Analysis of finite element method in balanced norms for singularly perturbed problems with two parameters

We consider the two-parameter singularly perturbed problems, and there are two exponential boundary layers in its solution. The purpose of this paper is to obtain the uniform convergence of the finite element method under the balanced norm for this problems. For this reason, a special balanced norm is defined so that this balanced norm can capture each exponential layer well. Based on this, we obtain the uniform convergence of $k$th($k\geq 1$) order finite element method with respect to both parameters on a Shishkin mesh. Moreover, the optimal order is proved and some discussions on this balanced norm are presented. Finally, we give some numerical examples to verify our theoretical results.