Abstract
This paper describes a parallel iterative solver for finite element discretisations of elliptic partial differential equations on 2D and 3D domains using unstructured grids. The discretisation of the PDE is assumed to be given in the form of element stiffness matrices and the solver is automatic in the sense that it requires minimal additional information about the PDE and the geometry of the domain. The solver parallelises matrix–vector operations required by iterative methods and provides parallel additive Schwarz preconditioners. Parallelisation is implemented through MPI. The paper contains numerical experiments showing almost optimal speedup on unstructured mesh problems on a range of four platforms and in addition gives illustrations of the use of the package to investigate several questions of current interest in the analysis of Schwarz methods. The package is available in public domain from the home page http://www.maths.bath.ac.uk/∼mjh/.
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References
R. E. Alcouffe, A. Brandt, J. E. Dendy, Jr and J. W. Painter, The multi-grid method for the diffusion equation with strongly discontinuous coefficients, SIAM J. Sci. Statist. Comput. 2 (1981) 430–454.
S. F. Ashby, R. D. Falgout, S. G. Smith and A. F. B. Tompson, Modeling groundwater flow on MPPS, in: Proceedings of the Scalable Parallel Libraries Conference (1993).
R. E. Bank and J. Xu, An algorithm for coarsening unstructured matrices, Numer. Math. 73 (1996) 1–36.
G. F. Carey and H. T. Dinh, Grading functions and mesh redistribution, SIAM J. Numer. Anal. 22 (1985) 1028–1040.
T. F. Chan, S. Go and J. Zou, Multilevel domain decomposition and multigrid methods for unstructured meshes: Algorithms and theory, in: Proceedings of the 8th International Conference on Domain Decomposition, eds. R. Glowinski, J. Périaux, Z.-C. Shi and O. B. Widlund (Wiley, Chichester, 1996).
T. F. Chan and T. P. Mathew, Domain decomposition algorithms, in: Acta Numerica, ed. A. Iserles (Cambridge University Press, Cambridge, 1994) pp. 61–143.
T. F. Chan, T. P. Mathew and J. P. Shao, Efficient variants of the vertex space domain decomposition algorithm, SIAM J. Sci. Comput. 15 (1994) 1349–1374.
T. F. Chan and J. P. Shao, Parallel complexity of domain decomposition methods and optimal coarse grid size, Parallel Comput. 21 (1995) 1033–1049.
T. F. Chan and B. Smith, Domain decomposition and multigrid algorithms for elliptic problems on unstructured meshes, Contemporary Mathematics 180 (1994) 175–189.
T. F. Chan, B. Smith and J. Zou, Overlapping Schwarz methods on unstructured meshes using non-matching coarse grids, Numer. Math. 73 (1996) 149–167.
T. F. Chan and J. Zou, Additive Schwarz domain decomposition methods for elliptic problems on unstructured meshes, Numer. Algorithms 8 (1994) 329–346.
R. K. Coomer and I. G. Graham, Domain decomposition methods for device modelling, in: Domain Decomposition Methods in Science and Engineering, eds. D. Keyes and J. Xu (Amer. Math. Soc., Providence, RI, 1995).
R. K. Coomer and I. G. Graham, Massively parallel methods for semiconductor device modelling, Comput. 56 (1996) 1–28.
I. S. Duff, A. M. Erisman and J. K. Reid, Direct Methods for Sparse Matrices (Oxford University Press, Oxford, 1986).
C. R. I. Emson, J. Simkin and C. W. Trowbridge, A status report on electromagnetic field computation, IEEE Trans. Magnetics 30 (1994) 1533–1540.
G. H. Golub and C. F. van Loan, Matrix Computations, 2nd ed. (Johns Hopkins University Press, Baltimore, MD, 1989).
I. G. Graham and M. J. Hagger, Additive Schwarz, CG and discontinuous coefficients, in: Proceedings of the 9th International Conference on Domain Decomposition Methods (1997).
I. G. Graham and M. J. Hagger, Unstructured additive Schwarz–CG method for elliptic problems with highly discontinuous coefficients, SIAM J. Sci. Comput. (1997, to appear).
H. Guillard, Node-nested multi-grid method with Delaunay coarsening, Technical Report, INRIA, Sophia Antipolis, France (1993).
M. J. Hagger, Iterative solution of large, sparse systems of equations, arising in groundwater flow models, Ph. D. thesis, University of Bath (1995).
B. Heise, Parallel solvers for FEM–BEM equations with applications to non-linear magnetic field problems, in: Numerical Treatment of Coupled Systems, Notes on Numerical Fluid Dynamics 51, eds. W. Hackbusch and G. Wittum (Vieweg, Braunschweig, 1995).
B. Heise and M. Kuhn, Parallel solves for linear and non-linear exterior magnetic field problems based on upon coupled FE/BE formulations, Technical Report, Institutsbericht Nr. 486, Johannes-Kepler-Universität, Linz (1995).
M. C. Hill, Solving groundwater flow problems by conjugate-gradient methods and the strongly implicit procedure, Water Resources Res. 26 (1990) 1961–1969.
G. Karypis and V. Kumar, METIS: Unstructured graph partitioning and sparse matrix ordering system, Department Computer Science, University of Minnesota, Minneapolis (August 1995).
Y. Saad and M. H. Shultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 7 (1986) 856–869.
B. F. Smith, An optimal domain decomposition preconditioner for the finite element solution of linear elasticity problems, SIAM J. Sci. Statist. Comput. 13 (1992) 364–378.
B. F. Smith, P. E. Bjorstad and W. D. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations (Cambridge University Press, Cambridge, 1996).
H. A. Van der Vorst, BI-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 13 (1992) 631–644.
D. F. Watson, Computing the n-dimensional Delaunay tessellation with application to Voronoi polytopes, Comput. J. 24 (1981) 167–172.
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Hagger, M. Automatic domain decomposition on unstructured grids (DOUG). Advances in Computational Mathematics 9, 281–310 (1998). https://doi.org/10.1023/A:1018997725374
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DOI: https://doi.org/10.1023/A:1018997725374