Upload
others
View
4
Download
0
Embed Size (px)
Citation preview
VEM’s for the numerical solution of PDE’s
F. Brezzi
IUSS-Istituto Universitario di Studi Superiori, Pavia, Italy
IMATI-C.N.R., Pavia, Italy
From Joint works with L. Beirao da Veiga, L.D. Marini, and A. Russo
Koper, 19. februarja 2015
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 1 / 92
Papers on VEMs from our group
L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D. Marini, A.Russo: Basic principles of Virtual Element Methods, M3AS 23 (2013)199-214.
F. Brezzi, L.D. Marini: Virtual Element Method for plate bending problems,CMAME 253 (2013) 455-462.
L. Beirao da Veiga, F. Brezzi, L.D. Marini: Virtual Elements for linearelasticity problems, SIAM JNA 51 (2013) 794-812
B. Ahmad, A. Alsaedi, F. Brezzi, L.D. Marini, A. Russo: Equivalentprojectors for virtual element methods, Comput. Math. Appl. 66 (2013)376-391
F. Brezzi, R.S. Falk, L.D. Marini: Basic principles of mixed Virtual ElementMethod, M2AN 48 ( 2014), 1227-1240
L. Beirao da Veiga, F. Brezzi, L.D. Marini, A. Russo: The hitchhiker guideto the Virtual Element Method, M3AS 24 (2014) 1541-1573
L. Beirao da Veiga, F. Brezzi, L.D. Marini, A. Russo: H(div) and H(curl)VEM, (submitted; arXiv preprint arXiv:1407.6822)
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 2 / 92
Outline1 Generalities on Scientific Computing
2 Variational Formulations and Functional Spaces
3 Galerkin approximations
4 Finite Element Methods
5 FEM approximations of various spaces
6 Difficulties with Finite Element Methods
7 Virtual Element Methods
8 Robustness of VEMs
9 Working with VEM’s
10 VEM Approximations of PDE’s
11 Some additional numerical results
12 Conclusions
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 3 / 92
Scientific Computing from a Mathematical point of view
The practical interest of Scientific Computing isknown to (almost) everybody.
Here I will discuss a (minor) part of the role ofMathematics in Scientific Computation
Within the M.S.O. (Modelization, Simulation,Optimization) paradigm I will focus on the ”S” part.
In particular, I will deal with ”basic instruments tocompute an approximate solution (as accurate asneeded) to a (system of) PDE’s”.
I apologize to Numerical Analysts for the first part ofthis lecture. I hope it will not be too boring.
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 4 / 92
Maxwell Equations
Basic physical laws
∇ ·D = ρ ∇ · B = 0
∂B
∂t+∇∧ E = 0
∂D
∂t−∇ ∧H = J
Phenomenological laws (material dependent)
D = εE B = µH
Compatibility of the right-hand sides
∂ρ
∂t+∇ · J = 0
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 5 / 92
Incompressible Navier-Stokes Equations
ε =1
2(∇+∇T )u σ = (2µε+ Iidp)
ρ∂u
∂t+ u · ∇u +∇ · σ = −f
∇ · u = 0
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 6 / 92
Variational formulations
Let us consider the simplest possible problem: Given apolygon Ω and f ∈ L2(Ω):
find u ∈ V such that −∆u = f in Ω,
where V ≡ H10 (Ω) ≡ v | v ∈ L2(Ω), gradv ∈ (L2(Ω))2
such that v = 0 on ∂Ω. The variational form of thisproblem consists in looking for a function u ∈ V suchthat: ∫
Ω
gradu · gradvdx =
∫Ω
f vdx ∀ v ∈ V .
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 7 / 92
Galerkin approximations
The Galerkin method consists in choosing a finitedimensional Vh ⊂ V and looking for uh ∈ Vh such that∫
Ω
graduh · gradvhdx =
∫Ω
f vhdx ∀ vh ∈ Vh.
