Preconditioning GMRES for discontinuous Galerkin approximations

  • Krzysztof Banaś Cracow University of Technology
  • Mary F. Wheeler The University of Texas

Abstract

The paper presents an implementation and the performance of several preconditioners for the discontinuous Galerkin approximation of diffusion dominated and pure diffusion problems. The preconditioners are applied for the restarted GMRES method and test problems are taken mainly from subsurface flow modeling. Discontinuous Galerkin approximation is implemented within an hp-adaptive finite element code that uses hierarchical 3D meshes. The hierarchy of meshes is utilized for multi-level (multigrid) preconditioning. The results of numerical computations show the necessity of using multi-level preconditioning and insufficiency of simple stationary preconditioners, like Jacobi or Gauss- Seidel. Successful preconditioners comprise a multi-level block ILU algorithm and a special multi-level block Gauss- Seidel method.

Keywords

References

[1] D. N. Arnold, F. Brezzi, B. Cockburn, D. Marini. Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM Journal on Numerical Analysis, 39: 1749-1779, 2001/2002.
[2] X. Feng, O. A. Karakashian. Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems. SIAM Journal on Numerical Analysis, 39: 1343-1365, 200l.
[3] F. Bassi, S. Rebay. GMRES discontinuous Galerkin solution of the compressible Navier-Stokes equations. In Discontinuous Galerkin methods, (Eds.), B. Cockburn, B. E. Karniadakis, C. W. Shu, volume 11 of Lecture Notes in Computational Science and Engineering, Springer Verlag, 197-208, Berlin, 2000.
[4] J. T. Oden, I. Babuska, C. E. Baumann. A discontinous hp finite element method for diffusion problems. Journal of Computational Physics, 146: 491-519, 1998.
[5] M. F. Wheeler B. Riviere, V. Girault. Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems, Part I. Computational Geosciences, 3: 337-360, 1999.
Published
Jan 18, 2023
How to Cite
BANAŚ, Krzysztof; WHEELER, Mary F.. Preconditioning GMRES for discontinuous Galerkin approximations. Computer Assisted Methods in Engineering and Science, [S.l.], v. 11, n. 1, p. 47-62, jan. 2023. ISSN 2956-5839. Available at: <https://cames.ippt.pan.pl/index.php/cames/article/view/1043>. Date accessed: 14 nov. 2024.
Section
Articles