Efficient spectral and spectral element methods for Sobolev equation with diagonalization technique

Abstract

In this paper, we first introduce a new series of Legendre basis functions by using the matrix decomposition technique, which are simultaneously orthogonal in both L2- and H1-inner products. Then, we construct efficient space-time spectral method for linear Sobolev equation using spectral approximation in space and multi-domain collocation approximation in time, which can be implemented in a synchronous parallel fashion. Next, we propose a novel Legendre spectral element method for solving nonlinear Sobolev equation with Burgers' type term, which reduce the non-zero entries of linear systems and computational cost. Some rigorous error estimates are carried out for one-dimensional Sobolev equation. Numerical experiments illustrate the effectiveness and accuracy of the suggested approaches.