Multidomain pseudospectral integration preconditioning matrices for the advection and the diffusion operators

Abstract

Single domain spectral/pseudospectral integration preconditioning matrices have been shown to be effective operators for solving differential equations. In this study we extend the integration precondition methodology to a multidomain computational framework, and construct global inverse matrices for the advection operator discretized by multidomain Gauss-Lobatto-Legendre pseudospectral methods. These inverse operators can be used as solution operators for solving first order boundary value problems even with piecewise continuous variable coefficients. Moreover, they are effective integration preconditioning matrices for the second order diffusion operator, in the sense that a model diffusion problem can be solved by an iterative method with the number of iteration steps being proportional to the number of subdomains but independent of the degree of the solution polynomial. Numerical experiments were conducted and we observed the performance of the inverse operators as expected.