Locally Refined Solution of Unsymmetric and Nonlinear Problems

• Peter Bastian
Part of the Notes on Numerical Fluid Mechanics (NNFM) book series (NNFM, volume 29)

Abstract

We describe a multigrid method with optimal computational work per cycle on locally refined grids. The method can be interpreted as a multiplicative variant of the BPX preconditioner but it is motivated from the viewpoint of the classical multigrid method. This has several advantages: In the case of quasi-uniform refinement the method is equivalent to the classical multigrid method. All well known smoothing algorithms can be used, including incomplete decompositions. In the nonlinear case the nonlinear multigrid method can be directly transferred to locally refined grids. Since no outer CG iteration is needed the method can also be applied directly to unsymmetric problems. Results will be presented for scalar, linear and nonlinear convection-diffusion equations.

Keywords

multigrid method unstructured locally refined grids nonlinear p.d.e unsymmetric linear systems

References

1. [1]
R. Bank, T. F. Dupont, H. Yserentant, The Hierarchical Basis Multigrid Method ,Numer. Math., 52, 427–458 (1988).
2. [2]
3. [3]
P. Bastian, ug 2.0 Short Manual ,Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, Univ. Heidelberg, 1992.Google Scholar
4. [4]
C. Becker, J.H. Ferziger, M. Peric, and G. Scheuerer, Finite Volume Multigrid Solution of the Twodimensional, Incompressible Navier-Stokes Equations , in Robust Multigrid Methods, Proceedings of the Fourth GAMM Seminar, Notes on Numerical Fluid Mechanics, Vieweg Verlag, Braunschweig, 23 (1988).Google Scholar
5. [5]
F. A. Bornemann, A Sharpened Condition Number Estimate for the BPX Preconditioner of Elleptic Finite Element Problems on Highly Nonuniform Triangulations ,Preprint SC 91, Konrad-Zuse-Zentrum, Berlin, 1991.Google Scholar
6. [6]
F.A. Bornemann, H. Yserentant, A Basic Norm Equivalence for the Theory of Multilevel Methods ,Preprint SC 92–1, Konrad-Zuse-Zentrum, Berlin, 1992Google Scholar
7. [7]
J. H. Bramble, J. E. Pasciak, and J. Xu, Parallel Multilevel Preconditioners ,Math. Com-put, 55, 1–22 (1990).
8. [8]
J. H. Bramble, J. E. Pasciak, New Estimates for Multilevel Algorithms Including the V Cycle ,Technical Report Brookhaven National Laboratory, BNL-46730, 1991.Google Scholar
9. [9]
A. Brandt, Multi-level Adaptive Solutions to Boundary Value Problems, Mathematics of Computation ,31, 333–390 (1977).
10. [10]
W. Hackbusch, Multigrid Methods and Applications ,Springer Verlag, Berlin, Heidelberg, 1985.Google Scholar
11. [11]
W. Hackbusch, Iterative Lösung groer schwachbesetzter Gleichungssysteme ,Teubner Verlag, Stuttgart, 1991.
12. [12]
W. Hackbusch, On First and Second Order Box Schemes ,Computing, 41, 277–296(1989).
13. [13]
M. Hortmann, M. Peric, and G. Scheuerer, Finite Volume Multigrid Prediction of Lami nar Natural Convection: Bench-Mark Solutions ,International Journal for Numerical Methods in Fluids, 11, 189–207 (1990).
14. [14]
J. A. Meijerink, H. A. van der Vorst, An Iterative Solution Method for Linear Systems of which the Coefficient Matrix is a Symmetric M-Matrix ,Mathematics of Computation, 31, 148–162 (1977).
15. [15]
P. Oswald, On Discrete Norm Estimates Related to Multilevel Preconditioners in the Fi nite Element Method ,Proc. Int. Conf. Constr. Theory of Functions, Varna, 1991, to appear.Google Scholar
16. [16]
G. E. Schneider, M. J. Raw, Control Volume Finite-Element Method for Heat Transfer and Fluid Flow Using Colocated VariablesL Computational Procedure ,Numerical Heat Transfer, 11, 363–390 (1987).
17. [17]
P. Wesseling, An Introduction to Multigrid Methods ,John Wiley &Sons, Chichester, 1992.
18. [18]
G. Wittum, Multigrid Methods for Stokes and Navier-Stokes Equations ,Numer. Math., 54 (1989).Google Scholar
19. [19]
G. Wittum, On the Robustness of ILU Smoothing ,SIAM J. Sci. Statist. Comput, 10, 699–717 (1989).
20. [20]
J. Xu, Iterative Methods by Space Decomposition and Subspace Correction: A Unifying Approach ,Report No. AM 67, Dep. of Math., Pennsylvania State University, 1990.Google Scholar
21. [21]
H. Yserentant, On the Multi-Level Splitting of Finite Element Spaces ,Numer. Math., 49, 379–412 (1986).
22. [22]
H. Yserentant, Two Preconditioners Based on Multi-Level Splitting of Finite Element Spaces ,Numer. Math., 58, 163–184 (1990).