basis
function numbers i and the local basis function numbers I is required.
( , )
( , ) .
i i I e
I I i e
== (5-5)
79
With the nodal coordinates denoted by 1 1( , )X Y , 2 2( , )X Y , 3 3( , )X Y , and 4 4( , )X Y , the
mapping
( ) ( )( ) ( )( )
( )( ) ( )( )( )
1 2
3 4
1, 1 1 1 1
4
1 1 1 1
k
xx Q X X
X X
= ξ η = −ξ −η + + ξ −η ++ + ξ + η + −ξ +η
(5-6)
( ) ( )( ) ( )( )
( )( ) ( )( )( )
1 2
3 4
1, 1 1 1 1
4
1 1 1 1
k
yy Q Y Y
Y Y
= ξ η = −ξ −η + + ξ −η ++ + ξ +η + −ξ +η
(5-7)
was used. Only straight-sided quadrilaterals were employed in this investigation.
Therefore the relationship between the polynomial shape functions IN and the basis
functions ( ),i x yΦ is given by
( ) ( ) ( ) ( ) ( )( ), , , ,e e
I i x yN Q Qξ η = Φ ξ η ξ η . (5-8)
The elemental stiffness matrices are computed by evaluating
( ) ( ) [ ] ( )( )* * , , 1, 2, , 2
st
Te e
IJ I J zk D N D N J t d d I J nΩ = Λ ξ η = ∫∫ … . (5-9)
where [ ]J is the Jacobian matrix
[ ] def
x y
Jx y
∂ ∂ ∂ξ ∂ξ = ∂ ∂ ∂η ∂η (5-10)
80
and *D is obtained from [ ]D by applying the chain rule to (5-11) and (5-12):
11 12J Jx y
∂ ∂ ∂= +∂ξ ∂ ∂ (5-11)
21 22J Jx y
∂ ∂ ∂= +∂η ∂ ∂ . (5-12)
Denoting the inverse of the Jacobian matrix by *J , *D is obtained by substituting
the following terms into [ ]D :
* *
11 12J Jx
∂ ∂ ∂= +∂ ∂ξ ∂η (5-13)
* *
21 22J Jy
∂ ∂ ∂= +∂ ∂ξ ∂η . (5-14)
Using Gaussian quadrature with GN points and weights rw and sw , the numerical
computation involves the evaluation of a double sum:
( ) ( ) [ ] * *
1 1
G G
r
s
N N Te
IJ r s I J z
r s
k w w D N D N J t ξ=ξ= = η=η = Λ ∑∑ . (5-15)
Analogously, the mass matrix terms ijm are
( ) 1 1
1 1
e
IJ I Jm N N J d d− −
= ξ η∫ ∫ . (5-16)
Integrating by the Gaussian quadrature, once again a double sum must be evaluated:
81
( ) ( ) ( )( )
1 1
, ,G G
r
s
N Ne
IJ r s I J z
r s
m w w N N J t ξ=ξη=η= == ξ η ξ η∑∑ . (5-17)
The convective term in equation (5-1),
c
c zh uvt dsΓ∫ (5-18)
modifies the stiffness matrix K and is computed by
( ) ( )( ) ( )( )
( )( ) ( )( )( )( ) ( )( )( )( ) ( )( )( )( ) ( )( )
2 2
11 22
2 2
21 22
,2 21
11 12
2 2
21 22
, if 1
, if 2
,
, if 3
, if 4
G
s s
i i
s sN def i i
s s s
IJ c i i I i J i s z ss si
i i
s s
i i
J P J P s
J P J P s
k h w N P N P t
J P J P s
J P J P s
=
+ = + == α α = + = + =
∑ (5-19)
and the integer s denotes the element side number and ( )s
iP is a vector quantity
denoting the coordinates of the thi Gauss point along side s .
