A Higher-Order Defect Correction Method Over an Adaptive Bakhvalov–Shishkin Mesh for Advection–Diffusion Equations

Abstract

The paper presents a defect correction method based on finite difference discretizations over a Bakhvalov–Shishkin mesh for a class of convection-diffusion equations. The technique combines a stable, low-order accurate, computationally inexpensive upwind difference scheme with a higher-order less stable modified central difference operator at several grid points. We prove that the method is second-order uniformly convergent at all mesh points. We do not use asymptotic expansions of discretization errors in our analysis. The method has no directional bias, and it can be used in an adaptive procedure to refine the mesh in the non-smooth parts. Numerical results are presented, which indicate that defect correction on layer-adapted meshes is an excellent method to achieve higher-order accuracy for singular perturbation problems.