Fitted finite element methods for singularly perturbed elliptic problems of convection-diffusion type

TL;DR

This paper develops fitted finite element methods using exponential splines and Shishkin meshes for 2D singularly perturbed convection-diffusion problems, achieving stable, parameter-uniform, and high-order convergence.

Contribution

It introduces a novel combination of exponential spline basis functions with Shishkin meshes to ensure stability and uniform convergence in singularly perturbed elliptic problems.

Findings

Methods satisfy a discrete maximum principle.

Second-order convergence in classical case.

First-order convergence uniformly for all perturbation parameters.

Abstract

Fitted finite element methods are constructed for a singularly perturbed convection-diffusion problem in two space dimensions. Exponential splines as basis functions are combined with Shishkin meshes to obtain a stable parameter-uniform numerical method. These schemes satisfy a discrete maximum principle. In the classical case, the numerical approximations converge, in the maximum pointwise norm, at a rate of second order and the approximations converge at a rate of first order for all values of the singular perturbation parameter.