Finally, the load vector r is composed from the boundary condition terms on the right
hand side of equation (5-1). Denoting the heat generation load vector component Qr ,
the flux component by qr , and the convective component by cr , we have:
Q q cr r r r= + + (5-20)
with
( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( )( )
1 1
, , , , , ,G G
s
r
N Ne e e e
Q r s x y z x y
r s
r w w Q Q Q t Q Q J ξ=ξ= = η=η= ξ η ξ η ξ η ξ η∑∑ , (5-21)
82
( ) ( )( )
( )( ) ( )( )( )( ) ( )( )( )( ) ( )( )( )( ) ( )( )
2 2
11 22
2 2
21 22
, , ,2 21
11 12
2 2
21 22
, if 1
, if 2
, where
, if 3
, if 4
G
s s
i i
s sN def i i
s s
q I n i i I i i s z ss si
i i
s s
i i
J P J P s
J P J P s
r q w N P t
J P J P s
J P J P s
=
+ = + == β β = + = + =
∑ (5-22)
and
( ) ( ) ( )( ) 11
22
, , , ,
1 11
22
, if 1
, if 2, where
, if 3
, if 4
GNs s s
c I c i c i i I i i s z s
i
J s
J sr h u w N P t
J s
J s
=
= == χ χ = = =∑ . (5-23)
5.2 The Discontinuous Galerkin Method, 2D
The changes to the mass matrix and stiffness matrix, as discussed in Section 5.1, apply
here as well. The boundary conditions were implemented in accordance with the
procedure described in Section 5.1.
5.3 L2 Projection of Initial Conditions at t = 0+
The 2L projection, introduced in Section 4.3, is the linear combination of shape
functions that, with respect to the particular discretization, optimally represents an initial
condition in the space 2L . In two dimensions, the coefficients of this linear combination
are obtained in three steps: First, the vertex modes, multiplied by the corresponding
values of the initial temperature distribution, are subtracted from the initial temperature
distribution function ( , )f x y :
83
4
1
( , ) ( , ) ( , )l l l
l
f x y f x y N f x y=
= −∑ . (5-24)
Then all element sides are L2-projected by using the set of all elemental side modes,
which is analogous to the procedure of Section 4.3:
( )1
1
( , ) ( , ) ( , ), , ,p
e
m m s s
m
f x y f x y a x y m x y−
== − Φ ∈ϒ ∈Γ∑ (5-25)
where sϒ is the set of indices of all side modes, ma are the side mode coefficients, and
( )e
sΓ are the corresponding element boundaries. Finally, all internal modes are projected,
which completes the operation:
( )
( , ) ( , ) ( , ), n , ,e
n n
n
f x y f x y b x y x y= − Φ ∈Ζ ∈Ω∑ (5-26)
where Ζ is the set of indices of all internal modes, nb are the internal mode coefficients,
and ( )eΩ are the corresponding element domains. The function ( , )f x y is the error
between the original function ( ),f x y and the projected function. These three steps
fully define the coefficient vector from which the initial condition is determined as a
linear combination of the global shape functions.
To illustrate the above procedure, the L-shaped domain from Chapter 3 was used and
imposed with the initial condition
( , ) sin( ) cos( )f x y x y= π π . (5-27)
84
Figures 5-1, 5-2, 5-3, 5-4, and 5-5 illustrate the initial condition, the results of the three
projection steps, and the final projected result. The error in energy norm with respect to
the projected solution is 2.6*10-5
% and the error in maximum norm is 1.5*10-3
%
FIGURE 5-1. Arbitrary Initial Condition ( , )f x y , 9 Elements, Refinement Level 1,
p = 8
85
FIGURE 5-2. Result of Step 1 of L2 Projection ( , )f x y : All Nodal Values are
Equal to Zero
FIGURE 5-3. Result of Step 2 of L2 Projection ( , )f x y : All Element Sides are
Equal to Zero
86
FIGURE 5-4. Result of Step 3 of L2 Projection f : Error in Energy Norm =
2.59*10-5
%, Error in Maximum Norm = 1.5*10-3
FIGURE 5-5. Final Result. Assembled L2 Projection by Linear Combination of
Projection Coefficients with the Global Shape Functions
87
5.4 Model Problems in 2D
The performance of the adaptive time solvers of Chapter 4 in 2D has been investigated
with respect to two model problems, as introduced in hp Discontinuous Galerkin Time
Stepping for Parabolic Problems [13]. The following transient heat conduction problem
was solved:
in
, 0 , 1
0 1
u u g Jt
x y x y
J t t
∂ −∆ = Ω×∂Ω = ≤ ≤
= ≤ ≤ (5-28)
0 on u J= ∂Ω× (5-29)
( ), , 0 ( , ,0) in u x y t f x y= = Ω (5-30)
where the heat generation term 0g = and ( , ,0)f x y is the initial condition. Temporal
error control was added to finite difference and discontinuous Galerkin methods and
comparisons with respect to speed and convergence were performed. The tests were
conducted using two initial conditions. To ensure compatibility with the finite element
space, these initial conditions were projected onto the space 2L . The constant
temperature boundary conditions were enforced by restriction at the domain boundaries.
