An efficient Legendre–Galerkin spectral element method for the steady flows in rectangular cavities

Abstract

An efficient Legendre–Galerkin spectral element method for the steady flows in rectangular cavities is proposed in this paper. Firstly, we eliminate the singularity of biharmonic equation in rectangular cavity at the corner by the singularity substraction technique. Then we construct some appropriate interior basis functions and interface basis functions which maintain -continuity. Consequently, the discrete variational formulation is reduced to a linear system with block diagonal and well-conditioned coefficient matrix, which can be efficiently solved by the conjugate gradient iteration method. Finally, several numerical examples are given to show the effectiveness of our numerical method. The present method is used to solve the creeping flows in rectangular cavities, the numerical results are compared well with the benchmark steady solutions provided by the finite difference method.