An efficient direct solver for multidimensional elliptic Robin boundary value problems using a Legendre spectral-Galerkin method

Abstract

In this paper, a Legendre–Galerkin method for solving second-order elliptic differential equations subject to the most general nonhomogeneous Robin boundary conditions is presented. The homogeneous Robin boundary conditions are satisfied exactly by expanding the unknown variable using a polynomial basis of functions which are built upon the Legendre polynomials. The direct solution algorithm here developed for the homogeneous Robin problem in two-dimensions relies upon a tensor product process. Nonhomogeneous Robin data are taken into account by means of a lifting. Such a lifting is performed in two successive steps, the first one to account for the data specified at the corners and the second one to account for the boundary values prescribed in the interior of the sides. Numerical results indicating the high accuracy and effectiveness of these algorithms are presented.