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Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)
Numerical ResultsConclusions and Future Work
Multiscale Basis Functions for Iterative Domain DecompositionProcedures
A. Francisco1, V. Ginting2, F. Pereira3 and J. Rigelo2
1Department Mechanical EngineeringFederal Fluminense University, Volta Redonda, RJ 27255-125, Brazil
afrancisco@metal.eeimvr.u!.br
2Department of MathematicsUniversity of Wyoming, Laramie, WY 82071-3036, USA
{vginting,jrigelo}@uwyo.edu
3Department of Mathematics and School of Energy ResourcesUniversity of Wyoming, Laramie, WY 82071-3036, USA
lpereira@uwyo.edu
Support: DOE: DE-FE0004832/DE-SC0004982; NSF: DMS-1016283;Center for Fundamentals of Subsurface Flow(UW).
Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)
Numerical ResultsConclusions and Future Work
Outline:
1 Motivation: Domain Decomposition Method
2 The Multiscale Mixed Method (MuMM)
3 Numerical Results
4 Conclusions and Future Work
Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)
Numerical ResultsConclusions and Future Work
Motivation
We are concerned withthe development ofnumerical proceduresfor the fast andaccurate approximationof subsurface flows thatcan take advantage ofheterogeneousprocessing units.
Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)
Numerical ResultsConclusions and Future Work
Motivation
Incorporate fine scale information into a coarse scalediscretization, without solving it directly.
Coarse Domain Decomposition
Our iterative procedure does not use MPI in each iteration.
Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)
Numerical ResultsConclusions and Future Work
Model Problem
Our model problem is a second order linear elliptic equationwritten as a first order system
!.u = f (x), where u = "k(x)!p in !, (1)
p = pb on "D , u.! = ub on "N . (2)
Here ! is a bounded domain with a Lipschitz boundary"! = "D # "N , "D $ "N = %.
Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)
Numerical ResultsConclusions and Future Work
Domain Decomposition
The domain ! is divided into a non-overlapping partition{!j}:
! =!M
j=1!j ; !j $ !k = %, j &= k .
Motivation: Non-overlapping iterative DDM based on theRobin boundary conditon.
J. Douglas, Jr., P. J. Paes Leme, J. E. Roberts, and J. Wang,A parallel iterative procedure applicable to the approximatesolution of second order partial di!erential equations by mixedfinite element methods, Numer. Math., 65 (1993) 95–108.
Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)
Numerical ResultsConclusions and Future Work
Domain Decomposition (cont.)
Mixed finite element space: hybridized Raviart-Thomas.
Procedure: subdomain = element.
Degrees of freedom (for each !j) :
p, u! and #! , $ = L,R ,B ,T .h
B
T
L R
Then for a single element, the discrete form of the Poisson’sequation is given by
uL + uR + uB + uT = fh, (3)
u! " 2
hk(p " #!) = 0. (4)
(5)
Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)
Numerical ResultsConclusions and Future Work
Domain Decomposition (cont.)
Robin Interface Condition:
#! = %!(u! + u!!) + #!! , where $ = L,R ,B ,T . (6)
!j !k $ - $!
L
T
R
T
B
R
L
B
"jk
Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)
Numerical ResultsConclusions and Future Work
Domain Decomposition (cont.)
Douglas, Jr. et al. parallel iterative scheme:
1 Set an initial guess: {p0, u0! , #0!}.2 For all red elements, update {p, u! , #!}, using [3, 4, 5].
3 For all black elements, compute {p, u! , #!}, by solving[3, 4, 5], using the updated values from the red elements.
4 Check for convergence.
new
old
old
oldold
Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)
Numerical ResultsConclusions and Future Work
Domain Decomposition (cont.)
o
o o
o
n
o
o o
o
n
subdomain: one element larger subdomain
Convergence is established.
Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)
Numerical ResultsConclusions and Future Work
The Multiscale Basis Functions Formulation
Consider a subdomain !j . Let &ji = (ui! , #i! , p
i )j , i = 1, ..., 4N,be the basis functions associated with this subdomain.
"%LuL+#L = 1 '0
0 0 0 0
0
0
0
0
0000
0
0
&j1
B. Ganis and I. Yotov, Implementation of a mortar mixed finiteelement method using a multiscale flux basis, Computer Methods inApplied Mechanics and Engineering, 198 (2009) 3989-3998.
Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)
Numerical ResultsConclusions and Future Work
The Multiscale Basis Functions Formulation
Given the Robin boundary values Aji , the solution for thePoisson equation is given by
S"j =4N"
i=1
Aji&ji j3
N
N
A
Aj1
j2A
where, for i = 1, ..., 4N,&ji = (ui! , #
i! , p
i )j are the canonical basis functions.
Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)
Numerical ResultsConclusions and Future Work
The Multiscale Basis Functions Formulation
Advantage :
Avoid the direct solution of the local problems.
Problem :
We have to compute 4N basis functions for each subdomain!
Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)
Numerical ResultsConclusions and Future Work
MuMM: A Modified iteration
Introduce an intermediate scale H, h ( H ( H.
Based on an average Robin condition:
Aji = "%LuTL +uBL
2 +"TL +"BL
2h
TB
Aji H
H
!j !k
Goal: To reduce the number of basis functions.
Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)
Numerical ResultsConclusions and Future Work
MuMM: A Modified iteration
The solution is given by, for example: S"j =
4N/2"
i=1
Aji &ji .
"%LuL+#L = 1 '
0
0 0
0 0
0
0
&j1
A
N
Aj2
j1
2D: Douglas, Jr. et al. iteration; 3D: CG preconditioned withthe AMG.
Solution in the fine grid: post-processing.
Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)
Numerical ResultsConclusions and Future Work
MuMM: A Modified iteration
Remarks :
Flux conservation is maintained in the H scale.
The balance between numerical accuracy and numericale#ciency is determined by the choice of
span{&ji} ) span{&ji}.
Extreme cases:
H = h: Douglas, Jr. et al. iteration.H = H: 4 basis functions/subdomain.
Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)
Numerical ResultsConclusions and Future Work
Example 1: kmax/kmin = 176
Example: 2D problem with a fine grid of 220* 60,coarse grid of 11* 3 (subdomains of 20* 20).
Permeability model: SPE10 model, where k(x) = exp(' ((x)).
20 40 60 80 100 120 140 160 180 200 220
5
10
15
20
25
30
35
40
45
50
55
60
5
10
15
20
25
The physical transport of fluids is given by solving:
)#c#t + u.!c = 0, with I .C .+ B .C . given.
Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)
Numerical ResultsConclusions and Future Work
H = H
H = H/4
H = H/2
fine grid
Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)
Numerical ResultsConclusions and Future Work
Tracer cut curves
The fraction of the tracer in the produced fluid is given by
F(t) =
#!"out
c u.n dS#!"out
u.n dS.
Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)
Numerical ResultsConclusions and Future Work
Relative Errors :
Relative error = !uMuMM"ufine!maxi,j!ufine! .
Figure :From top to bottom:
4, 8 and 16 basis functions.
Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)
Numerical ResultsConclusions and Future Work
Example 2: kmax/kmin = 1408.
Example: 2D problem with a fine grid of 220* 60,coarse grid of 11* 3 (subdomains of 20* 20).
Permeability model: SPE10 model. We consider 16 basisfunctions.
MuMM fine grid
Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)
Numerical ResultsConclusions and Future Work
Example 2: Tracer cut curve and permeability field
Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)
Numerical ResultsConclusions and Future Work
Conclusions
Properties :
uL + uR + uB + uT = fh holds in the fine grid.
Sources and sinks are naturally incorporated in the procedure.
All local problems are positive definite.
Global information is not needed.
Straightforward implementation in 2 and 3D.
Fits well in CPU-GPU clusters.
Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)
Numerical ResultsConclusions and Future Work
Future Work
3D implementation on GPUs.
Extension to multiphase/Compositional flows.
Adaptivity (basis functions not altered).
Enrichment of basis functions.
Thank you!!
Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)
Numerical ResultsConclusions and Future Work
References
J. Douglas, Jr., P. J. Paes Leme, J. E. Roberts, and J. Wang,A parallel iterative procedure applicable to the approximatesolution of second order partial di!erential equations by mixedfinite element methods, Numer. Math., 65 (1993) 95–108.
B. Ganis and I. Yotov, Implementation of a mortar mixed finiteelement method using a multiscale flux basis, ComputerMethods in Applied Mechanics and Engineering, 198 (2009)3989-3998.
Vegard Kippe . Jorg E. Aarnes. Knut-Andreas Lie, Acomparison of multiscale methods for elliptic problems inporous media flow, Comput Geosci, (2008) 12:377-398.
Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)
Numerical ResultsConclusions and Future Work
Variational Formulation
The pressure and velocity spaces for the global problem [1, 2] are:
W = L2(!) and Vr = {v + H(div ;!)| v.! = r on "N},
where H(div ;!) = {v + (L2(!))2| div v + L2(!)}.
The global weak form is giving by finding {p,u} + W * Vr suchthat
(K"1u, u)" " (p, div u)" = 0, u + V0, (7)
(div u, p)" = (f , p)", p + W . (8)
Motivation: Domain Decomposition MethodThe Multiscale Mixed Method (MuMM)
Numerical ResultsConclusions and Future Work
Variational Formulation
Similarly, define the spaces for each subdomain !j by
Wj = {w |!j | w + W (!)},
Vr ,j = {v + H(div ;!j) | v.!j = r on "!j $ "N}.
The weak formulation are given by seeking {pj ,uj} + Wj * Vr ,j
such that
(div u, p)"j = (f , p)"j , p + Wj ,
(K"1u, u)"j " (p, div u)"j +"
j #=k
< p, u.!j >#jk=
"M"
j
< pb, u.!j >#"j$#D , u + V0,j ,
where "jk = "kj = "!j $ "!k .
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