Abstract
We present the implementation of two hierarchically preconditioned methods for the fast solution of mesh equations that approximate 2D-elliptic boundary value problems on unstructured quasi uniform triangulations. Based on the fictitious space approach the original problem can be embedded into an auxiliary one, where both the hierarchical grid information and the preconditioner are well defined. We implemented the corresponding Yserentant preconditioned conjugate gradient method as well as thebpx-preconditioned cg-iteration having optimal computational costs. Several numerical examples demonstrate the efficiency of the artificially constructed hierarchical methods which can be of importance in industrial engineering, where often only the nodal coordinates and the element connectivity of the underlying (fine) discretization are available.
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References
Aubin, J. P.: Approximation of elliptic boundary value problems. New York London Sydney Toronto: Wiley-Interscience, 1972.
Bank, R. E., Xu, J.: The hierarchical basis multigrid method and incomplete LU decomposition. Cont. Math.180, 163–174 (1994).
Bank, R. E., Xu, J.: An algorithm for coarsening unstructured meshes. Numer. Math.73, 1–36 (1996).
Bramble, J. H., Pasciak, J. E., Xu, J.: Parallel multilevel preconditioners. Math. Comp.55, 1–22 (1990).
Chan, T. F., Smith, B. F.: Domain decomposition and multigrid algorithms for elliptic problems on unstructured meshes. Cont. Math.180, 175–190 (1994).
Ciarlet, Ph.: The finite element method for elliptic problems. Amsterdam: North-Holland, 1977.
Globisch, G.: Robuste Mehrgitterverfahren für einige elliptische Randwertaufgaben in zweidimensionalen Gebieten. Technische Universität Chemnitz, Dissertation, Chemnitz, 1993.
Globisch, G.: parmesh — a parallel mesh generator. Parallel Comput.21, 509–524 (1995).
Globisch, G.: The hierarchical preconditioning having unstructured threedimensional grids. Preprint SFB393/97_25, Technische Universität Chemnitz, Chemnitz, 1997.
Groh, U.: FEM auf irregulären hierarchischen Dreiecksnetzen. Preprint SFB393/97_05, Technische Universität Chemnitz, Chemnitz, 1997.
Haase, G., Langer, U., Meyer, A.: Parallelisierung und Vorkonditionierung des CG-Verfahrens durch Gebietszerlegung. In: Bader, G., Rannacher, R., Wittum, G. (eds.) Numerische Algorithmen auf Transputer-Systemen. Teubner-Skripten zur Numerik. Stuttgart: Teubner-Verlag, 1992.
Haase, G., Hommel, Th., Meyer, A., Pester, M.: Bibliotheken zur Entwicklung paralleler Algorithmen. Preprint SPC 95_20, Technische Universität Chemnitz-Zwickau, Chemnitz, 1995.
Heise, B.: Analysis of a fully discrete finite element method for a nonlinear magnetic field problem. SIAM J. Numer. Anal.31, 745–759 (1994).
Matsokin, A. M., Nepomnyaschikh, S. V.: The fictitious domain method and explicit continuation operators. Zh. Vychisl. Mat. Mat. Fiz.33, 45–59 (1993).
Meyer, A.: A parallel preconditioned conjugate gradient method using domain decomposition and inexact solvers on each subdomain. Computing45, 217–234 (1990).
Meyer, A., Pester, M.: Verarbeitung von Sparse-Matrizen in Kompaktspeicherform KLZ/KZU. Preprint SPC 94_12, Technische Universität Chemnitz-Zwickau, Chemnitz, 1994.
Nepomnyaschikh, S. V.: Method of splitting into subspaces for solving elliptic boundary value problems in complex-form-domains. Sov. J. Numer. Anal. Math. Model.6, 151–168 (1991).
Nepomnyaschikh, S. V.: Mesh theorems of traces, normalization of function traces and their inversion. Sov. J. Numer. Anal. Model.6, 223–242 (1991).
Nepomnyaschikh, S. V.: Fictitious space method on unstructured meshes. East-West J. Numer. Math.3, 71–79 (1995).
Nepomnyaschikh, S. V.: Preconditioning operators on unstructured grids. In: Nelson, N. D., Manteuffel, T. A., McCormick, S. F., Douglas, C. C. (eds.) Proceedings of the Seventh Copper Mountain Conference on Multigrid Methods. NASA-Conference Publication,3339, 607–621 (1996).
Oganesyan, L. A., Ruchovets, L. A.: Variational difference methods for solving elliptic equations. Izdat. Akad. Nauk Arm. SSR, Erevan (1979) (in Russian).
Oswald, P.: Multilevel finite element approximation: Theory and applications. Teubner Skripten zur Numerik. Stuttgart: B. G. Teubner, 1994.
Queck, W. (ed.): femgp (Finite Element Multigrid Package). Programmdokumentation, Technologieberatungszentrum Parallele Informationsverarbeitung GmbH (TBZ* PARIV), Chemnitz (1993).
Xu, J.: Iterative methods by space decomposition and subspace correction. SIAM Rev.34, 581–613 (1992).
Xu, J.: The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids. Computing56, 215–235 (1996).
Yakovlev, G. N.: On traces of piecewise smooth surfaces of functions from the spaceWp/l. Mat. Sbornik74, 526–543 (1967).
Yserentant, H.: On the multi-level splitting of finite element spaces. Numer. Math.49, 379–412 (1986).
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Globisch, G., Nepomnyaschikh, S.V. The hierarchial preconditioning on unstructured grids. Computing 61, 307–330 (1998). https://doi.org/10.1007/BF02684383
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DOI: https://doi.org/10.1007/BF02684383