Boundary layer preconditioners for finite-element discretizations of singularly perturbed reaction-diffusion problems

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...

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Veröffentlicht in:Numerical algorithms 2018-09, Vol.79 (1), p.281-310
Hauptverfasser: Nhan, Thái Anh, MacLachlan, Scott, Madden, Niall
Format: Artikel
Sprache:eng
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Zusammenfassung: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.
ISSN:1017-1398
1572-9265
DOI:10.1007/s11075-017-0437-3