The initial conditions in (5-31) and (5-33) will be referred to as Model Problems 4 and
5.
Since exact solutions need to be calculated to assess convergence, the following
problems had to be calculated on a rectangular domain. One p-element was used for all
numerical experiments.
88
5.4.1 Model Problem 4
( )4
( , ,0) sin( )sin( ), 0u x y x y g= π π = (5-31)
As stated in hp Discontinuous Galerkin Time Stepping for Parabolic Problems [13],
( )4( , ,0)u x y is actually the first eigenfunction of the Laplacian and compatible with
being in 1
0 ( )H Ω . The corresponding exact solution ( )4
( , , )EXu x y t is smooth in space and
time:
( )4 2( , , ) exp( 2 )sin( )sin( )EXu x y t t x y= − π π π (5-32)
5.4.2 Model Problem 5
( )5
( , ,0) (1 ) (1 ), 0u x y x x y y g= − − = (5-33)
The initial condition ( )5
( , ,0)u x y is also compatible with being in 1
0 ( )H Ω . The exact
solution ( )5
( , , )EXu x y t is represented as a Fourier series with coefficients 5,kla :
( ) 2 2 25 ( )
5,
1 1
( , , ) sin( )sin( )t l k
EX kl
l k
u x y t a e l x k y∞ ∞ −π += =
= π π∑∑ (5-34)
5, 3 3 6
(1 cos( ))(1 cos( ))16kl
l ka
l k
− π − π= π . (5-35)
5.5 The Initial Solution f(x,y,0) = sin(πx)sin(πy)
One element with 8p = was used to solve this problem. The projection error in energy
norm amounts 1.57*10-7
% at a maximum norm of 5.0*10-5
. The Figures 5-6, 5-7, 5-8,
89
and 5-9 illustrate the solution domain, the initial solution, the projection error, and the
projected initial solution, respectively.
FIGURE 5-6. Solution Domain
FIGURE 5-7. Initial Solution
90
FIGURE 5-8. Projection Error for the Initial Condition, Error in Energy Norm:
1.57*10-7
%, Error in Maximum Norm = 5.0*10-5
FIGURE 5-9. Projected Initial Solution
91
5.5.1 The Finite Difference Time Solver
At 8p = , the finite difference solver reduces the error in energy norm to 10-4
by
implementing more than 400 time steps within 17.24 seconds. The convergence path is
illustrated in Figure 5-10.
100
101
102
103
10-5
10-4
10-3
10-2
10-1
100
101
102
103
DOF in time [1]
inte
gra
l err
or:
100* ∫ e r(t
)dt
[%]
Time Convergence, Spatial DOF = 14
100* ∫ er(t)dt
FIGURE 5-10. Model Problem 4: Convergence of the Finite Difference Method
5.5.2 The Discontinuous Galerkin Time Solvers
At 8p = and m4 and r 8mr = = were used, requiring total CPU-times of 6.61 and 1.61
seconds to converge to an error below 10-4
% (h-DGFEM.) Respectively, convergence
is illustrated in Figures 5-11 and 5-12. Faster than algebraic convergence and only
algebraic convergence was observed when using 4 and 8m mr r= = , respectively. Again,
this is explained by the interplay between spatial and temporal errors. The DGFEM
converges faster than algebraic as long as the temporal error is larger than the spatial
error. As the spatial error becomes dominant, the solver performance returns to
92
algebraic (see Appendix C). A few temporal solution steps of the most accurate overall
time solution are shown in Figure 5-13.
