Convergence and superconvergence analysis of finite element methods on graded meshes for singularly and semisingularly perturbed reaction–diffusion problems

Abstract

The bilinear finite element methods on appropriately graded meshes are considered both for solving singular and semisingular perturbation problems. In each case, the quasi-optimal order error estimates are proved in the -weighted H 1 -norm uniformly in singular perturbation parameter , up to a logarithmic factor. By using the interpolation postprocessing technique, the global superconvergent error estimates in -weighted H 1 -norm are obtained. Numerical experiments are given to demonstrate validity of our theoretical analysis.

Convergence and superconvergence analysis of an anisotropic nonconforming finite element methods for singularly perturbed reaction–diffusion problems

“…Whether meshes have superconvergence results or not is now a question. In addition, we have obtained similar results for conforming bilinear finite elements in [28].…”

“…Here we give a short discussions between the present paper with papers [27,28]. In [27], the authors gave a graded meshes, analogous to the ones used in this paper, but constructed independently of the perturbation parameter ε, and got the same quasi-optimal error estimate in the energy norm for bilinear finite elements.…”

Superconvergence analysis of a finite element method for a two-parameter singularly perturbed problem

“…It is known that finite element solutions of partial differential equations can have superconvergence property in some subregions of the domain (see [9,31,33,34], for example and references therein). A typical task is to find conditions on the finite element mesh that, together with smoothness assumptions on the partial differential equation, guarantee the existence of superconvergence.…”