Recent Development of Fast Numerical Solver for Elliptic Problem

Abstract

Most elliptic solvers developed by researchers need long processing time to be solved. This is due to the complexity of the methods. The objective of this paper is to present new finite difference and finite element methods to overcome the problem. Solving scientific problems mathematically always involved partial differential equations. Two recommended common numerical methods are mesh-free solutions (Belytschko et al, 1996; Zhu 1999; Yagawa & Furukawa, 2000) and mesh-based solutions. The mesh-based solutions can be further classified as finite difference method, finite element method, boundary element method, and finite volume method. These methods have been widely used to construct approximation equations for scientific problems. The developments of numerical algorithms have been actively done by researchers. Evans and Biggins (1982) have proposed an iterative four points Explicit Group (EG) for solving elliptic problem. This method employed blocking strategy to the coefficient matrix of the linear system of equations. By implementing this strategy, four approximate equations are evaluated simultaneously. This scenario speed up the computation time of solving the problem compared to using point based algorithms. At the same time, Evans and Abdullah (1982) utilized the same concepts to solve parabolic problem. Four years later, the concept has been further extended to develop two, nine, sixteen and twenty five points EG (Yousif & Evans, 1986a). These EG schemes have been compared to one and two lines methods. As the results of comparison, the EG solve the problem efficiently compared to the lines methods. Utilizing higher order finite difference approximation, a method called Higher Order Difference Group Explicit (HODGE) was developed (Yousif & Evans, 1986b). This method have higher accuracy than the EG method. Abdullah (1991) modified the EG method by using rotated approximation scheme. The rotated scheme is actually rotate the ordinary computational molecule by 45° to the left. By rearranging the new computational molecule on the solution domain, only half of the total nodes are solved iteratively. The other half can be solved directly using the ordinary computational molecule. This method was named