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

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 reference tetrahedron as basis functions with their various recurrence relations and differentiation properties being explored. The method leads to well-conditioned linear systems whose entries can either be calculated directly by orthogonality of generalized constant coefficients or evaluated efficiently via our algorithm problems variable coefficients. Clenshaw algorithms evaluation any polynomial in expansion also designed boost efficiency method. Finally, numerical experiments carried out illustrate effectiveness proposed spectral