It is an easy exercise to show that such a uh exists and isunique in Vh, and satisfies the estimate∫
Ω
|grad(u − uh)|2dx ≤ C infvh∈Vh
∫Ω
|grad(u − vh)|2dx
bounding the error ‖u − uh‖ with the best approximationthat could be given of u within the subspace Vh.
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 8 / 92
Sequences of approximations
More generally, the analysis, from the mathematical pointof view, of these procedures assumes that we are given asequence of subspaces Vhh and proves, under suitableassumptions on the subspaces, that the sequence ofsolutions uhh converges to the exact solution u when htends to 0.As far as possible, one also tries to connect the speed ofthis convergence with suitable properties of the sequenceVhh, and hence to find what are the sequences ofsubspaces that would provide the best speed, plus possiblyother convenient properties (e.g.computability, positivity,conservation of physical quantities, etc.).
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 9 / 92
Finite Element Methods (FEM)
In the FEM’s one decomposes the domain Ω in smallpieces and takes Vh as the space of functions that arepiece-wise polynomials. The most classical case is that ofdecompositions in triangles
Figure: Triangulations of a square domain: non-uniform or uniform
taking then Vh as the space of functions that arepolynomials of degree ≤ 1 in each triangle.
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 10 / 92
Higher order methods
Instead of p.w. polynomials of degree ≤ 1 one can takepiecewise polynomials of degree ≤ k (k = 2, 3, ...).
For the analysis we consider a sequence of decompositionsThh, and piecewise polynomials of degree ≤ k , and tryto express the speed of convergence (of uh to u) in termsof k , of h = biggest diameter among the elements of Th,and of some additional geometric property θ (e.g. theminimum angle of all triangles of all decompositions):
‖grad(u − uh)‖L2(Ω) ≤ Cθ hk ‖Dk+1u‖L2(Ω).
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 11 / 92
Lagrange FEM’s
k=1 k=2
k=3 k=4
Triangular elements and their degrees of freedom.
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 12 / 92
Typical Functional Spaces
Here are the functional spaces most commonly used invariational formulations of PDE problems
L2(Ω) (ex. pressures, densities)
H(div; Ω) (ex. fluxes, D, B)
H(curl; Ω) (ex. vector potentials, E, H )
H(grad; Ω) (H1) (ex. displacements, velocities)
H(D2; Ω) (H2) (ex. in K-L plates, Cahn-Hilliard)
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 13 / 92
Continuity requirements
For a piecewise smooth vector valued function, at thecommon boundary between two elements,
in order to belong to you need to match
L2(Ω) nothing
H(div; Ω) normal component
H(curl; Ω) tangential components
H(grad; Ω) all the components
H(D2; Ω) w , wx wy
Note that the freedom you gain by relaxing the continuityproperties can be used to satisfy other properties
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 14 / 92
Elegance of FEM spaces
P1 := v = a + c · x with a ∈ R and c ∈ R3
(1 d.o.f. per node)H(grad; Ω) ∼ v ∈ H(grad; Ω) s.t. v |T ∈ P1 ∀T ∈ Th.
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 15 / 92
Elegance of FEM spaces
N0 := ϕ = a + c ∧ x with a ∈ R3 and c ∈ R3
(1 d.o.f. per edge)H(curl; Ω) ∼ ϕ ∈ H(curl; Ω) s.t. ϕ|T ∈ N0 ∀T ∈ Th.
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 16 / 92
Elegance of FEM spaces
RT0 := τ = a + cx with a ∈ R3 and c ∈ R(1 d.o.f. per face)
H(div; Ω) ∼ τ ∈ H(div; Ω) s.t. τ |T ∈ ∀T ∈ Th.Franco Brezzi (vv) VEM Koper, 19. februarja 2015 17 / 92
Elegance of FEM spaces
P0 := constants (1 d.o.f. per element)
L2(Ω) ∼ q ∈ L2(Ω) such that q|T ∈ P0 ∀T ∈ Th.
