Scaled discrete derivatives of singularly perturbed elliptic problems

Abstract

Numerical approximations to the solution of a singularly perturbed elliptic convection-diffusion problem in two space dimensions are generated using a monotone finite difference operator on a tensor product of piecewise-uniform Shishkin meshes. The bilinear interpolants of these numerical approximations are parameter-uniformly convergent to the solution of the continuous problem, in the pointwise maximum norm. In this article, discrete approximations to the first derivatives of the solution are shown to be globally first-order (up to logarithmic factors) uniformly convergent, when the errors are scaled within the analytical layers of the continuous problem. Numerical results are presented to illustrate the theoretical error bounds established in an appropriated weighted C 1 -norm.

Numerical approximations to the scaled first derivatives of the solution to a two parameter singularly perturbed problem

“…In this paper, we will establish parameter-uniform bounds on approximations to the scaled first derivative of the solution of a two parameter singularly perturbed boundary value problem, where we simply scale (by appropriate factors) within the analytical layer regions only. Our method of proof is based on the analysis in [5,6,7], which dealt with singularly perturbed parabolic and elliptic problems containing a single perturbation parameter.…”

“…Our method of proof is based on the analysis in [5,6,7], which dealt with singularly perturbed parabolic and elliptic problems containing a single perturbation parameter.…”