100
101
102
10-5
10-4
10-3
10-2
10-1
100
101
DOF in time [1]
inte
gra
l err
or:
100* ∫ e r(t
)dt
[%]
Time Convergence, Spatial DOF = 14
100* ∫ er(t)dt
FIGURE 5-11. Convergence of the h-DGFEM for rm = 4
100
101
102
10-7
10-6
10-5
10-4
10-3
10-2
10-1
DOF in time [1]
inte
gra
l err
or:
100* ∫ e r(t
)dt
[%]
Time Convergence, Spatial DOF = 14
100* ∫ er(t)dt
FIGURE 5-12. Convergence of the h-DGFEM for rm = 8
93
FIGURE 5-13. Temporal Solutions Corresponding to the Most Accurate Solution
of Figure 5-12; Time Starts at Top Left (t = 0), Progresses Row by Row and Ends at
Bottom Right (t = 1.0)
At 8p = , the p-DGFEM required 6.92 seconds to converge to an error in energy norm
of 10-4
. The convergence behavior is plotted in Figure 5-14. The last iteration of this
solution illustrates again departure from faster than algebraic convergence, which is
discussed in Appendix C. The hp-DGFEM requires 7.59 seconds to reduce the relative
error in energy norm to 10-4
%. Convergence is illustrated in Figure 5-15. Figure 5-16
illustrates a few temporal solutions.
94
100
101
10-4
10-3
10-2
10-1
100
101
102
DOF in time [1]
inte
gra
l err
or:
100* ∫ e r(t
)dt
[%]
Time Convergence, Spatial DOF = 14
100* ∫ er(t)dt
FIGURE 5-14. Convergence of the p-DGFEM
100
101
102
10-5
10-4
10-3
10-2
10-1
100
101
102
DOF in time [1]
inte
gra
l err
or:
100* ∫ e r(t
)dt
[%]
Time Convergence, Spatial DOF = 14
100* ∫ er(t)dt
FIGURE 5-15. Convergence of the hp-DGFEM
95
FIGURE 5-16. Temporal Solutions Corresponding to the Most Accurate Solution
of Figure 5-14; Time Starts at Top Left (t = 0), Progresses Row by Row and Ends at
Bottom Right (t = 1.0)
5.6 The Initial Solution f(x,y,0) = x(1-x)y(1-y)
One element with 8p = was used to solve this problem. Since this initial solution lies
in in the space 2L , the projection error in energy norm amounts to only 1.3*10-28
% and
a maximum norm of 8.0*10-17
. Figures 5-17, 5-18, and 5-19 illustrate the initial
solution, the projection error, and the projected initial solution, respectively.
96
FIGURE 5-17. Initial Solution
FIGURE 5-18. Projection Error for the Initial Condition, Error in Energy Norm =
1.3*10-28
%, Error in Maximum Norm = 8.0*10-17
97
FIGURE 5-19. Projected Initial Solution
5.6.1 The Finite Difference Time Solver
At 8p = , the finite difference solver reduces the error to less than 10-4
% within 39.22
seconds. The convergence path behavior is illustrated in Figure 5-20.
100
101
102
103
10-5
10-4
10-3
10-2
10-1
100
101
102
103
DOF in time [1]
inte
gra
l err
or:
100* ∫ e r(t
)dt
[%]
Time Convergence, Spatial DOF = 14
100* ∫ er(t)dt
FIGURE 5-20. Convergence of the Finite Difference Method
98
5.6.2 The Discontinuous Galerkin Time Solvers
At 8p = and 8mr = , the h-DGFEM solver requires a total CPU-time of 15.58 seconds
to reduce the error in energy norm to less than 10-4
%. The convergence path is
illustrated in Figure 5-21. p-DGFEM and hp-DGFEM computed for 21.93 and 31.93
seconds, respectively. Their convergence plots are shown in Figures 5-22 and 5-23,
respectively. The last five iterations of Figure 5-22 again exemplify the interplay
between spatial and temporal errors, which is discussed in Appendix C. Corresponding
to the most accurate solution of Figure 5-21, Figure 5-24 illustrates a few temporal
solutions.