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 18 / 92
Difficulties with FEM’s: distorted elements
Distorted quads can degenerate in many ways:
YES
NO
NO
NO
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 19 / 92
More difficulties: FE approximations of H2(Ω)
There are relatively few C 1 Finite Elements on themarket. Here are some:
Bell
HCT reduced HCT
Argyris
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 20 / 92
Programming C 1 elements
Cod liver oil(Olio di fegato di merluzzo, Ribje olje)
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 21 / 92
A flavor of VEM’s
For a decomposition in general sub-polygons, FEM’s faceconsiderable difficulties. With VEM, instead, you cantake a decomposition like
having four elements with 8 12 14, and 41 nodes,respectively! Can we work in 3D as well?
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 22 / 92
A flavor of VEM’s
WE CAN !! These are three possible 3D elements
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 23 / 92
Polygonal and Polyhedral elements
There is a wide literature on Polygonal and PolyhedralElements
Rational Polynomials (Wachspress, 1975, 2010)
Voronoi tassellations (Sibson, 1980; Hiyoshi-Sugihara,1999; Sukumar et als, 2001)
Mean Value Coordinates (Floater, 2003)
Metric Coordinates (Malsch-Lin-Dasgupta, 2005)
Maximum Entropy (Arroyo-Ortiz, 2006;Hormann-Sukumar, 2008)
Harmonic Coordinates (Joshi et als 2007; Martin etals, 2008; Bishop 2013)
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 24 / 92
Related Polygonal Methods
Today, there are several similar methods that generalizeFinite Element Methods to Polygonal/Polyhedral elements
HDG Methods (e.g. Bernardo Cockburn, JayGopalakhrisnan),
Weak Galerkin Methods (e.g. Junping Wang, Xiu Ye),
Discrete Gradient Reconstruction (e.g. Daniele DiPietro, Alexandre Ern ).
There are also a strong connections between thesemethods and
Mimetic Finite Diffrences (e.g. Mikhail Shashkov,Konstantin Lipnikov),
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 25 / 92
Why Polygonal/Polyhedral Elements
There are several types of problems where Polygonal andPolyhedral elements are used:
Crack propagation and Fractured materials (e.g. T.Belytschko, N. Sukumar)
Topology Optimization (e.g. O. Sigmund, G.H.Paulino)
Computer Graphics (e.g. M.S. Floater)
Fluid-Structure Interaction (e.g. W.A. Wall)
Complex Micro structures (e.g. N. Moes)
Two-phase flows (e.g. J. Chessa)
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 26 / 92
Voronoi Meshes
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 27 / 92
Local refinements and hanging nodes
The ”interface” big squares are treated as polygons with11 edges.
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 28 / 92
Possible Inclusions
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
1512 polygons, 2849 vertices
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 29 / 92
Robustness of VEMs - General elements
Note that the pink element is a polygon with 9 edges,while the blue element is a polygon (not simplyconnected) with 13 edges. We are exact on linears...
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 30 / 92
The exact solution of the PDE
0
0.2
0.4
0.6
0.8
1
00.2
0.4
0.60.8
1
−2
−1
0
1
2
max |u−uh| = 0.008783
For reasons of ”glastnost”, we take as exact solution
w = x(x − 0.3)3(2− y)2 sin(2πx) sin(2πy) + sin(10xy)
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 31 / 92
Robustness of the method
0
0.2
0.4
0.6
0.8
1
00.2
0.4
0.60.8
1
−2
−1
0
1
2
max |u−uh| = 0.074424
Mesh of 512 (16× 16× 2) elements. Max-Err=0.074
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 32 / 92
Finer grids
0
0.2
0.4
0.6
0.8
1
00.2
0.4
0.60.8
1
−2
−1
0
1
2
max |u−uh| = 0.019380
Mesh of 2048 (32× 32× 2) elements. Max-Err=0.019
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 33 / 92
Solution on the finer grid
0
0.2
0.4
0.6
0.8
1
00.2
0.4
0.60.8
1
−2
−1
0
1
2
max |u−uh| = 0.005035
Mesh of 8192 (64× 64× 2) elements. Max-Err=0.005Note the O(h2) convergence in L∞.
