Improvements of generalized finite difference method and comparison with other meshless method

Abstract

One of the most universal and effective methods, in wide use today, for approximately solving equations of mathematical physics is the finite difference (FD) method. An evolution of the FD method has been the development of the generalized finite difference (GFD) method, which can be applied over general or irregular clouds of points. The main drawback of the GFD method is the possibility of obtaining ill-conditioned stars of nodes. In this paper a procedure is given that can easily assure the quality of numerical results by obtaining the residual at each point. The possibility of employing the GFD method over adaptive clouds of points increasing progressively the number of nodes is explored, giving in this paper a condition to be accomplished to employ the GFD method with more efficiency. Also, in this paper, the GFD method is compared with another meshless method the, so-called, element free Galerkin method (EFG). The EFG method with linear approximation and penalty functions to treat the essential boundary condition is used in this paper. Both methods are compared for solving Laplace equation. © 2003 Elsevier Inc. All rights reserved.