P000010
Sparse spectral-Galerkin method on an arbitrary tetrahedron using generalized Koornwinder polynomials
*Lueling Jia (Shandong Normal University, Beijing Computational Science Research Center)
Huiyuan Li (State Key Laboratory of Computer Science/Laboratory of Parallel Computing, Institute of Software, Chinese Academy of Sciences)
Zhimin Zhang (Department of Mathematics, Wayne State University)
In this talk, we propose a sparse spectral-Galerkin method for solving the second-order partial differential equations on an arbitrary tetrahedron. Generalized Koornwinder polynomials are introduced on the reference tetrahedron as basis functions with their various recurrence relations and differentiation properties being explored. The method leads to well-conditioned and sparse linear systems whose entries can either be calculated directly by the orthogonality for differential equations with constant coefficients or be evaluated efficiently via our recurrence algorithm for problems with variable coefficients. Clenshaw algorithms for the evaluation of any polynomial in an expansion of the generalized Koornwinder basis are also designed to boost the efficiency of the method. Finally, numerical experiments are carried out to illustrate the effectiveness of the proposed Koornwinder spectral method.