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 34 / 92
The next steps? (by M.C. Escher)
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 35 / 92
The next steps? (by M.C. Escher)
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 36 / 92
Going berserk
−0.5 0 0.5 1 1.5
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.41 polygons, 82 vertices
The first step: a pegasus-shaped polygon with 82 edges.
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 37 / 92
Going berserk
−0.5 0 0.5 1 1.5
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1 polygons, 82 vertices
1
2 3
4 5
6
7
8 9 10
1112
13
1415
16 17
18
1920
21
22
23
24
2526 27
2829
3031
3233
34
3536
37 38
39
404142
43
44
45
46
4748
4950
51
5253
545556
57
585960
61 62
63 64 65
66
6768
6970
71
7273
74 75
7677
787980
81
82
The second step: local numbering of the 82 nodes.
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 38 / 92
Going berserk
−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4−0.2
0
0.2
0.4
0.6
0.8
1
1.2
4 polygons, 243 vertices
The third step: a mesh of 2× 2 pegasus.
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 39 / 92
Going totally berserk
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
400 cells, 16821 vertices
A mesh of 20× 20 pegasus.
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 40 / 92
Going totally berserk
0
0.2
0.4
0.6
0.8
1
00.2
0.4
0.60.8
1
−2
−1
0
1
2
max |u−uh| = 0.077167
Solution on a 20× 20-pegasus mesh. Max-Err=0.077
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 41 / 92
Going totally berserk
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
11600 cells, 65641 vertices
A mesh of 40× 40 pegasus.
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 42 / 92
Going totally berserk
0
0.2
0.4
0.6
0.8
1
00.2
0.4
0.60.8
1
−2
−1
0
1
2
max |u−uh| = 0.026436
Solution on a 40× 40-pegasus mesh. Max-Err=0.026
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 43 / 92
The main features of VEM
As for other methods on polyhedral elements
the trial and test functions inside each element arerather complicated (e.g. solutions of suitable PDE’s orsystems of PDE’s).
Contrary to other methods on polyhedral elements,
they do not require the approximate evaluation oftrial and test functions at the integration points.
In most cases they satisfy the patch test exactly (upto the computer accuracy).
We have now a full family of spaces.
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 44 / 92
The general philosophy
In every element, to define the generic (scalar or vectorvalued) element v of our VEM space:
You start from the boundary d.o.f. and use a 1Dedge-by edge reconstruction
Then you define v inside as the solution of a (systemof) PDE’s, typically with a polynomial right-hand side.
The construction is such that all polynomials of acertain degree belong to the local space. In generalthe local space also contains some additional elements.
Let us see some examples.
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 45 / 92
Nodal 2D elements
We take, for every integer k ≥ 1
V Eh = v | v|e ∈ Pk(e)∀ edge e and ∆v ∈ Pk−2(E )
It is easy to see that the local space will contain all Pk .As degrees of freedom we take:
the values of v at the vertices,
the moments∫e v pk−2de on each edge,
the moments∫E v pk−2dE inside.
It is easy to see that these d.o.f. are unisolvent.
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 46 / 92
The L2-projection
A fantastic trick (sometimes called The Three CardMonte trick), often allows the exact computation of themoments of order k − 1 and k of every v ∈ V E
h .
This is very useful for dealing with the 3D case.
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 47 / 92
The Three Card Monte Trick is hard to believe
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 48 / 92
Example: Degrees of freedom of nodal VEM’s in 2D
k=4
k=1 k=2
k=3
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 49 / 92
More general geometries k = 1
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 50 / 92
More general geometries k = 2
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 51 / 92
Approximations of H1(Ω) in 3D
For a given integer k ≥ 1, and for every element E , we set
V Eh = v ∈ H1(E )| v|e ∈ Pk(e)∀ edge e, v|f ∈ V f
h ∀ face f , and ∆v ∈ Pk−2(E )with the degrees of freedom:• values of v at the vertices,• moments
∫e v pk−2 (e) on each edge e,
• moments∫f v pk−2 (f ) on each face f , and
• moments∫E v pk−2 (E ) on E .
