General interior Hermite collocation methods for second-order elliptic partial differential equations

Abstract

The method of collocation based on C 1 piecewise polynomials has been shown (by Houstis et al. (1985)) to be an efficient approach to solving general elliptic partial differential equations (PDEs). The iterative solution of the discrete collocation equations is a very challenging problem and only recently it has been fully resolved by the authors (1992) in the case of rectangular domains. For the case of general PDE domains, the iterative solution of the corresponding discrete equations is still an open problem. In this paper we generalize the method of interior collocation for PDEs defined on rectilinear regions, study the structure of these equations under different ordering schemes, and apply AOR and CG-type iterative solvers to the discrete equations. One of the ordering schemes introduced here has been successfully applied to iterative solvers for the general discrete collocation equations. A number of numerical experiments are reported that indicate the applicability and effectiveness of the AOR and CG iterative schemes. Moreover, we have identified experimentally the appropriate range of the semi-optimal acceleration parameters and some effective preconditioning matrices. These preliminary results indicate that iterative approaches are efficient in solving the general Hermite collocation equations on general domains.