A parameter-uniform Schwarz method for a singularly perturbed reaction-diffusion problem with an interior layer. (English) Zbl 0967.65086
Authors’ abstract: We consider numerical methods for a singularly perturbed reaction-diffusion problem with a discontinuous source term. We show that such a problem arises naturally in the context of models of simple semiconductor devices. We construct a numerical method consisting of a standard finite difference operator and a non-standard piecewise-uniform mesh. The mesh is fitted to the boundary and interior layers that occur in the solution of the problem. We show by extensive computations that, for this problem, this method is parameter-uniform in the maximum norm, in the sense that the numerical solutions converge in the maximum norm uniformly with respect to the singular perturbation parameter.
MSC:
| 65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 34E15 | Singular perturbations for ordinary differential equations |
| 65L50 | Mesh generation, refinement, and adaptive methods for ordinary differential equations |
| 65L12 | Finite difference and finite volume methods for ordinary differential equations |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
| 82D37 | Statistical mechanics of semiconductors |
Keywords:
convergence; grid generation; Schwarz method; reaction-diffusion equation; semiconductor device equation; singular perturbation; interior layer; finite differenceReferences:
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