Ex: for k = 3 the number of degrees of freedom wouldbe: the number of vertices, plus 2× the number of edges,plus 3× the number of faces, plus 4. On a cube thismakes 8 + 24 + 18 + 4 = 54 against 64 for Q3.
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 52 / 92
The spaces Gk , G⊥k , Rk , and R⊥k
In the sequel it will be convenient to introduce thefollowing notation
Gk := grad(Pk+1)
Rk := rot(Pk+1) (in 2 dimensions)
Rk := curl(P3k+1) (in 3 dimensions)
Moreover, for every vector valued polynomial spacePk(E ) ⊂ Pd
k (E ) we denote
P⊥k (E ) := q ∈ Pd(E ) s.t. (q,p)0,E = 0∀p ∈ Pk(E )
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 53 / 92
VEM approximations of H(div; Ω) in 2d and in 3d
In each element E , and for each integer k , we define
Bk(∂E ) := g | g|e∈Pk ∀ edge e ∈ ∂E in 2d
Bk(∂E ) := g | g|f ∈Pk ∀ face f ∈ ∂E in 3d.
Then we define, in 2 dimensions:
V k(E ) = τ |τ · n∈Bk(∂E ),∇divτ ∈Gk−2, rotτ ∈Pk−1
and in 3 dimensions
V k(E ) = τ |τ · n∈Bk(∂E ),∇divτ ∈Gk−2, curlτ ∈Rk−1.
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 54 / 92
Variants of VEMs in H(div; Ω)
For k , r and s integer, we define, in 2 dimensions:
V (k,r ,s)(E ) = τ |τ · n∈Bk(∂E ),∇divτ ∈Gr−1, rotτ ∈Ps
and in 3 dimensions
V (k,r ,s)(E ) = τ |τ ·n∈Bk(∂E ),∇divτ ∈Gr−1, curlτ ∈Rs.
In general we might say that
V k ≡ V (k,k−1,k−1) ' BDMk and V (k,k,k−1) ' RTk
On a triangle: V (0,0,−1) = RT0. We point out that ∀k ≥ 0
(Pk)d ⊂ V (k ,k−1,k−1) and ∇(Pk+1) ⊂ V (k,k−1,−1)
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 55 / 92
Degrees of freedom in V k ≡ V (k ,k−1,k−1)(E ) in 2d
In V k ≡ V (k,k−1,k−1) we can take the following d.o.f.∫e
τ · n qkde ∀qk ∈ Pk(e) ∀ edge e∫E
τ · gradqk−1dE ∀qk−1 ∈ Pk−1∫E
τ · g⊥k dE ∀g⊥k ∈ G⊥k
with natural variants for other spaces of the type V (k,s,r).
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 56 / 92
Degrees of freedom for V (0,0,−1)
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 57 / 92
Degrees of freedom for V (1,1,0)
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 58 / 92
Degrees of freedom in V k ≡ V (k ,k−1,k−1)(E ) in 3d
In V k ≡ V (k,k−1,k−1) we can take the following d.o.f.∫f
τ · n qkdf ∀qk ∈ Pk(f ) ∀ face f∫E
τ · gradqk−1dE ∀qk−1 ∈ Pk−1∫E
τ · g⊥k dE ∀g⊥k ∈ G⊥k
with natural variants for other spaces of the type V (k,s,r).
