Uniform convergence of finite element methods on Bakhvalov-type meshes in the case of N−1 ≤ ε

Abstract

It has been reported that convergence stalls on Bakhvalov-Shishkin mesh in the case of N−1≤ε, where ε is the singular perturbation parameter and N is the number of mesh intervals. To analyze these phenomena, we present uniform convergence analysis of finite element methods on Bakhvalov-type meshes, one popular kind of graded meshes closely related to Bakhvalov-Shishkin mesh, when N−1≤ε. An optimal order of convergence is proved and this result is used for the improvement of Bakhvalov-Shishkin mesh. These theoretical results are verified by numerical experiments.