WIT Press

Matrix Decomposition MFS Algorithms


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WIT Press


A. Karageorghis & Y.-S. Smyrlis


We describe the application of the Method of Fundamental Solutions (MFS) to elliptic boundary value problems in rotationally symmetric problems. In particular, we show how efficient matrix decomposition MFS algorithms can be developed for such problems. The efficiency of these algorithms is optimized by using Fast Fourier Transforms (FFTs). 1 Introduction The Method of Fundamental Solutions (MFS) is a meshless boundary method which has become popular in recent years primarily because of its simplicity. Implementational details as well as a wide range of applications of theMFS can be found in the survey papers [1–3]. The MFS has recently been used for the solution of several elliptic boundary value problems in rotationally symmetric problems. At first, the MFS was used to solve the axisymmetric version of the governing equations [4–6]. However, the fundamental solutions of these equations involve the potentially troublesome evaluation of complete elliptic integrals. Further, when the boundary conditions of the problem under consideration are not axisymmetric, this approach requires the solution of a sequence of problems in order to approximate a finite Fourier sum. More recently, a different approach has been suggested, in which the three dimensional version of the governing equations is considered. In this work, matrix decomposition algorithms are developed for the efficient solution of the resulting systems. (An overview of matrix decomposition algorithms can be found in [7]). The algorithms proposed in this approach make use of Fast Fourier Transforms (FFTs). The basic ideas for the solution of two-dimensional harmonic problems in a disk subject to Dirichlet boundary conditions can be found in [8] while the numerical analysis of these problems is carried out in [9]. The implemen-