Superconvergence analysis of FEM and SDFEM on graded meshes for a problem with characteristic layers

TL;DR

This paper analyzes the superconvergence properties of FEM and SDFEM methods on graded meshes for singularly perturbed convection-diffusion problems with boundary layers, demonstrating improved convergence results.

Contribution

It provides new superconvergence analysis for FEM and SDFEM on graded meshes, showing advantages over Shishkin meshes for boundary layer problems.

Findings

FEM achieves almost uniform convergence and superconvergence on graded meshes.

SDFEM attains almost uniform estimates in the SD norm on graded meshes.

Graded meshes improve numerical analysis of boundary layer problems.

Abstract

We consider a singularly perturbed convection-diffusion with exponential and characteristic boundary layers. The problem is numerically solved by the FEM and SDFEM method with bilinear elements on a graded mesh. For the FEM we prove almost uniform convergence and superconvergence. The use of graded mesh allows for the SDFEM to prove almost uniform esimates in the SD norm, which is not possible for Shishkin type meshes.