Abstract
A semilinear reaction-diffusion equation with multiple solutions is considered in a smooth two-dimensional domain. Its diffusion parameter ε2 is arbitrarily small, which induces boundary layers. Constructing discrete sub- and super-solutions, we prove existence and investigate the accuracy of multiple discrete solutions on layer-adapted meshes of Bakhvalov and Shishkin types. It is shown that one gets second-order convergence (with, in the case of the Shishkin mesh, a logarithmic factor) in the discrete maximum norm, uniformly in ε for ε ≤ Ch. Here h > 0 is the maximum side length of mesh elements, while the number of mesh nodes does not exceed Ch-2. Numerical experiments are performed to support the theoretical results.
Original language | English |
---|---|
Pages (from-to) | 631-646 |
Number of pages | 16 |
Journal | Mathematics of Computation |
Volume | 76 |
Issue number | 258 |
DOIs | |
Publication status | Published - Apr 2007 |
Keywords
- Bakhvalov mesh
- Maximum norm error estimate
- Second order
- Semilinear reaction-diffusion
- Shishkin mesh
- Singular perturbation
- Z-field