Efficient implementation of the MFS: The three scenarios

• Published in: Journal of computational and applied mathematics Vol. 227; no. 1; pp. 83 - 92
• Main Authors: Smyrlis, Yiorgos-Sokratis, Karageorghis, Andreas
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Kidlington Elsevier B.V 01.05.2009; Elsevier
• Subjects: Calculus of variations and optimal control; Circulant matrices; Elliptic boundary value problems; Exact sciences and technology; Least-squares method; Mathematical analysis; Mathematics; Method of fundamental solutions; Numerical analysis; Numerical analysis. Scientific computation; Over-determined systems; Partial differential equations; Partial differential equations, boundary value problems; Sciences and techniques of general use; Under-determined systems; Variational methods for elliptic equations; Least-squares method; Method of fundamental solutions; Elliptic boundary value problems; Variational methods for elliptic equations; Over-determined systems; Circulant matrices; Under-determined systems; Fourier transformation; Approximation; Fast Fourier transformation; Fourier analysis; Boundary condition; Algorithm; Partial differential equation; Implementation; Discretization method; Fundamental solution; Numerical analysis; Elliptic equation; Boundary value problem; Variational equation; Applied mathematics; Least squares method; Dirichlet problem
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2008.07.010

In this study we investigate the approximation of the solutions of harmonic problems subject to Dirichlet boundary conditions by the Method of Fundamental Solutions (MFS). In particular, we study the application of the MFS to Dirichlet problems in a disk. The MFS discretization yields systems which possess special features which can be exploited by using Fast Fourier transform (FFT)-based techniques. We describe three possible formulations related to the ratio of boundary points to sources, namely, when the number of boundary points is equal, larger and smaller than the number of sources. We also present some numerical experiments and provide an efficient MATLAB implementation of the resulting algorithms.