Effective Condition Number of Finite Difference Method for Poisson's Equation Involving Boundary Singularities

Abstract

For solving the linear algebraic equations Ax = b with the symmetric and positive definite matrix A, the effective condition number Cond_eff is defined in [6, 10] by following Chan and Foulser [2] and Rice [14]. The Cond_eff is smaller, or much smaller, than the traditional condition number Cond. Besides, the simplest condition number Cond_EE is also defined in [6, 10]. This article studies a popular model of Poisson's equation involving the boundary singularities by the finite difference method using the local refinements of grids. The bounds of Cond_EE are derived to display theoretically that the effective condition number is significantly smaller than the Cond. In this article, by exploring local refinement properties, we derive the bounds of effective condition numbers up to O(1) and at least o(h −1/2) for the maximal step size h. They are significant improvements compared with the bound O(h −3/2), which is established in [6, 10]. Therefore, the study of effective condition number in this article reaches a new comprehensive and advanced level.