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 . The generality of our techniques is showed by investigations of high-order problems, steady and nonsteady semilinear problems.
Li, J. (2000). Convergence analysis of finite element methods for singularly perturbed problems. Computers and Mathematics with Applications, 40(6), 735–745. https://doi.org/10.1016/S0898-1221(00)00192-9