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A Dirichlet-to-Neumann (DtN)Multigrid Algorithm for Locally Conservative Methods Sandia National Laboratories is a multi program laboratory managed and.

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Presentation on theme: "A Dirichlet-to-Neumann (DtN)Multigrid Algorithm for Locally Conservative Methods Sandia National Laboratories is a multi program laboratory managed and."— Presentation transcript:

1 A Dirichlet-to-Neumann (DtN)Multigrid Algorithm for Locally Conservative Methods Sandia National Laboratories is a multi program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.. 1 Mary F. Wheeler The University of Texas at Austin – ICES Tim Wildey Sandia National Labs SIAM Conference Computational and Mathematical Issues in the Geosciences March 21-24, 2011 Long Beach, CA

2 Motivation: Multinumerics Coupling of mixed and DG using mortars – G. Pencheva Local grid refinement around wells Advantages in using weak coupling (mortars)

3 Motivation: Multinumerics

4 Motivation: General Framework Both MFEM and DG are locally conservative. Multiscale mortar domain decomposition methods: Arbogast, Pencheva, Wheeler, Yotov 2007 Girault, Sun, Wheeler, Yotov 2008 General a posteriori error estimation framework: Vohralik 2007, 2008 Ern, Vohralik 2009, 2010 Pencheva, Vohralik, Wheeler, Wildey 2010 Is there a multilevel solver applicable to both MFEM and DG? Can it be applied to the case of multinumerics? Can it be used for other locally conservative methods?

5 Outline I.Interface Lagrange Multipliers – Face Centered Schemes II.A Multilevel Algorithm III.Multigrid Formulation IV.Applications V.Conclusions and Future Work

6 Mixed methods yield linear systems of the form: Hybridization of Mixed Methods

7 Mixed methods yield linear systems of the form: Hybridization of Mixed Methods

8 Introduce Lagrange multipliers on the element boundaries: Hybridization of Mixed Methods

9 Introduce Lagrange multipliers on the element boundaries: Hybridization of Mixed Methods

10 Reduce to Schur complement for Lagrange multipliers: Hybridization of Mixed Methods

11 Existing Multilevel Algorithms

12 Mathematical Formulation 12

13 13

14 Assumptions on Local DtN Maps 14

15 Defining Coarse Grid Operators X

16 A Multilevel Algorithm

17 A Multilevel Direct Solver Given a face-centered scheme

18 A Multilevel Direct Solver Given a face-centered scheme 1.Identify interior DOF

19 A Multilevel Direct Solver Given a face-centered scheme 1.Identify interior DOF  Eliminate

20 A Multilevel Direct Solver Given a face-centered scheme 1.Identify interior DOF  Eliminate 2.Identify new interior DOF

21 A Multilevel Direct Solver Given a face-centered scheme 1.Identify interior DOF  Eliminate 2.Identify new interior DOF  Eliminate Continue …

22 Advantages: Only involves Lagrange multipliers No upscaling of parameters Applicable to hybridized formulations as well as multinumerics Can be performed on unstructured grids Easily implemented in parallel Disadvantage: Leads to dense matrices A Multilevel Direct Solver

23 An Alternative Multilevel Algorithm Given a face-centered scheme

24 1.Identify interior DOF An Alternative Multilevel Algorithm

25 Given a face-centered scheme 1.Identify interior DOF  Coarsen

26 An Alternative Multilevel Algorithm Given a face-centered scheme 1.Identify interior DOF  Coarsen  Eliminate

27 An Alternative Multilevel Algorithm Given a face-centered scheme 1.Identify interior DOF  Coarsen  Eliminate 2.Identify new interior DOF

28 An Alternative Multilevel Algorithm Given a face-centered scheme 1.Identify interior DOF  Coarsen  Eliminate 2.Identify new interior DOF  Coarsen

29 An Alternative Multilevel Algorithm Given a face-centered scheme 1.Identify interior DOF  Coarsen  Eliminate 2.Identify new interior DOF  Coarsen  Eliminate Continue …

30 How to use these coarse level operators? An Alternative Multilevel Algorithm

31 Multigrid Formulation

32 A Multigrid Algorithm

33

34

35

36

37

38 Theorem A Multigrid Algorithm

39 Numerical Results

40 Laplace Equation - Mixed LevelsDOFV-cyclesMG Factor 322480.19 496080.22 5396890.23 61612890.24 76502490.24

41 Laplace Equation – Symmetric DG LevelsDOFV-cyclesMG Factor 322480.20 496080.21 5396880.21 61612880.21 76502480.21

42 Laplace Equation – Symmetric DG LevelsDOFV-cyclesMG Factor 322450.08 496050.08 5396850.08 61612850.08 76502450.08

43 Laplace Equation – Nonsymmetric DG LevelsDOFV-cyclesMG Factor 322470.16 496070.17 5396870.17 61612870.17 76502470.17

44 Laplace Equation – Nonsymmetric DG LevelsDOFV-cyclesMG Factor 322480.18 496080.18 5396880.19 61612880.19 76502480.19

45 Laplace Equation – Multinumerics

46 LevelsDOFV-cyclesMG Factor 322480.19 496080.19 5396880.20 61612880.20

47 Advection - Diffusion LevelsDOFV-cyclesMG FactorPGMRES Iters 4960100.237 5396870.116 61612880.115 76502490.145

48 Poisson Equation – Unstructured Mesh

49 Single Phase Flow with Heterogeneities

50 Conclusions and Future Work Developed an optimal multigrid algorithm for mixed, DG, and multinumerics. No subgrid physics required on coarse grids  only local Dirichlet to Neumann maps. No upscaling of parameters. Only requires solving local problems (of flexible size). Applicable to unstructured meshes. Physics-based projection and restriction operators. Extends easily to systems of equations (smoothers?) ? Analysis for nonsymmetric operators/formulations ? Algebraic approximation of parameterization

51 Thank you for your attention! Questions?

52 Poisson Equation - Full Tensor LevelsDOFV-cyclesMG FactorPCG Iters 3224130.367 4960170.468 53968190.499 616128200.4810 765024210.4710

53 Poisson Equation - Jumps in Permeability LevelsDOFV-cyclesMG FactorPGMRES Iters 3224150.3510 4960310.618 53968290.598 616128270.568 765024250.528

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