Abstract
We describe and analyze the parallel implementation of a novel domain decomposition preconditioner for the fast iterative solution of linear systems of algebraic equations arising from the discretization of elliptic partial differential equations (PDEs) in three dimensions. In previous theoretical work, [3], this preconditioner has been proved to be optimal for symmetric positive-definite (SPD) linear systems. In this paper we provide details of our 3-d parallel implementation and demonstrate that the technique may be generalized to the solution of non-symmetric algebraic systems, such as those arising when convection-diffusion problems are discretized using either Galerkin or stabilized finite element methods (FEMs), [9].
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Ashby, S. F., Manteuffel, T. A., Taylor, P. E.: A Taxonomy for Conjugate Gradient Methods. SIAM J. on Numer. Anal. 27 (1990) 1542–1568.
Bank, R. E., Jimack, P. K.: A New Parallel Domain Decomposition Method for the Adaptive Finite Element Solution of Elliptic Partial Differential Equations. Concurrency and Computation: Practice and Experience 13 (2001) 327–350.
Bank, R. E., Jimack, P. K., Nadeem, S. A., Nepomnyaschikh, S. V.: A Weakly Overlapping Domain Decomposition for the Adaptive Finite Element Solution of Elliptic Partial Differential Equations. Submitted to SIAM J. on Sci. Comp. (2001).
Cai, X.-C., Sarkis, M.: A Restricted Additive Schwarz Preconditioner for General Sparse Linear Systems. SIAM J. on Sci. Comp. 21 (1999) 792–797.
Chan, T., Mathew, T.: Domain Decomposition Algorithms. Acta Numerica 3 (1994) 61–143.
Farhat, C., Mandel, J., Roux, F. X.: Optimal Convergence Properties of the FETI Domain Decomposition Method. Comp. Meth. for Appl. Mech. and Eng. 115 (1994) 365–385.
Gropp, W. D., Keyes, D. E.: Parallel Performance of Domain-Decomposed Preconditioned Krylov Methods for PDEs with Locally Uniform Refinement. SIAM J. on Sci. Comp. 13 (1992) 128–145.
Hodgson, D. C., Jimack, P. K.: A Domain Decomposition Preconditioner for a Parallel Finite Element Solver on Distributed Unstructured Grids. Parallel Computing 23 (1997) 1157–1181.
Johnson, C.: Numerical Solutions of Partial Differential Equations by the Finite Element Method. Cambridge University Press (1987).
Message Passing Interface Forum: MPI: A Message-Passing Interface Standard. Int. J. Supercomputer Appl. 8 (1994) no. 3/4.
Oswald, P.: Multilevel Finite Element Approximation: Theory and Applications. Teubner Skripten zur Numerik, B. G. Teubner (1994).
Saad, Y.: SPARSEKIT: A Basic Tool Kit for Sparse Matrix Computations, Version 2. Technical Report, Center for Supercomputing Research and Development, University of Illinois at Urbana-Champaign, Urbana, IL, USA (1994).
Smith, B., Bjorstad, P., Gropp, W.: Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press (1996).
Speares, W., Berzins, M.: A 3-D Unstructured Mesh Adaptation Algorithm for Time-Dependent Shock Dominated Problems. Int. J. for Numer. Meth. in Fluids 25 (1997) 81–104.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Jimack, P.K., Nadeem, S.A. (2001). Parallel Application of a Novel Domain Decomposition Preconditioner for the Stable Finite Element Solution of Three-Dimensional Convection-Dominated PDEs. In: Sakellariou, R., Gurd, J., Freeman, L., Keane, J. (eds) Euro-Par 2001 Parallel Processing. Euro-Par 2001. Lecture Notes in Computer Science, vol 2150. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44681-8_85
Download citation
DOI: https://doi.org/10.1007/3-540-44681-8_85
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42495-6
Online ISBN: 978-3-540-44681-1
eBook Packages: Springer Book Archive