Boundary layer preconditioners for finite-element discretizations of singularly perturbed reaction-diffusion problems |
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Authors: | Thái Anh Nhan Scott MacLachlan Niall Madden |
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Institution: | 1.Department of Mathematics,Ohlone College,Fremont,USA;2.Department of Mathematics and Statistics,Memorial University of Newfoundland,St John’s,Canada;3.School of Mathematics, Statistics and Applied Mathematics,National University of Ireland Galway,Galway,Ireland |
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Abstract: | We consider the iterative solution of linear systems of equations arising from the discretization of singularly perturbed reaction-diffusion differential equations by finite-element methods on boundary-fitted meshes. The equations feature a perturbation parameter, which may be arbitrarily small, and correspondingly, their solutions feature layers: regions where the solution changes rapidly. Therefore, numerical solutions are computed on specially designed, highly anisotropic layer-adapted meshes. Usually, the resulting linear systems are ill-conditioned, and so, careful design of suitable preconditioners is necessary in order to solve them in a way that is robust, with respect to the perturbation parameter, and efficient. We propose a boundary layer preconditioner, in the style of that introduced by MacLachlan and Madden for a finite-difference method (MacLachlan and Madden, SIAM J. Sci. Comput. 35(5), A2225–A2254 2013). We prove the optimality of this preconditioner and establish a suitable stopping criterion for one-dimensional problems. Numerical results are presented which demonstrate that the ideas extend to problems in two dimensions. |
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