Exponentially Refined Mesh for Singularly Perturbed Interior Layer Problem

Abstract

The main aim of this article is to demonstrate the efficiency of the exponentially refined a priori mesh $$(H(\ell )$$) on capturing the interior layer. Upwind scheme on the proposed $$H(\ell )$$ mesh is considered for a class of singularly perturbed differential equation with discontinuous convection coefficient. We have estimated that the algorithm is perturbation parameter ($$\epsilon $$) uniformly convergent with error asymptotic to $$O(N^{-1}(g^{\ell }(N))^2)$$ where $$g^\ell $$ is an arithmetic function and $$\ell \in {\mathbb {N}}$$. The numerical estimates based on $$H(\ell )$$ mesh and other layer adapted meshes like B-mesh, Shishkin mesh and $$S(\ell )$$ mesh are compared demonstrating the efficiency of the $$H(\ell )$$ mesh on capturing the interior layer.