Sparse Spectral-Galerkin Method on An Arbitrary Tetrahedron Using Generalized Koornwinder Polynomials

TL;DR

This paper introduces a sparse spectral-Galerkin method using generalized Koornwinder polynomials for solving PDEs on arbitrary tetrahedra, emphasizing efficiency, conditioning, and applicability to variable coefficients.

Contribution

The paper develops a novel spectral-Galerkin scheme on tetrahedra using generalized Koornwinder polynomials, including recurrence relations, differentiation properties, and efficient evaluation algorithms.

Findings

Produces well-conditioned sparse linear systems

Efficient evaluation via recurrence algorithms

Numerical experiments confirm effectiveness

Abstract

In this paper, we propose a sparse spectral-Galerkin approximation scheme 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 of the generalized Koornwinder polynomials 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.