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

Chadha, Naresh M.; Kopteva, Natalia

IMA J. Numer. Anal. 31, No. 1, 188-211 (2011).

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.

Reviewer: Abdallah Bradji (Annaba)

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

Keywords:

second order semilinear equation; equidistribution method; second order accurate; grid equidistribution; reaction-diffusion; singular perturbation; adaptive mesh; finite differences: maximum norm; a posteriori estimate; algorithm