On the Iterative Solution of Hermite Collocation Equations

Abstract

Collocation methods based on bicubic Hermite piecewise polynomials have been proven to be effective techniques for solving general second order linear elliptic partial differential equations (PDEs) with mixed boundary conditions [ACM Trans. Math. Software, 11 (1985), pp. 379–412]. The corresponding system of discrete collocation equations is generally nonsymmetric and nondiagonally dominant. Methods for their iterative solution are not known and are currently solved using Gauss elimination with scaling and partial pivoting. Point iterative methods like those in ITPACK [Tech. Report CNA-216, Center for Numerical Analysis, Univ. of Texas at Austin, April 1988] do not converge even for the collocation equations obtained from the discretization of model PDE problems. The development of efficient iterative solvers for these collocation equations is necessary for the case of three-dimensional PDE problems and their parallel solution, since direct solvers tend to be space bound and their parallelization is difficult. In this paper block iterative methods are developed and analyzed for the collocation equations corresponding to elliptic PDEs defined on a rectangle and subject to uncoupled mixed boundary conditions. For these types of PDE problems certain boundary degrees of freedom of the collocation approximation can be predetermined symbolically [Houstis, Mitchell, and Rice]. The remaining equations are called “interior” collocation equations. The system of all discrete equations is referred to as “general” collocation equations. Papatheodorou [Math. Comp., 41 (1983), pp. 511-525] was first to determine the exact parameters of accelerated overrelaxation (AOR)-type iterative methods for the case of “interior” collocation equations associated with a model problem. This paper generalizes the results of Papatheodorou for the “interior” collocation equations and presents new results for a particular class of “general” collocation equations. Specifically, in the case of a model elliptic PDE problem with uncoupled mixed boundary conditions, analytic expressions are derived for the eigenvalues of the block Jacobi iteration matrix based on a new partitioning of the interior collocation matrix, and the optimal overrelaxation factors are determined for the block successive overrelaxation (SOR) iterative method. A number of numerical results are presented to verify the theoretical analysis of the block SOR method and to compare its convergence behavior with those of the block Jacobi, Gauss–Seidel and the optimal AOR of Papatheodorou. Furthermore, the authors compare the time and memory complexity of the block SOR, UNPACK Band GE, and generalized minimal residual (GMRES) mathematical software for solving the Hermite collocation equations obtained from the discretization of several PDE problems. The numerical results indicate that the block SOR is an efficient method for solving these equations.