100
101
102
10-5
10-4
10-3
10-2
10-1
DOF in time [1]
inte
gra
l err
or:
100* ∫ e r(t
)dt
[%]
Time Convergence, Spatial DOF = 14
100* ∫ er(t)dt
FIGURE 5-21. Convergence of the h-DGFEM
99
100
101
102
10-4
10-3
10-2
10-1
100
101
102
DOF in time [1]
inte
gra
l err
or:
100* ∫ e r(t
)dt
[%]
Time Convergence, Spatial DOF = 14
100* ∫ er(t)dt
FIGURE 5-22. Convergence of the p-DGFEM
100
101
102
10-5
10-4
10-3
10-2
10-1
100
101
102
DOF in time [1]
inte
gra
l err
or:
100* ∫ e r(t
)dt
[%]
Time Convergence, Spatial DOF = 14
100* ∫ er(t)dt
FIGURE 5-23. Convergence of the hp-DGFEM
100
FIGURE 5-24. Temporal Solutions Corresponding to the Most Accurate Solution
of Figure 5-21; Time Starts at Top Left (t = 0), Progresses Row by Row and Ends at
Bottom Right (t = 1.0)
101
Chapter 6
Conclusions
This work was concerned with the numerical solution of transient heat diffusion
problems in the light of a new technique that has gained much popularity in the Applied
Mathematics community in recent years. The method is called the hp-Discontinuous
Galerkin FEM (hp-DGFEM) and promises, upon selecting optimal combinations of h
and p, faster than algebraic, and even exponential, convergence, a fact that has long been
established for steady state hp-FEM applications. High convergence rates are of interest
in engineering applications from a perspective of time saving. In addition, the hierarchy
of the hp method makes it feasible to use a posteriori error estimates to assess solution
quality.
Since hp-DGFEM is in the research stage, the topic was approached from the point of
view of implementation. Steady state hp-FEM in one and two dimensions was
implemented. The algorithmic structure of the implementation is outlined in Chapters 1
and 2. Chapter 3 presents the first set of examples in which spatial errors are controlled
by systematic mesh refinement in problem areas. This is shown by the construction of a
very singular problem on the L-shaped domain and solving the same via successive
refinement in its reentrant corner. Theoretically predicted convergence rates were
achieved which provided verification of the developed code.
Next, the work focused on the time domain. The initial conditions required special
treatment. For compatibility with the finite element formulation, initial conditions need
102
to either lie in the energy space or they need to be projected into the energy space prior
to time integration. This was accomplished by the development of an 2L projection
algorithm, which ensured that any numerical problem was transitioned consistently into
the FEM. The early parts of Chapters 4 and 5 discuss this issue in great detail. It is
shown that arbitrarily high accuracy of an incompatible solution in one dimension can
be obtained.
Following this preliminary investigation, work on the actual time solvers was
performed. First the backward Euler and the Crank-Nicolson finite difference solvers
were implemented. Next, the more complicated hp-Discontinuous Galerkin solver was
realized. Eventually, a program structure resulted which enabled the investigation of
temporal solver performance in one and two-dimensions.
The research software was written in MATLAB®
and designed for comparing various
time solvers. In this framework, the Backward Euler Finite Difference Method, the
Crank-Nicolson Finite Difference Method, and the Discontinuous Galerkin Method with
h-, p-, and hp-extensions were developed and investigated experimentally. In addition,
the built-in MATLAB®
ODE Solver Suite was included as a benchmark. Solver
performance was evaluated on the basis of speed and accuracy with respect to known
exact solutions of special model problems. These model problems are popular
benchmarks in the research of time dependent problems and represent an interface to
other studies.
The current hp-DGFEM research code delivers a much better performance than the
finite difference methods and is comparable with the professionally developed ode15s,
although performance optimization was not an objective of this work. In addition, it was
demonstrated that the DG solvers converge faster than algebraic.