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 59 / 92
VEM approximations of H(rot; Ω) in 2d
In each element E , and for each integer k , we recall
Bk(∂E ) := g | g|e ∈ Pk ∀ edge e ∈ ∂E in 2d
Then we set
V k(E )=ϕ|ϕ · t∈Bk(∂E ), divϕ∈Pk−1, rotrotϕ∈Rk−2
and for integers k , r , and s
V (k,r ,s)(E )=ϕ|ϕ·t∈Bk(∂E ), divϕ∈Pr , rotrotϕ∈Rs−1
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 60 / 92
Degrees of freedom in V k ≡ V (k ,k−1,k−1)(E ) in 2d
In V k ≡ V (k,k−1,k−1) in 2d we can take the followingd.o.f.∫
e
ϕ · t qkde ∀qk ∈ Pk(e) ∀ edge e∫E
ϕ · rotqk−1dE ∀qk−1 ∈ Pk−1∫E
ϕ · r⊥k dE ∀r⊥k ∈ R⊥k
with natural variants for other spaces of the type V (k,r ,s).
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 61 / 92
Degrees of freedom for V (0,−1,0)
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 62 / 92
Degrees of freedom for V (1,0,1)
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 63 / 92
VEM approximations of H(curl; Ω) in 3d
In each element E , and for each integer k , we set
Bk(∂E ) := ϕ| ϕ|f ∈ V k(f )∀ face f ∈ ∂E and
ϕ · te is single valued at each edge e ∈ ∂E
Then we set
V k(E ) = ϕ| such that ϕ · t ∈ Bk(∂E ),
divϕ ∈ Pk−1, curlcurlϕ∈Rk−2
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 64 / 92
Degrees of freedom in V k(E ) ≡ V (k ,k−1,k−1) in 3d
for every edge e
∫e
ϕ · t qkde ∀qk ∈ Pk(e)
for every face f∫f
ϕ · rotqk−1df ∀qk−1 ∈ Pk−1(f )∫f
ϕ · r⊥k df ∀r⊥k ∈ R⊥k (f )
and inside E∫E
ϕ · curlqk−1dE ∀qk−1 ∈ (Pk−1(E ))3∫E
ϕ · r⊥k dE ∀r⊥k ∈ R⊥k (E )
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 65 / 92
VEM approximations of L2(Ω)
In each element E , and for each integer k , we set
V k(E ) := Pk(E ),
and then obviously
V k(Ω) = v | such that v|E ∈ Pk(E ),∀E in Th
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 66 / 92
Degrees of freedom in V k(E )
As degrees of freedom in V k(E ) we can obviously choose∫E
v qkdE ∀qk ∈ (Pk(E ))3
. k = 0 k = 1
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 67 / 92
VEM approximations of H2(Ω)
For r , s, k , with
r ≥ k ,
s ≥ k − 1
we set
Vh := v ∈ V : v ∈ Pr(e), vn ∈ Ps(e) ∀ edge e,
and ∆2v ∈ Pk−4(E ) ∀ element E
It is clear that for every element E the restriction V Eh of
Vh to E contains all the polynomials of degree ≤ k .
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 68 / 92
Degrees of freedom for K-L plate elements
We had:
Vh := v ∈ V : v ∈ Pr(e), vn ∈ Ps(e)
∀ edge e and ∆2v ∈ Pk−4(E ) ∀ element E
In each E the functions in V Eh are identified by
their value and the value of their derivatives on ∂E ,
(for k > 3) the moments up to the order k − 4 in E
Hence we have to worry only for the boundary degrees offreedom.
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 69 / 92
Example: r = 3, s = 1
On each vertex we assign v , vx , vy .
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 70 / 92
Example: r = 4, s = 3 (Pac-Plate)
On each vertex we assign v , vx , vy . On each midpoint weassign v , vn, vnt .
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 71 / 92
Manipulating VEM’s
When we deal with VEM, we cannot manipulate them aswe please. As we don’t want to use approximate solutionsof the PDE problems in each element, we have to use onlythe degrees of freedom and all the information that youcan deduce exactly from the degrees of freedom.
In a sense, is like doing Robotic Surgery
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 72 / 92
Example of manipulation
For instance if you know a function v on ∂E and its meanvalue in E you can compute∫
E
∇v · q1dE =
∫∂E
v q1 · nds −∫E
v divq1dE
for every vector valued polynomial q1 ∈ (P1)2.
