A higher-order uniformly convergent defect correction method for singularly perturbed convection-diffusion problems on an adaptive mesh

Abstract

This paper presents a defect correction scheme based on finite-difference discretization for a singularly perturbed convection-dominated diffusion problem. The solution of this class of problems exhibits a multiscale character. There are narrow regions in which the solution grows exponentially and reveals layer behaviour. The defect correction method that we propose improves the efficiency of a numerical solution through iterative improvement and generates a stable higher-order method over an adaptive non-uniform polynomial-Shishkin mesh. An extensive theoretical analysis is presented, which establishes that the method is second-order uniformly convergent and highly stable. The convergence obtained is optimal because it is free from any logarithmic term. The numerical result for two model problems is presented, agreeing with the theoretical estimates. Furthermore, we compare the results with those of other non-uniform mesh found in the literature.

Keywords

• Singular perturbation
• Upwind method
• Finite difference
• Non-uniform mesh
• Defect-Correction method