An adaptive local grid refinement method for 2D diffusion equation with variable coefficients based on block-centered finite differences

In this paper, an adaptive local grid refinement method based on block-centered finite differences is proposed for 2D diffusion equation with Neumann boundary condition. The method first identifies the regions with large local error, then these regions are divided until a prescribed tolerance is satisfied. The numerical solutions of unknown variable along with its first derivatives are obtained simultaneously. Theoretical analysis shows that the proposed method is second order accurate both on uniform and non-uniform meshes. Some high gradient problems are carried out to verify the efficiency and reliability of the adaptive local grid refinement method.