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A Dirichlet -to-Neumann ( DtN ) Multigrid Algorithm for Locally Conservative Methods

A Dirichlet -to-Neumann ( DtN ) Multigrid Algorithm for Locally Conservative Methods . 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.

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A Dirichlet -to-Neumann ( DtN ) Multigrid Algorithm for Locally Conservative Methods

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  1. A Dirichlet-to-Neumann (DtN)Multigrid Algorithm for Locally Conservative Methods 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 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. .

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

  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 • Interface Lagrange Multipliers – Face Centered Schemes • A Multilevel Algorithm • Multigrid Formulation • Applications • Conclusions and Future Work

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

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

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

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

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

  11. Existing Multilevel Algorithms

  12. Mathematical Formulation 12

  13. Mathematical Formulation 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 Identify interior DOF

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

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

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

  22. A Multilevel Direct Solver 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

  23. An Alternative Multilevel Algorithm Given a face-centered scheme

  24. An Alternative Multilevel Algorithm Given a face-centered scheme Identify interior DOF

  25. An Alternative Multilevel Algorithm • Given a face-centered scheme • Identify interior DOF • Coarsen

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

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

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

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

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

  31. Multigrid Formulation

  32. A Multigrid Algorithm

  33. A Multigrid Algorithm

  34. A Multigrid Algorithm

  35. A Multigrid Algorithm

  36. A Multigrid Algorithm

  37. A Multigrid Algorithm

  38. A Multigrid Algorithm Theorem

  39. Numerical Results

  40. Laplace Equation - Mixed

  41. Laplace Equation – Symmetric DG

  42. Laplace Equation – Symmetric DG

  43. Laplace Equation – Nonsymmetric DG

  44. Laplace Equation – Nonsymmetric DG

  45. Laplace Equation – Multinumerics

  46. Laplace Equation – Multinumerics

  47. Advection - Diffusion

  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

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