Name
Title | ||
|---|---|---|
Boundary layer preconditioners for finite-element discretizations of singularly perturbed reaction-diffusion problems. |
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. |
Year | DOI | Venue |
|---|---|---|
2018 | 10.1007/s11075-017-0437-3 | Numerical Algorithms |
Keywords | Field | DocType |
Singularly perturbed, Layer-adapted meshes, Robust multigrid, Preconditioning | Discretization,Differential equation,Polygon mesh,Linear system,Preconditioner,Mathematical analysis,Finite element method,Boundary layer,Reaction–diffusion system,Mathematics | Journal |
Volume | Issue | ISSN |
79 | 1 | 1017-1398 |
Citations | PageRank | References |
0 | 0.34 | 3 |
Authors | ||
3 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Thái Anh Nhan | 1 | 0 | 1.01 |
| Scott MacLachlan | 2 | 78 | 8.09 |
| Niall Madden | 3 | 29 | 7.41 |