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

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 View the MathML source, where the diagonal tensor a=(ε2,1) or a=(ε2,ε2). Global uniform convergence rates of O(N−2) for u in the L2-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 L2-norm for a1/2∇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.