Numerical analysis of the MFS for certain harmonic problems (original) (raw)
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Numerical analysis of the method of fundamental solution for harmonic problems in annular domains
Numerical Methods for Partial Differential Equations, 2006
In this study, we investigate the application of the Method of Fundamental Solutions (MFS) to the Dirichlet problem for Laplace's equation in an annular domain. We examine the properties of the resulting coefficient matrix and its eigenvalues. The convergence of the method is proved for analytic boundary data. An efficient matrix decomposition algorithm using Fast Fourier Tranforms (FFTs) is developed for the computation of the MFS approximation. We also tested the algorithm numerically on several problems confirming the theoretical predictions. 2000 Mathematics Subject Classification. Primary 65N12, 65N38; Secondary 65N15,65T50,65Y99. Key words and phrases. Method of fundamental solutions, Laplace's equation, annular domains, error bounds and convergence of the MFS.
Applicability of the method of fundamental solutions
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 differential 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 differential 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 effective condition number. Three numerical examples are presented in which various boundary value problems for the Laplace equation are solved. We show that the effective 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 effective 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.
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In this work, the use of the Method of Fundamental Solutions (MFS) for solving elliptic partial differential equations is investigated, and the performance of various least squares routines used for the solution of the resulting minimization problem is studied. Two modified versions of the MFS for harmonic and biharmonic problems with boundary singularities, which are based on the direct subtraction of the leading terms of the singular local solution from the original mathematical problem, are also examined. Both modified methods give more accurate results than the standard MFS and also yield the values of the leading singular coefficients. Moreover, one of them predicts the form of the leading singular term.
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Engineering Analysis With Boundary Elements, 2004
The method of external source collocation is used to solve a discretised boundary value problem (BVP) 7 2 U ¼ 0; where U is the potential in a two-dimensional simply connected region D, subject to a mixture of Neumann and Dirichlet boundary conditions. We clarify previous discussion on how many sources are required to give sufficient accuracy and where they should be placed. A general proof shows that a sequence of solutions to the BVP converges as the sources are moved further from the boundary, provided that certain conditions are satisfied. Numeric computations provide guidelines on how many sources should be used, and where they should be placed, in order to achieve sufficient accuracy. q
The Method of Fundamental Solutions: A Weighted Least-Squares Approach
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We investigate the Method of Fundamental Solutions (MFS) for the solution of certain elliptic boundary value problems. In particular, we study the case in which the number of collocation points exceeds the number of singularities. In such a case, the resulting linear system is overdetermined and the proposed algorithm chooses the approximate solution for which the error, when restricted on the boundary, minimizes a suitably defined discrete Sobolev norm. This is equivalent to a weighted least-squares treatment of the resulting over-determined system. We prove convergence of the method in the case of the Laplace's equation with Dirichlet boundary data in the disk. We develop an alternative way of implementing the numerical algorithm which avoids the inherent ill-conditioning of the MFS matrices. Finally, we present numerical experiments suggesting that introduction of Sobolev weights improves the approximation.