Outline: 1 Motivation: Domain Decomposition Method 2 3 4

Transcription

1 Multiscale Basis Functions for Iterative Domain Decomposition Procedures A. Francisco 1, V. Ginting 2, F. Pereira 3 and J. Rigelo 2 1 Department Mechanical Engineering Federal Fluminense University, Volta Redonda, RJ , Brazil afrancisco@metal.eeimvr.uff.br 2 Department of Mathematics University of Wyoming, Laramie, WY , USA {vginting,jrigelo}@uwyo.edu 3 Department of Mathematics and School of Energy Resources University of Wyoming, Laramie, WY , USA lpereira@uwyo.edu Support: DOE: DE-FE /DE-SC ; NSF: DMS ; Center for Fundamentals of Subsurface Flow(UW).

2 Outline: 1 Motivation: Domain Decomposition Method 2 3 4

3 Motivation We are concerned with the development of numerical procedures for the fast and accurate approximation of subsurface flows that can take advantage of heterogeneous processing units.

4 Motivation Incorporate fine scale information into a coarse scale discretization, without solving it directly. Coarse Domain Decomposition Our iterative procedure does not use MPI in each iteration.

5 Model Problem Our model problem is a second order linear elliptic equation written as a first order system.u = f (x), where u = k(x) p in Ω, (1) p = p b on Γ D, u.ν = u b on Γ N. (2) Here Ω is a bounded domain with a Lipschitz boundary Ω = Γ D Γ N,Γ D Γ N =.

6 Domain Decomposition The domain Ω is divided into a non-overlapping partition {Ω j }: Ω = M j=1 Ω j ; Ω j Ω k =, j k. Motivation: Non-overlapping iterative DDM based on the Robin boundary conditon. J. Douglas, Jr., P. J. Paes Leme, J. E. Roberts, and J. Wang, A parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed finite element methods, Numer. Math., 65 (1993)

7 Domain Decomposition (cont.) (5) Mixed finite element space: hybridized Raviart-Thomas. Procedure: subdomain = element. Degrees of freedom (for each Ω j ): p, u β and l β, β = L, R, B, T. h Then for a single element, the discrete form of the Poisson s equation is given by L T B R u L + u R + u B + u T = fh, (3) u β 2 h k(p l β) = 0. (4)

8 Domain Decomposition (cont.) Robin Interface Condition: l β = χ β (u β +ũ β )+ l β, where β = L, R, B, T. (6) Ω j Ω k β - β R L B T L R T B Γ jk

9 Domain Decomposition (cont.) Douglas, Jr. et al. parallel iterative scheme: 1 Set an initial guess: {p 0, u 0 β,l0 β }. 2 For all red elements, update {p, u β,l β }, using [3, 4, 5]. 3 For all black elements, compute {p, u β,l β }, by solving [3, 4, 5], using the updated values from the red elements. 4 Check for convergence. old old new old old

10 Domain Decomposition (cont.) o o o n o o n o o o subdomain: one element larger subdomain Convergence is established.

11 The Multiscale Basis Functions Formulation Consider a subdomain Ω j. Let ψ ji =(u i β,li β, pi ) j, i =1,..., 4N, be the basis functions associated with this subdomain. χ L u L +l L = ψ j1 B. Ganis and I. Yotov, Implementation of a mortar mixed finite element method using a multiscale flux basis, Computer Methods in Applied Mechanics and Engineering, 198 (2009)

12 The Multiscale Basis Functions Formulation Given the Robin boundary values A ji, the solution for the Poisson equation is given by S Ωj = 4N i=1 A ji ψ ji Aj1 Aj2 Aj3 where, for i =1,..., 4N, ψ ji =(u i β,li β, pi ) j are the canonical basis functions. N N

13 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!

14 MuMM: A Modified iteration Introduce an intermediate scale H, h H H. Based on an average Robin condition: ul A ji = χ T +ub L L 2 + lt L +lb L 2 A ji T B h H H Ω j Ω k Goal: To reduce the number of basis functions.

15 MuMM: A Modified iteration The solution is given by, for example: S Ωj = χ L u L +l L = ψ j1 Aj1 A j2 4N/2 i=1 A ji ψ ji. N 2D: Douglas, Jr. et al. iteration; 3D: CG preconditioned with the AMG. Solution in the fine grid: post-processing.

16 MuMM: A Modified iteration Remarks : Flux conservation is maintained in the H scale. The balance between numerical accuracy and numerical efficiency is determined by the choice of span{ψ ji } span{ψ ji }. Extreme cases: H = h: Douglas, Jr. et al. iteration. H = H: 4 basis functions/subdomain.

17 Example 1: k max /k min = 176 Example: 2D problem with a fine grid of , coarse grid of 11 3 (subdomains of 20 20). Permeability model: SPE10 model, where k(x) = exp(δ ξ(x)) The physical transport of fluids is given by solving: φ c t + u. c = 0, with I.C. + B.C. given.

18 H = H H = H/2 H = H/4 fine grid

19 Tracer cut curves The fraction of the tracer in the produced fluid is given by Ω F(t) = out c u.n ds Ω out u.n ds.

20 Motivation: Domain Decomposition Method Relative Errors : Relative error =!u MuMM u fine!. maxi,j!u fine! Figure : From top to bottom: 4, 8 and 16 basis functions.

21 Example 2: k max /k min = Example: 2D problem with a fine grid of , coarse grid of 11 3 (subdomains of 20 20). Permeability model: SPE10 model. We consider 16 basis functions. MuMM fine grid

22 Example 2: Tracer cut curve and permeability field

23 Conclusions Properties : u L + u R + u B + u T = 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.

24 Future Work 3D implementation on GPUs. Extension to multiphase/compositional flows. Adaptivity (basis functions not altered). Enrichment of basis functions. Thank you!!

25 References J. Douglas, Jr., P. J. Paes Leme, J. E. Roberts, and J. Wang, A parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed finite element methods, Numer. Math., 65 (1993) B. Ganis and I. Yotov, Implementation of a mortar mixed finite element method using a multiscale flux basis, Computer Methods in Applied Mechanics and Engineering, 198 (2009) Vegard Kippe. Jorg E. Aarnes. Knut-Andreas Lie, A comparison of multiscale methods for elliptic problems in porous media flow, Comput Geosci, (2008) 12:

26 Variational Formulation The pressure and velocity spaces for the global problem [1, 2] are: W = L 2 (Ω) and V r = {v H(div; Ω) v.ν = r on Γ N }, where H(div; Ω) = {v (L 2 (Ω)) 2 div v L 2 (Ω)}. The global weak form is giving by finding {p, u} W V r such that (K 1 u, ǔ) Ω (p, div ǔ) Ω =0, ǔ V 0, (7) (div u, ˇp) Ω =(f, ˇp) Ω, ˇp W. (8)

27 Variational Formulation Similarly, define the spaces for each subdomain Ω j by W j = {w Ω j w W (Ω)}, V r,j = {v H(div;Ω j ) v.ν j = r on Ω j Γ N }. The weak formulation are given by seeking {p j, u j } W j V r,j such that (div u, ˇp) Ωj =(f, ˇp) Ωj, ˇp W j, (K 1 u, ǔ) Ωj (p, div ǔ) Ωj + j k < p, ǔ.ν j > Γjk = M < p b, ǔ.ν j > Ωj Γ D, ǔ V 0,j, j where Γ jk =Γ kj = Ω j Ω k.