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The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids

Die Hilfsraummethode und optimale Mehrgitter-Präkonditionierungstechniken für unstrukturierte Gitter

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Abstract

An abstract framework ofauxiliary space method is proposed and, as an application, an optimal multigrid technique is developed for general unstructured grids. The auxiliary space method is a (nonnested) two level preconditioning technique based on a simple relaxation scheme (smoother) and an auxiliary space (that may be roughly understood as a nonnested coarser space). An optimal multigrid preconditioner is then obtained for a discretized partial differential operator defined on an unstructured grid by using an auxiliary space defined on a more structured grid in which a furthernested multigrid method can be naturally applied. This new technique makes it possible to apply multigrid methods to general unstructured grids without too much more programming effort than traditional solution methods. Some simple examples are also given to illustrate the abstract theory and for instance the Morley finite element space is used as an auxiliary space to construct a preconditioner for Argyris element for biharmonic equations. Some numerical results are also given to demonstrate the efficiency of using structured grid for auxiliary space to precondition unstructured grids.

Zusammenfassung

Das abstrakte Konzept derHilfsraummethode wird vorgeschlagen, und als Anwendung eine optimale Mehrgittertechnik für allgemeine unstrukturierte Gitter entwickelt. Die Hilfsraummethode ist eine (nichtgeschachtelte) Zweigitter-Präkonditionierungstechnik, bestehend aus einem einfachen Glättungsverfahren und einem Hilfsgitter, das etwa einem nichtgeschachtelten Grobgitter entspricht. Zu einer partiellen Differentialgleichung, die auf einem unstrukturierten Gitter diskretisiert ist, wird eine optimale Mehrgittermethode konstruiert, die einen Hilfsraum mit einem besser strukturierten Gitter verwendet. In diesem ist ein Mehrgitterverfahren mit geschachtelten Räumen anwendbar. Die neue Technik ermöglicht es, Mehrgittermethoden für unstrukturierte Gitter anzuwenden, die nur wenig mehr Programmieraufwand erfordern als traditionelle Methoden. Einige einfache Beispiele sollen die abstrakte Theorie illustrieren. Als Beispiel wird der Morley-FE-Raum als Hilfsraum zur Präkonditionierung der Argyris-Elemente für die biharmonische Gleichung benutzt. Einige numerische Resultate demonstrieren die Effizienz des Einsatzes des strukturierten Hilfsraumes.

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References

  1. Bank, R., Xu, J.: An algorithms for coarsening unstructured meshes. Numer Math. (to appear).

  2. Bramble, J. H.: Multigrid methods. Pitman Research Notes in Mathematical Sciences, Vol. 294. Essex: Longman 1993.

    Google Scholar 

  3. Bramble, J. H., Pasciak, J. E.: New estimates for multilevel algorithms including the V cycle. Math. Comp.60, 447–471 (1993).

    Google Scholar 

  4. Bramble, J. H., Pasciak, J. E. Xu, J.: Parallel multilevel preconditioners. Math. Comp.55, 1–22 (1990).

    Google Scholar 

  5. Bramble, J. H., Pasciak, J. E. Xu, J.: The analysis of multigrid algorithms with nonnested spaces or noninherited quadratic forms. Math. Comp.56, 1–34 (1991).

    Google Scholar 

  6. Bramble, J. H., Xu, J.: Some estimates for a weightedl 2 projection. Math. Comp.56, 463–476 (1991).

    Google Scholar 

  7. Brandt, A.: Multi-level adaptive solutions to boundary-value problems. Math. Comp.31, 333–390 (1977).

    Google Scholar 

  8. Brenner, S. C.: Two-level additive schwarz preconditioners for nonconforming finite element methods (preprint).

  9. Brenner, S. C.: An optimal order nonconforming multigrid method for the biharmonic equation. SIAM J. Numer. Anal.26, 1124–1138 (1989).

    Article  Google Scholar 

  10. Hackbusch, W.: Multigrid methods and applications Vol. 4. Computational mathematics. Berlin Heidelberg New York: Springer 1985.

    Google Scholar 

  11. Kornhuber, R., Yserentatn, H.: Multilevel methods for elliptic problems on domains not resolved by the coarse grid. In: Domain Decomposition Methods in Science and Engineering. Proceedings of the Seventh International Conference on Domain Decomposition, October 27–30, 1993, The Pennsylvania State University (Keyes, D. E., Xu, J., eds.). Providence: AMS 1995.

    Google Scholar 

  12. McCormick, S. F.: Multilevel adaptive methods for partial differential equations. Frontiers in Applied Mathematics, Vol. 6., Philadelphia: SIAM 1989.

    Google Scholar 

  13. Nepomnyaschikh, S. V.: Mesh theorems of traces, normalizations of function traces and their inversion. Sov. J. Numer. Anal. Math. Modeling6, 1–25 (1991).

    Google Scholar 

  14. Nepomnyaschikh, S. V.: Decomposition and fictitious domains methods for elliptic boundary value problems. In: Fifth International Symposium on Domain Decomposition Methods for Partial Differential Equations (Keyes, D. E., Chan, T. F., Meurant, G., Scroggs, J. S., Voigt, R. G., eds.), pp. 62–72. Philadelphia: SIAM 1992.

    Google Scholar 

  15. Peisker, P., Braess, D.: A conjugate gradient method and a multigrid method for Morley's finite element approximation of the biharmonic equation. Numer. Math.50, 567–586 (1987).

    Article  Google Scholar 

  16. Raviart, P.-A., Thomas, J. M.: Primal hybrid finite element methods for 2nd order elliptic equations. Math. Comp.31, 391–413, 1977.

    Google Scholar 

  17. Smith, B., Chan, T., Zou, J.: Overlapping schwarz methods on unstructured meshes using non-matching coarse grids. Technical Report 94-8, Dept. of Math, UCLA, Feb. 1994.

  18. Xu, J.: Theory of multilevel methods. PhD thesis, Cornell University, 1989.

  19. Xu, J.: Convergence estimates for some multigrid algorithms. In: Third International Symposium of Domain Decomposition Methods for Partial Differential Equations (Chan, T. F., Glowinski, R., Périaux, J., Widlund, O. B., eds.), pp. 174–187. Philadelphia: SIAM 1990.

    Google Scholar 

  20. Xu, J.: Iterative methods by space decomposition and subspace correction. SIAM Rev.34, 581–613 (1992).

    Article  Google Scholar 

  21. Xu, J.: Multigrid and domain decomposition methods. In: Numerical analysis in applied sciences (Cai, W., Shi, Z., Shu, J., Xu, J., eds.). Singapore: Science Press, 1995.

    Google Scholar 

  22. Zhang, S.: Optimal order non-nested multigrid methods for solving finite element equations I–III. Math. Comp., 1990, 1991, 1994.

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Authors and Affiliations

  1. Department of Mathematics, The Pennsylvania State University, 16802, University Park, PA, USA

    J. Xu

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  1. J. Xu
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This work was partially supported by National Science Foundation, DMS94-03915-1.

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Xu, J. The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids. Computing 56, 215–235 (1996). https://doi.org/10.1007/BF02238513

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