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A numerical investigation on the interplay amongst geometry, meshes, and linear algebra in the finite element solution of elliptic PDEs

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Abstract

In this paper, we study the effect of the choice of mesh quality metric, preconditioner, and sparse linear solver on the numerical solution of elliptic partial differential equations (PDEs). We smooth meshes on several geometric domains using various quality metrics and solve the associated elliptic PDEs using the finite element method. The resulting linear systems are solved using various combinations of preconditioners and sparse linear solvers. We use the inverse mean ratio and radius ratio metrics in addition to conditioning-based scale-invariant and interpolation-based size-and-shape metrics. We employ the Jacobi, SSOR, incomplete LU, and algebraic multigrid preconditioners and the conjugate gradient, minimum residual, generalized minimum residual, and bi-conjugate gradient stabilized solvers. We focus on determining the most efficient quality metric, preconditioner, and linear solver combination for the numerical solution of various elliptic PDEs with isotropic coefficients. We also investigate the effect of vertex perturbation and the effect of increasing the problem size on the number of iterations required to converge and on the solver time. In this paper, we consider Poisson’s equation, general second-order elliptic PDEs, and linear elasticity problems.

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Acknowledgments

The authors would like to thank Anirban Chatterjee and Padma Raghavan for interesting the third author in this area of research and Nicholas Voshell for helpful discussions. This work was funded in part by NSF Grant CNS 0720749 and an Institute for Cyberscience grant from The Pennsylvania State University. This work was supported in part through instrumentation funded by the National Science Foundation through Grant OCI-0821527. They also wish to thank the two anonymous referees for their careful reading of the paper and for their helpful suggestions which strengthened it.

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  1. The Pennsylvania State University, University Park, PA, 16802, USA

    Jibum Kim, Shankar Prasad Sastry & Suzanne M. Shontz

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  1. Jibum Kim
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  2. Shankar Prasad Sastry
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  3. Suzanne M. Shontz
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Correspondence to Suzanne M. Shontz.

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Kim, J., Sastry, S.P. & Shontz, S.M. A numerical investigation on the interplay amongst geometry, meshes, and linear algebra in the finite element solution of elliptic PDEs. Engineering with Computers 28, 431–450 (2012). https://doi.org/10.1007/s00366-011-0231-0

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  • DOI: https://doi.org/10.1007/s00366-011-0231-0

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