Abstract
The solution of linear systems of equations that arise when singularly perturbed partial differential equations are discretized can be difficult: direct solvers scale poorly, but are also known not to be robust with respect to the perturbation parameter, while the design of parameter robust preconditioners is not trivial, primarily due to the specialised layer adapted meshes used for such problems; see MacLachlan and Madden (SIAM J Sci Comput 35:A2225–A2254, 2013). Here we present a multigrid solver strategy that circumvents this problem by using a robust patched mesh method proposed by de Falco and O’Riordan (BAIL 2008—Boundary and Interior Layers vol. 69, pp. 117–127. Springer, Berlin, 2009), as well as permitting parallelization. Numerical results demonstrate the efficiency of the method.
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Acknowledgements
The research of J.L. Gracia was partly supported by the Institute of Mathematics and Applications, the project MTM2013-40842-P and the Diputación General de Aragón. The research of T.A. Nhan is supported by the Irish Research Council under Grant No. RS/2011/179. The authors are grateful to the anonymous referee for their insightful comments.
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Gracia, J.L., Madden, N., Nhan, T.A. (2017). Applying a Patched Mesh Method to Efficiently Solve a Singularly Perturbed Reaction-Diffusion Problem. In: Bock, H., Phu, H., Rannacher, R., Schlöder, J. (eds) Modeling, Simulation and Optimization of Complex Processes HPSC 2015 . Springer, Cham. https://doi.org/10.1007/978-3-319-67168-0_4
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DOI: https://doi.org/10.1007/978-3-319-67168-0_4
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