103
The interplay between spatial and temporal errors was often manifest during this study
and is discussed in various sections. It is concluded that optimal solver performance
will depend on a sound balance between these two errors. From the numerical
experiments, it became evident that solver optimality will also depend on the balance
between the h and p discretization of the time solver. Therefore, solver optimality
studies are one of the logical extensions of this work. Since the main focus of this work
rested on the implementation of the hp-DGFEM algorithm, such an optimal time/space
solver algorithm was not investigated. However, time solver optimality was considered
with respect to geometric grading. While the grading factor of 0.14 worked best in a
specific example, much more work needs to be done to obtain general results.
From an applied engineering perspective, future work may consider a posteriori error
estimators, optimal time/space solver algorithms, the addition of the third geometric
dimension, and convection as well as radiation boundary conditions in the hp-DGFEM
context.
Furthermore, the numerical computation of equation (4-26) could be accelerated by
decoupling as described in hp Discontinuous Galerkin Time Stepping for Parabolic
Problems [13]. Furthermore, replacement of critical MATLAB®
interpreter functions
with dynamic link libraries written in C/C++
would improve software performance. For
example, the interpreted MATLAB®
function ‘kron.m’ currently handles the block
matrix assembly of (4-26). It is expected that the performance of this function revealed
that its performance could be enhanced by orders of magnitude.
104
Appendix A
Conventions
It should be mentioned that the material properties λ , c , and ρ generally depend on
direction and temperature. However, all numerical calculations presented herein were
limited to isotropic and constant material properties. All quantities are given in SI units,
whereas the units themselves are generally omitted. Table A-1 references the physical
quantities used and their respective units.
TABLE A-1. Physical Quantities and their Units as Used in this Paper
Physical Quantity Variable SI Units Units in Words
Time t [s] Seconds
Distance , , x y z [m] Meters
Area A [m2] Square Meters
Temperature u [ºC], [K] Degrees Celsius, Kelvin
Density ρ [kg/m3] Kilograms per Cubic Meter
Specific Heat C [J/(kg*K)] Joules per Kilogram and
Kelvin
Thermal Conductivity λ [W/(m*K)] Watts per Meters and Kelvin
Thermal Diffusivity α [m2/s] Square Meters per Second
105
Appendix B
Shape Functions in 2D for Quadrilateral
Elements with p = 8
FIGURE B-1. 2D Shape Functions. Top Row: Vertex Modes, Four Left Columns
without First Row: Side Modes, Columns 5 through 9: Internal Modes
106
Appendix C
Appropriate Spatial Discretizations
for Model Problems 1, 2, and 3
Figures C-1 through C-5 illustrate the convergence of Model Problem 1,
( ) sin( ), (0) 0, (1) 0f x x f f= π = = , with respect to an increasingly refined discretization.
To reach an error in energy norm of less than 10-1
%, it is concluded by inspection that
one element with 8p = refines the solution domain sufficiently. Figures C-3, C-4, and
C-5 are virtually identical, indicating that the spatial solution has converged.
100
101
102
103
104
100
101
102
103
Comparison of Convergence Performance, DOF in Space = 2
rela
tive e
rror
e r = 1
00*|
|uFE-u
EX|| L
2 /
||u
EX|| L
2 [
%]
DOF in time [1]
1
2
34
5
6
7
8
9 10 11
1
2
3
4
5
67 8 9 10 11 12
1
2
3
4
56 7 8 9
Backward Euler, p=8
