Convergence analysis of finite element methods for singularly perturbed problems

In this paper, a unified convergence analysis is presented for solving singularly perturbed problems by using the standard Galerkin finite element method on a nontraditional Shishkin-type mesh, which separates the boundary layers totally from other subregions. The results obtained show that the error estimates on such nontraditional Shishkin-type mesh are much easier to prove than on the traditional Shishkin-type mesh. However, both meshes give comparable error estimates, which justifies the conjecture of Roos [1]. The generality of our techniques is showed by investigations of high-order problems, steady and nonsteady semilinear problems.