Department of Mathematics
The condition number of a matrix is commonly used for investigating the stability of solutions to linear algebraic systems. Recent meshless techniques for solving partial di®erential equations have been known to give rise to ill-conditioned matrices, yet are still able to produce results that are close to machine accuracy. In this work, we consider the method of fundamental solutions (MFS), which is known to solve, with extremely high accuracy, certain partial di®erential equations, namely those for which a fundamental solution is known. To investigate the applicability of the MFS, either when the boundary is not analytic or when the boundary data is not harmonic, we examine the relationship between its accuracy and the e®ective condition number. Three numerical examples are presented in which various boundary value prob- lems for the Laplace equation are solved. We show that the e®ective condition number, which estimates system stability with the right-hand side vector taken into account, is roughly inversely proportional to the maximum error in the numerical approximation. Unlike the proven theories in literature, we focus on cases when the boundary and the data are not analytic. The e®ective condition number numerically provides an estimate of the quality of the MFS solution without any knowledge of the exact solution and allows the user to decide whether the MFS is, in fact, an appropriate method for a given problem, or what is the appropriate formulation of the given problem
Laplace equation, Method of Fundamental Solutions, E®ective condition number, Error estimation
Source Publication Title
Engineering Analysis with Boundary Elements
Link to Publisher's Edition
Drombosky, Tyler . , Ashley Meyer, and L. Ling. "Applicability of the method of fundamental solutions." Engineering Analysis with Boundary Elements 33.5 (2009): 637-643.