CNM, p=8h-DGFEM, p=8, r
m=0
FIGURE C-1. Convergence of Model Problem 1, 1 Element in Space, p = 3
107
100
101
102
103
104
105
10-2
10-1
100
101
102
103
Comparison of Convergence Performance, DOF in Space = 3
rela
tive e
rror
e r = 1
00*|
|uFE-u
EX|| L
2 /
||u
EX|| L
2 [
%]
DOF in time [1]
1
2
34
5
6
7
8
9
10
11
12
13
1415
1
2
3
4
5
6
7
89
12
3
4
5
6
7
8
Backward Euler, p=8
CNM, p=8h-DGFEM, p=8, r
m=0
FIGURE C-2. Convergence of Model Problem 1, 1 Element in Space, p=4
100
101
102
103
104
10-2
10-1
100
101
102
103
Comparison of Convergence Performance, DOF in Space = 7
rela
tive e
rror
e r = 1
00*|
|uFE-u
EX|| L
2 /
||u
EX|| L
2 [
%]
DOF in time [1]
1
2
34
5
6
7
8
9
10
11
12
13
14
1
2
3
4
5
6
7
8
12
3
4
5
6
7
8
Backward Euler, p=8
CNM, p=8h-DGFEM, p=8, r
m=0
FIGURE C-3. Convergence of Model Problem 1, 1 Element in Space, p=8
108
100
101
102
103
104
10-2
10-1
100
101
102
103
Comparison of Convergence Performance, DOF in Space = 79
rela
tive e
rror
e r = 1
00*|
|uFE-u
EX|| L
2 /
||u
EX|| L
2 [
%]
DOF in time [1]
1
2
34
5
6
7
8
9
10
11
12
13
14
1
2
3
4
5
6
7
8
12
3
4
5
6
7
8
Backward Euler, p=8
CNM, p=8h-DGFEM, p=8, r
m=0
FIGURE C-4. Convergence of Model Problem 1, 10 Spatial Elements, p=8
100
101
102
103
104
10-2
10-1
100
101
102
103
Comparison of Convergence Performance, DOF in Space = 159
rela
tive e
rror
e r = 1
00*|
|uFE-u
EX|| L
2 /
||u
EX|| L
2 [
%]
DOF in time [1]
1
2
34
5
6
7
8
9
10
11
12
13
14
1
2
3
4
5
6
7
8
12
3
4
5
6
7
8
Backward Euler, p=8
CNM, p=8h-DGFEM, p=8, r
m=0
FIGURE C-5. Convergence of Model Problem 1, 10 Spatial Elements, p=16
109
Figures C-6 through C-8 illustrate the convergence of Model Problem 2,
( ) (1 ), (0) 0, (1) 0f x x x f f= − = = , with respect to an increasingly refined discretization.
To reach an error in energy norm of less than 10-1
%, it is concluded by inspection that
one element with 8p = refines the solution domain sufficiently. Figures C-7 and C-8
are virtually identical, indicating that the spatial solution has converged.
100
101
102
103
104
100
101
102
103
Comparison of Convergence Performance, DOF in Space = 2
rela
tive e
rror
e r = 1
00*|
|uFE-u
EX|| L
2 /
||u
EX|| L
2 [
%]
DOF in time [1]
1
2
34
5
6
7
8
9 10 11
1
2
3
4
5
6 7 8 9 10 11
1
2
3
4
56 7 8 9
Backward Euler, p=8
CNM, p=8h-DGFEM, p=8, r
m=0
FIGURE C-6. Convergence of Model Problem 2, 1 Spatial Element, p=3
110
100
101
102
103
104
10-2
10-1
100
101
102
103
Comparison of Convergence Performance, DOF in Space = 7
rela
tive e
rror
e r = 1
00*|
|uFE-u
EX|| L
2 /
||u
EX|| L
2 [
%]
DOF in time [1]
1
2
34
5
6
7
8
9
10
11
12
13
14
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
Backward Euler, p=8
CNM, p=8h-DGFEM, p=8, r
m=0
FIGURE C-7. Convergence of Model Problem 2, 1 Spatial Element, p=8
100
101
102
103
104
10-2
10-1
100
101
102
103
Comparison of Convergence Performance, DOF in Space = 79
rela
tive e
rror
e r = 1
00*|
|uFE-u
EX|| L
2 /
||u
EX|| L
2 [
%]
DOF in time [1]
1
2
34
5
6
7
8
9
10
11
12
13
14
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
Backward Euler, p=8
CNM, p=8h-DGFEM, p=8, r
m=0
FIGURE C-8. Convergence of Model Problem 2, 10 Spatial Elements, p=8
111
Figures C-9 and C-10 illustrate the convergence of Model Problem 3,
( ) 1, (0) 0, (1) 0f x f f= = = , with respect to an increasingly refined discretization. To
reach an error in energy norm of less than 10-1
%, it is concluded by inspection that five
geometrically graded elements with 8p = are required to refine the solution domain
sufficiently. The steady algebraic convergence of Figure C-10 supports this claim.
