Investigation of Triangle Element Analysis for the Solutions of 2D Poisson Equations via AOR method

In earlier studies of iterative approaches, the accelerated over relaxation (AOR) method has been pointed out to be much faster as compared to the existing successive over re- laxation (SOR) and Gauss Seidel (GS) methods. Due to the effectiveness of this method, the foremost goal of this paper is to demonstrate the use of the AOR method together with triangle element solutions based on the Galerkin scheme method. The effectiveness of this method has been shown via results of numerical experiments, which are logged and examined. The findings show that the AOR method is superior as compared with the existing SOR and GS methods

The explicit group TOR method

The numerical methods for solving partial differential equations have been one of the significant achievements made possible by the digital computers. With the advent of parallel computers, many studies have been performed and a number of new techniques have been investigated in order to develop new methods that are suitable for these computers. One of these techniques is the explicit group iterative methods which have been extensively studied and analysed in the last two decades. The explicit group iterative methods for the numerical solution of self-adjoint elliptic partial differential equations have been introduced (Evans & Biggins, 1982; Yousif & Evans, 1986) and has been shown to be computationally superior in comparison with other iterative methods. These methods were found to be suitable for parallel computers as they possess independent tasks (Evans & Yousif, 1990). Martins, Yousif & Evans (2002) introduced a new explicit 4-points group accelerated overrelaxation (EGAOR) iterative method, a comparison with the point AOR method has shown its computational advantages. The point TOR method was developed and a number of papers related to the TOR method and its convergence have been presented (Kuang & Ji, 1988; Chang, 1996; Chang, 2001; Martins, Trigo & Evans 2003). In this paper, we formulate a new group method from the TOR family, the explicit 4-points group overrrelaxation (EGTOR) iterative method, the derivation of the new method is presented. Numerical experiments have been carried out and the results obtained confirm the superiority of the new method when compared to the point TOR method

The Analysis of Iterative Elliptic PDE Solvers Based on The Cubic Hermite Collocation Discretization

Abstract. Collocation methods based on bicubic Hermite piecewise polynomials have been proven effective t.echniques for solving second-order linear elliptic PDEs with mixed boundary conditions. The corresponding linear system is in general non-symmetric and non-diagonally dominant. Iterative methods for their solution arc not known and they aTC currently solved using Gauss elimination with scaling and partial pivoting. Point iterat.ive methods do not convcrge even for the collocation equations obtained from model PDE problems. The del/elopment of efficient iterative solvers for these equations is necessary for three-dimensional problems and their parallel solution, since direct solvers tend to be space bound and their parallelization is difficult. In this thesis, we develop block iterative methods for the collocation equations of elliptic PDEs defined on a rectangle and subject to uncoupled mixed boundary conditions. For model problems of this type, we derive analytic expressions for the eigenvalues of the block Jacobi iteration matrix: and determine the optimal parameter for the block SOR method. For the case of general domains, the iterative solution of tile collocation equations is still an open problem. We address this open problem b