A New Insight into the Differential Quadrature Method in Solving 2-D Elliptic PDEs

Abstract

When the local differential quadrature (LDQ) has been successfully applied to solve two-dimensional problems, the global method of DQ still has a problem by requiring to solve the inversions of ill-posed matrices. Previously, when one uses (n 1)th order polynomial test functions to determine the weight- ing coefficients with n grid points, the resultant n n Vandermonde matrix is highly ill-conditioned and its inversion is hard to solve. Now we use (m 1)th order poly- nomial test functions by n grid points that the size of Vandermonde matrix is m n, of which m is much less than n. We find that the (m 1)th order polynomial test functions are accurate enough to express the solutions, and the novel method sig- nificantly improves the ill-condition of algebraic equations. Such a new DQ as being combined with FTIM (Fictitious Time Integration Method) can solve 2-D elliptic type PDEs successfully. There are some examples tested in this paper and the numerical errors are found to be very small.