Higher-order layer-adapted finite difference schemes for singularly perturbed Robin problems

Abstract

In this paper, we construct the midpoint upwind scheme on the Shishkin mesh and on the Bakhvalov-Shishkin mesh respectively for solving the singularly perturbed Robin boundary value problem. The central divided difference is used to discretize the first derivative in the Robin boundary condition to achieve the higher-order uniform convergence. The elaborate ε-uniform pointwise error estimates O(N-1 ln N) for 1≤i≤pmN and O(N-2) for pmN <i<N with pm = 1/(4e)+1/4 on the Shishkin mesh and O(N-1) for 1≤i≤N/2 and O(N-2) N/2 <i<N on the Bakhvalov-Shishkin mesh are proved. We also construct the hybrid finite difference scheme that combines the midpoint upwind scheme on the coarse part with the central difference scheme on the fine part on the Shishkin mesh, and prove a better uniform convergence of orders O(N-2 ln2 N) for 1≤i≤phN and O(N-2) for phN <i<N with ph = 1/(2e). Finally, a numerical experiment illustrates that these error estimates are sharp and the convergence is uniform with respect to the perturbation parameter.