Comparison of eigenvalue ratios in artificial boundary perturbation and Jacobi preconditioning for solving Poisson equation

The Shortley-Weller method is a standard nite di erence method for solving the Poisson equation with Dirichlet boundary condition. Unless the domain is rectangular, the method meets an inevitable problem that some of the neighboring nodes may be outside the domain. The function values at outside nodes are extrapolated by quadratic polynomial, and the extrapolation becomes unstable, that is, some of the extrapolation coe cient increases rapidly when the grid nodes are very near the boundary. A practical remedy, which we call arti cial perturbation, is to treat grid nodes very near the boundary as boundary points. The aim of this paper is to reveal the adverse e ects of the arti cial perturbation on the condition number of the matrix and the convergence of the solution. We show that the matrix is nearly symmetric so that the ratio of its minimum and maximum eigenvalues can be referenced as the measure of its condition number. Our analysis shows that the arti cial perturbation results in a small enhancement of the condition number from O(1/(h ·hmin) to O(h−3) and triggers an oscillatory order of convergence. Instead, we suggest using Jacobi or ILU-type preconditioner on the matrix without applying the arti cial perturbation. According to our analysis, the preconditioning not only reduces the condition number from O(1/(h · hmin) to O(h−2), but also keeps the sharp second order convergence.