Effective condition number and superconvergence of the Trefftz method coupled with high order FEM for singularity problems

Abstract

The new results of this paper are twofold. (a) For solving the linear algebraic equations Ax= b resulting from elliptic equations, we propose the new simplified effective condition number defined by Cond _ E = | | b | | / ( | | b | | 2 − β n 2 / Cond . 2 ) + β n 2 , where β n = u n T b , u n is the eigenvector of the minimal eigenvalue of matrix A, and Cond. is the traditional condition number. The Cond_E may provide a better upper bound of relative errors of x resulting from the perturbation of A and b. (b) To solve the Poisson equation with singularities, the penalty plus hybrid Trefftz method coupled with high order finite element methods (FEM) is used. New superconvergence being the best or almost the best is proven, and verified by numerical experiments for Motz's problem. Compared with the hybrid techniques in Li and Huang [Li ZC, Huang HT. Global superconvergence of simplified hybrid combinations of the Ritz-Galerkin and FEMs of elliptic equations with singularities. II. Lagrange elements and Adini's elements. Appl Numer Math 43;2002:253–73.], the accuracy of the solution is higher, and the numerical stability is fairly worse, based on the numerical values of the computed Cond_E. Hence, the new comparisons are made for solving Motz's problem, to indicate that the penalty plus hybrid Trefftz combination is superior to the hybrid Trefftz combination in Li and Huang [Li ZC, Huang HT. Global superconvergence of simplified hybrid combinations of the Ritz-Galerkin and FEMs of elliptic equations with singularities. II. Lagrange elements and Adini's elements. Appl Numer Math 43;2002:253–73.]. This paper integrates both the error and the stability analyses, to show a significance of the new simplified effective condition number.