Uniform Convergence Analysis for Singularly Perturbed Elliptic Problems with Parabolic Layers

Abstract

In this paper, using Lin's integral identity technique, we prove the optimal uniform convergence $O(N_x^{-2}\ln^2N_x+N_y^{-2}\ln^2N_y)$ in the $L^2$-norm for singularly perturbed problems with parabolic layers. The error estimate is achieved by bilinear finite elements on a Shishkin type mesh. Here $N_x$ and $N_y$ are the number of elements in the $x$- and $y$-directions, respectively. Numerical results are provided supporting our theoretical analysis.