Analysis of finite element method in balanced norms for two-parameter singularly perturbed problems

Abstract

We consider the two-parameter singularly perturbed problems, and there are two exponential boundary layers in its solution. In general, the energy norm is used in error estimations of finite element method for singularly perturbed problems. However, it is difficult to capture each layer simultaneously in this norm. In this paper, a special balanced norm is defined, which can capture each exponential layer well. Based on the balanced norm, we obtain the uniform convergence of any order finite element method with respect to both parameters on a Shishkin mesh. Moreover, the optimal order is proved and some discussions in this balanced norm are presented. Finally, we give some numerical examples to verify our theoretical findings.