100
101
102
103
104
10-1
100
101
102
103
Comparison of Convergence Performance, DOF in Space = 14
rela
tive e
rror
e r = 1
00*|
|uFE-u
EX|| L
2 /
||u
EX|| L
2 [
%]
DOF in time [1]
1
2
3 45
6
7
8
9
10
11
12
1
2
3
4
5
67
CNM, p=8h-DGFEM, p=8, r
m=0
FIGURE C-9. Convergence of Model Problem 3, 2 Spatial Elements, p=8
112
100
101
102
103
104
10-1
100
101
102
103
Comparison of Convergence Performance, DOF in Space = 39
rela
tive e
rror
e r = 1
00*|
|uF
E-u
EX|| L
2 /
||u
EX|| L
2 [
%]
DOF in time [1]
1
2
3 45
6
7
8
9
10
11
12
13
1
2
3
4
5
6
7
CNM, p=8h-DGFEM, p=8, r
m=0
FIGURE C-10. Convergence of Model Problem 3, Geometrically Graded Mesh
with 5 Elements, p=8, (Identical to Figure 4-8)
113
Appendix D
MATLAB® ODE Solver: ode15s
MATLAB The Language of Technical Computing [12] states that “ode15s is a variable
order solver based on the numerical differentiation formulas (NDFs). Optionally, it uses
the backward differentiation formulas (BDFs, also known as Gear's method) that are
usually less efficient. ode15s is a multi-step solver.” More information about this solver
is available in Matlab program documentation and on the Internet by simply using the
keyword ode15s. In addition, http://www.mathworks.com/ is another resource.
From MATLAB The Language of Technical Computing [12] :
ODE15S is a quasi-constant step size implementation in
terms of backward differences of the Klopfenstein-
Shampine family of Numerical Differentiation Formulas
of orders 1-5. The natural "free" interpolants are used.
Local extrapolation is not done. By default, Jacobians are
generated numerically. Details are to be found in The
MATLAB ODE Suite, L. F. Shampine and M. W.
Reichelt, SIAM Journal on Scientific Computing, 18-1,
1997, and in Solving Index-1 DAEs in MATLAB and
Simulink, L. F. Shampine, M. W. Reichelt, and J. A.
Kierzenka, SIAM Review, 41-3, 1999.
114
References
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[2] Babuška, I. and W.C. Rheinboldt. 1978. Error Estimates for Adaptive Finite
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[3] Kwon, Young W. and H. Bang. 1997. The Finite Element Method using
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116
Vita
A. Konrad Juethner
Born on 14 Mar 1968 in Dortmund, North-Rhine Westfalia, Germany
Professional Accomplishments
• Establishment and Leadership of Numerical Solutions Team at Watlow Inc
• Co-inventor of US-Patents 5,786,838, 6,124,579, and 6,147,335
• COOP Student with Watlow Inc from May 1995 through December 1996
• Mandatory Military Service in German Army (Deutsche Bundeswehr),
August 1988 through October 1989
Professional Society Memberships
• National Society of Professional Engineers (NSPE), EIT Certificate
• Society for Industrial and Applied Mathematics (SIAM)
Academic Accomplishments
• Acceptance of Contributed Talk to a SIAM Conference in Berlin Germany in
August 2001
• Bachelor of Science in Mechanical Engineering from Washington University
in Saint Louis, Missouri, December 1996
• Pi Tau Sigma, Honorary Mechanical Engineering Fraternity, April 1996
• Bachelor of Arts, Major in Physics, Minor in Mathematics, Drury University
in Springfield, Missouri, May 1994
• Kappa Mu Epsilon, Mathematics Honor Society, February 1994
117
• Pre-Clinical Program (9 Courses), Ludwig-Maximillians Universität Medical
School, July 1991
• “Abitur”, German High School Diploma, Gymnasium Bad Aibling in Bad
Aibling, Bavaria, Germany, July 1988
December, 2001
Short Title: hp-Discontinuous Galerkin FEM A. Konrad Juethner, M.S. 2001