Nonuniform FDM for Poisson-Type Equations on a Disk with Singular Solutions

摘要

We are concerned with the Dirichlet boundary value problem of Poisson-type equations on a disk. Matsunaga and Yamamoto [88. N. Matsunaga and T. Yamamoto ( 1999 ). Convergence of Swartztrauber-Sweet's approximation for the Poisson-type equation on a disk . Numer. Funct. Anal. Optimz. 20 : 917 - 928 . [Taylor & Francis Online], [Web of Science ®]View all references] proved that if the exact solution u is very smooth over the closure of the disk, then the approximate solution by the Swartztrauber-Sweet scheme with uniform partition is second order accurate. In this article, it is assumed that the exact solution performs singular properties such that its derivatives go to infinity at the boundary of the disk. We use a stretching polynomial-like function with a parameter to construct a local grid refinement and consider the Swartztrauber-Sweet scheme over the non-uniform partition. The effects of the parameter are analyzed completely by carrying out convergence analysis and numerical results show that there exists an optimal value for the parameter to have a best approximated solution.