×

How we solve PDEs. (English) Zbl 0982.65113

Summary: A finite difference method on an unstructured finite element mesh which we call finite difference element method (FDEM) is presented. The FDEM program package will be a black-box solver for systems of nonlinear elliptic and parabolic partial differential equations (PDEs) with mesh refinement and automatic control of the consistency order in each space grid point. In this paper we present the solution method (with examples) for 2-D systems of elliptic PDEs.

MSC:

65N06 Finite difference methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations

Software:

LINSOL; PLTMG
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abgrall, R.; Lanteri, S.; Sonar, T., ENO approximations for compressible fluid dynamics, Z. Angew. Math. Mech., 79, Suppl. 1, 3-28 (1999) · Zbl 0917.76060
[2] Bank, R. E., PLTMG: A Software Package for Solving Elliptic Partial Differential Equations (1994), SIAM: SIAM Philadelphia · Zbl 0860.65113
[3] P. Bastian et al., A parallel software platform for solving problems of partial differential equations using unstructured grids and adaptive multigrid methods, in: E. Krause, W. jäger (Eds.), High Performance Computing in Science and Engineering ’98, Springer, Berlin, 1999, PP. 3260-339.; P. Bastian et al., A parallel software platform for solving problems of partial differential equations using unstructured grids and adaptive multigrid methods, in: E. Krause, W. jäger (Eds.), High Performance Computing in Science and Engineering ’98, Springer, Berlin, 1999, PP. 3260-339. · Zbl 0928.65120
[4] Blom, J. G.; Trompert, R. A.; Verwer, J. G., VLUGR2. a vectorizable adaptive grid solver for PDEs in 2 D, Report NM-R9403 (1994), CWI: CWI Amsterdam · Zbl 0884.65094
[5] J.G. Blom, J.G. Verwer, VLUGR3: A vectorizable adaptive grid solver for PDEs in 3 D. I. Algorithmic aspects and applications, Appl. Numer. Math. 16 (1994) 129-156 and CWI Report NM-R9404.; J.G. Blom, J.G. Verwer, VLUGR3: A vectorizable adaptive grid solver for PDEs in 3 D. I. Algorithmic aspects and applications, Appl. Numer. Math. 16 (1994) 129-156 and CWI Report NM-R9404. · Zbl 0816.65049
[6] Blom, J. G.; Verwer, J. G., VLUGR3: A vectorizable adaptive grid solver for PDEs in 3D. II. Code description, Report NM-R9405 (1994), CWI: CWI Amsterdam · Zbl 0816.65049
[7] L. Grosz, C. Roll, W. Schönauer, A black box solver for the numerical solution of general nonlinear functional equations by mixed FEM, in: M. Krizek, P. Nettaanmaeki, R. Steenberg (Eds.), Fifty Years of the Courant Element, Marcel Dekker, New York 1994, pp. 225-234. This paper and further documents are available under the URL: http://www.rz.uni-karlsruhe.de/Uni/RZ/Forschung/Numerik/vecfem/docs.html.; L. Grosz, C. Roll, W. Schönauer, A black box solver for the numerical solution of general nonlinear functional equations by mixed FEM, in: M. Krizek, P. Nettaanmaeki, R. Steenberg (Eds.), Fifty Years of the Courant Element, Marcel Dekker, New York 1994, pp. 225-234. This paper and further documents are available under the URL: http://www.rz.uni-karlsruhe.de/Uni/RZ/Forschung/Numerik/vecfem/docs.html. · Zbl 0812.65097
[8] L. Grosz, A-posteriori error estimates for the finite element solution of non-linear variational problems, Dissertation Universität Karlsruhe, 1997, this dissertation is available under the URL given in [1].; L. Grosz, A-posteriori error estimates for the finite element solution of non-linear variational problems, Dissertation Universität Karlsruhe, 1997, this dissertation is available under the URL given in [1].
[9] H. Häfner, W. Schönauer, R. Weiss, The parallel and portable linear solver package LINSOL, in: H. Lederer, F. Hertweck (Eds.), Proceedings of the Fourth European SGI/Cray MPP Workshop, Max-Planck-Institut für Plasmaphysik, Garching bei München, Germany, 1998, pp. 242-251.; H. Häfner, W. Schönauer, R. Weiss, The parallel and portable linear solver package LINSOL, in: H. Lederer, F. Hertweck (Eds.), Proceedings of the Fourth European SGI/Cray MPP Workshop, Max-Planck-Institut für Plasmaphysik, Garching bei München, Germany, 1998, pp. 242-251.
[10] Häfner, H.; Schönauere, W.; Weiss, R., Parallelization and integration of the LU and ILL algorithm in the LINSOL program package, (Malyskin, V., Parallel Computing Technologies, Springer Lecture Notes in Computer Science, Vol. 1662 (1999), Springer: Springer Berlin), 417-427
[11] Rannacher, H., Error control in finite element computations, (Bulgak, C.; Zenger, C., Error Control and Adaptivity in Scientific Computing (1999), Kluwer Academic Publishers, Dordrecht: Kluwer Academic Publishers, Dordrecht Netherlands), 247-278 · Zbl 0943.65123
[12] Schmauder, M.; Schönauer, W., CADSOL — A fully vectorized black box solver for 2-D and 3-D partial differential equations, (Vichnevetsky, R.; Knight, D.; Richter, G., Advances in Computer Methods for Partial Differential Equations — VII (1992), IMACS: IMACS New Brunswick), 639-645
[13] Schönauer, W., Scientific Computing on Vector Computers (1987), North-Holland: North-Holland Amsterdam
[14] W. Schönauer, H. Häfner, R. Weiss, LINSOL, a parallel iterative linear solver package of generalized CG-type for sparse matrices, in: M. Heath et al. (Eds.), Proceedings of the Eighth SIAM Conference on Parallel Processing for Scientific Computing, SIAM, Philadelphia 1997, CD-ROM (ISBN 0-89871-395-1), 8 pp.; W. Schönauer, H. Häfner, R. Weiss, LINSOL, a parallel iterative linear solver package of generalized CG-type for sparse matrices, in: M. Heath et al. (Eds.), Proceedings of the Eighth SIAM Conference on Parallel Processing for Scientific Computing, SIAM, Philadelphia 1997, CD-ROM (ISBN 0-89871-395-1), 8 pp.
[15] W. Schönauer, Generation of difference and error formulae of arbitrary consistency order on an unstructured grid, to appear in Z. Angew. Math. Mech. 78 (Suppl. 3) (1998) 1061-1062 (special issue of the GAMM-Tagung 1997).; W. Schönauer, Generation of difference and error formulae of arbitrary consistency order on an unstructured grid, to appear in Z. Angew. Math. Mech. 78 (Suppl. 3) (1998) 1061-1062 (special issue of the GAMM-Tagung 1997).
[16] W. Schönauer, Scientific supercomputing, available in the internet by the URL: http://www.uni-karlsruhe.de/Uni/RZ/Personen/rz03/book/.; W. Schönauer, Scientific supercomputing, available in the internet by the URL: http://www.uni-karlsruhe.de/Uni/RZ/Personen/rz03/book/.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.