A robust grid equidistribution method for a one-dimensional singularly perturbed semilinear reaction-diffusion problem.(English)Zbl 1211.65099

The authors consider a one dimensional second order semilinear singularly perturbed equation. The diffusion parameter is denoted by $$\varepsilon^2$$. On an arbitrary mesh and thanks to a posteriori error estimate, it is remarked that it is possible to obtain second order accuracy uniformly after a suitable choice of the mesh. To get this convenient mesh, the authors use a monitor function equidistribution. The aim of this paper is then to resolve two questions. The fisrt question is to discuss the existence of a solution to the equidistribution problem, and the second question is to suggest an algorithm which yields second order accuracy uniformly w.r.t. the singular parameter for the discrete solution.
First, the existence of a solution to the equidistribution problem is established. This is done in a framework which can be applied to a more general equidistribution problem. An algorithm is suggested which yields second order accuracy uniformly, when the equation is linear and under further mild assumptions, after $$O(|\,\ln\varepsilon|/\ln N)$$ iterations, where $$N+1$$ is the number of mesh points.

MSC:

 65L10 Numerical solution of boundary value problems involving ordinary differential equations 65L70 Error bounds for numerical methods 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 34B15 Nonlinear boundary value problems for ordinary differential equations 65L11 Numerical solution of singularly perturbed problems involving ordinary differential equations 34E15 Singular perturbations for ordinary differential equations
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