Jacobi spectral Galerkin method for the integrated forms of second-order differential equations

Efficiently direct solvers based on the Jacobi–Galerkin method for the integrated forms of second-order elliptic equations in one and two space variables are presented. They are based on appropriate base functions for the Galerkin formulation which lead to discrete systems with specially structured matrices that can be efficiently inverted. The homogeneous Dirichlet boundary conditions are satisfied exactly by expanding the unknown variable into a polynomial basis of functions which are built upon the Jacobi polynomials. The direct solution algorithm here developed for the homogeneous Dirichlet problem in two-dimensions relies upon a diagonalization process. Numerical results indicating the high accuracy and effectiveness of these algorithms are presented.