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 73 / 92
A very useful property
We observe that the classical differential operators grad ,curl , and div send these VEM spaces one into the other(up to the obvious adjustments for the polynomialdegrees). Indeed:
grad(VEM , nodal) ⊆ VEM , edge
curl(VEM , edge) ⊆ VEM , face
div(VEM , face) ⊆ VEM , volume
Ri−→ V ver
k (Ω)grad−−−−→ V edg
k−1(Ω)curl−−−→ V fac
k−2(Ω)div−−−→ V vol
k−3(Ω)o−→ 0
.
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 74 / 92
The crucial feature
The crucial feature common to all these choices is thepossibility to construct (starting from the degrees offreedom, and without solving approximate problems in theelement) a symmetric bilinear form [u, v]h such that,on each element E , we have
[pk , v]Eh =
∫E
pk ·vdE ∀pk ∈ (Pk(E ))d , ∀v in the VEM space
and ∃α∗ ≥ α∗ > 0 independent of h such that
α∗‖v‖2L2(E ) ≤ [v, v]Eh ≤ α∗‖v‖2
L2(E ), ∀v in the VEM space
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 75 / 92
The crucial feature - 2
In other words: In each VEM space (nodal, edge, face,volume) we have a corresponding inner product[· , ·
]VEM,nodal
,[· , ·
]VEM,edge
,[· , ·
]VEM,face
,[· , ·
]VEM,volume
that scales properly, and reproduces exactly the L2
inner product when at least one of the two entries is apolynomial of degree ≤ k .
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 76 / 92
General idea on the construction of Scalar Products
First note that you can always integrate a polynomial∫E
x3dE =
∫∂E
x4
4nxds.
You construct Πk : V E → (Pk(E ))d defined by∫E (v − Πkv) · pkdE = 0∀pk ,
and then set, for all u and v in V E[u, v
]E
:=∫E (Πku · Πkv)dE + S(u− Πku, v − Πkv)
where the stabilizing bilinear form S is for instance themeasure of E times the Euclidean inner product in RN .
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 77 / 92
Strong formulation of Darcy’s law
p = pressure
u = velocities (volumetric flow per unit area)
f = source
K = material-depending (full) tensor
u = −K∇p (Constitutive Equation)
div u = f (Conservation Equation)
−div(K∇p) = f in Ω,
BBBBBBBBBBBBBBBp = 0 on ∂ Ω, for simplicity.
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 78 / 92
Variational formulation and VEM approximation
The variational formulation of Darcy problem is:find p ∈ H1
0 (Ω) such that∫Ω
K∇p · ∇q dx =
∫Ω
f qdx ∀q ∈ H10 (Ω).
and as VEM approximate problem we can take:find ph ∈ VEM,nodal such that:
[K∇ph,∇qh]VEM,edge = [f , qh]VEM,nodal ∀qh ∈ VEM,nodal
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 79 / 92
Variational formulation - Mixed
Darcy problem, in mixed form, is instead:find p ∈ L2(Ω) and u ∈ H(div; Ω) such that:∫
Ω
K−1u · vdV =
∫Ω
p divvdV ∀v ∈ H(div; Ω)
and ∫Ω
divu qdV =
∫Ω
f q dV ∀q ∈ L2(Ω).
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 80 / 92
Approximation of Darcy - Mixed
The approximate mixed formulation can be written as:
find ph ∈ VEM , volume and uh ∈ VEM , face such that
[K−1uh, vh]VEM,face = [ph, divvh]VEM,volume ∀vh ∈VEM,face
and
[divuh, qh]VEM,volume = [f , qh]VEM,volume ∀qh ∈VEM,volume.
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 81 / 92
Strong formulation of Magnetostatic problem
j = divergence free current density
µ = magnetic permeability
u = vector potential with the gauge div u = 0
B = curl u = magnetic induction
H = µ−1B = µ−1curl u =magnetic field
curlH = j
The classical magnetostatic equations become now
curlµ−1curl u = j and div u = 0 in Ω,
u× n = 0 on ∂Ω.
