Convergence and superconvergence analysis of finite element methods on highly nonuniform anisotropic meshes for singularly perturbed reaction–diffusion problems

Abstract

The standard finite element methods on one type of highly nonuniform rectangular meshes, which is different from the extensively discussed Shishkin type meshes, are considered for solving the singular perturbation problem − div(a∇u)+bu=f , where the diagonal tensor a=(ε 2,1) or a=(ε 2,ε 2) . Global uniform convergence rates of O (N −2) for u in the L 2 -norm are obtained in both cases for bilinear rectangular finite elements, where N is the number of intervals in both the x- and y-directions. The pointwise interior (away from the boundary layers) convergence rates of O (N −1) for u are also proved. Global superconvergence rates of O (N −2) in the L 2 -norm for a 1/2∇u are obtained by a postprocessing. Numerical experiments supporting the theoretical analysis show that the new type anisotropic meshes give equivalent performance as the Shishkin type meshes.