Higher-Order Layer-Adapted Finite Difference Schemes For Singularly Perturbed Robin Problems

In recent years,the numerical solution of singularly perturbed differential equation by using layer-adapted method attracts many scholars.Generally,two aspects are considered to improve the accuracy of numerical solution:the construction of grid functions and choosing the appropriate difference scheme to discretize singularly perturbed boundary value problems.In order to find a suitable layer-adapted mesh and a difference scheme matching the mesh,we need to consider the exact solution and decomposition properties of singularly perturbed Robin boundary value problem.At the same time,we need to deal with Robin boundary conditions to improve the overall error order.This paper starts with the above aspects and does the following work:(1)The exact solution of singularly perturbed Robin boundary value problem and its decomposition properties are obtained by using the comparison principle and constructing obstacle function.(2)In this paper,the singularly perturbed Robin boundary value problem is solved to achieve a higher convergence order from the above two aspects.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 ?-uniform pointwise error estimates O(N-1 ln N)for 1?i?p1N and O(N-2)for p1N<i<N with p1=1/(4e)+1/4 on the Shishkin mesh and O(N-1)for 1?i?N/2 and O(N-2)for N/2<i<N on the Bakhvalov-Shishkin mesh are proved.(3)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 ln N)for 1?i?p2N and O(N-2 for p2N<i<N with p2=1/(2e).Finally,a numerical experiment illustrates that these error estimates are sharp and the convergence is uniform with respect to the perturbation parameter.