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 82 / 92
Variational formulation of the magnetostatic problem
The variational formulation of the magnetostatic problem(setting B = µH = curl u) is :
Find u ∈ H0(curl,Ω) and p ∈ H10 (Ω) such that :
(µ−1curl u, curl v)− (∇p, v) = (j, v)∀ v ∈ H0(curl; Ω)
(u,∇q) = 0 ∀ q ∈ H10 (Ω),
and the VEM approximation can be chosen as:Find uh ∈ VEM,edges and ph ∈VEM,nodal such that:
[µ−1curl uh, curl vh]VEM,face − [gradph, vh]VEM,edge
= [j, vh]VEM,edge ∀ vh ∈ VEM , edge,
[u, gradqh]VEM,edge = 0 ∀ qh ∈ VEM , nodal .
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 83 / 92
Numerical results for Darcy problem with ”BDM-like” VEM
Mesh of squares 4x4, 8x8, ...,64x64Exact solution p=sin(2x)cos(3y)
00.2
0.40.6
0.81
0
0.2
0.4
0.6
0.8
1−1
−0.5
0
0.5
1
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 84 / 92
Numerical results-Squares
10−1
10−3
10−2
L2 −error
log h
||p−
ph||
0
BDM1
h2
VEM1
10−1
10−3
10−2
L2 −error
log h
||u−
hh||
0
BDM1
h2
VEM1
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 85 / 92
Numerical results-Voronoi Meshes
Voronoi polygons 88,...,7921Exact solution p=sin(2x)cos(3y)
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
triangle0.0188 polygons, h
max = 0.174116, h
mean = 0.149291
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
triangle0.00017921 polygons, h
max = 0.019754, h
mean = 0.015641
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 86 / 92
Numerical results-Voronoi
10−2
10−1
100
10−4
10−3
10−2
10−1
h1
h2
||p−ph|| in L
2
10−2
10−1
100
10−4
10−3
10−2
10−1
h1
h2
||σ−Πaσ
h|| in L
2
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 87 / 92
Numerical results-Distorted Quads meshes
Mesh of distorted quads: 10x10; 20x20; 40x40Exact solution: p = sin(2x) cos(3y)
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
quadrati−distorti−0.5−10x10100 polygons, h
max = 0.234445, h
mean = 0.171046
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
quadrati−distorti−0.5−40x401600 polygons, h
max = 0.065760, h
mean = 0.043012
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 88 / 92
Numerical results–Distorted Quads
10−2
10−1
100
10−4
10−3
10−2
10−1
h1
h2
||p−ph|| in L
2
10−2
10−1
100
10−3
10−2
10−1
h1
h2
||σ−Πaσ
h|| in L
2
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 89 / 92
Numerical results-Winged horses meshes
Mesh of horses: 4x4; 8x8; 10x10; 16x16Exact solution: p = sin(2x) cos(3y)
−0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4−0.2
0
0.2
0.4
0.6
0.8
1
1.2
4 polygons, 243 vertices
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
400 cells, 16821 vertices
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 90 / 92
Numerical results–Winged horses
10−2
10−1
100
10−4
10−3
10−2
10−1
h1
h2
||p−ph|| in L
2
10−2
10−1
100
10−3
10−2
10−1
h1
h2
||σ−Πaσ
h|| in L
2
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 91 / 92
Conclusions
Virtual Elements are a new method, and a lot of workis needed to assess their pros and cons.
Their major interest is on polygonal and polyhedralelements, but their use on distorted quads, hexa, andthe like, is also quite promising.
For triangles and tetrahedra the interest seems to beconcentrated in higher order continuity (e.g. plates).
The use of VEM mixed methods seems to be quiteinteresting, in particular for their connections withother methods for polygonal/polyhedral elements.
Franco Brezzi (vv) VEM Koper, 19. februarja 2015 92 / 92