Stability and convergence of the method of fundamental solutions for Helmholtz problems on analytic domains

Barnett, Alex H.; Betcke, Timo

doi:10.1016/j.jcp.2008.04.008

Cited by 159 publications

(181 citation statements)

References 29 publications

Supporting

Mentioning

174

Contrasting

Order By: Relevance

“…Trefftz [21] is usually credited with this idea which was further developed by Bergman and Vekua in the 1940's (see [11] for a review up to the mid 1980s). Recent work in this area includes [3,4]. More recently, in an attempt to avoid ill-conditioning and slow convergence in some situations, methods have been developed that use complete families locally on small sub-regions of the domain.…”

Section: Introductionmentioning

confidence: 99%

Error estimates for the Ultra Weak Variational Formulation of the Helmholtz equation

Buffa¹,

Monk²

2008

ESAIM: M2AN

106

View full text Add to dashboard Cite

Abstract. The Ultra Weak Variational Formulation (UWVF) of the Helmholtz equation provides avariational framework suitable for discretization using plane wave solutions of an appropriate adjoint equation. Currently convergence of the method is only proved on the boundary of the domain. However substantial computational evidence exists showing that the method also converges throughout the domain of the Helmholtz equation. In this paper we exploit the fact that the UWVF is essentially an upwind discontinuous Galerkin method to prove convergence of the solution in the special case where there is no absorbing medium present. We also provide some other estimates in the case when absorption is present, and give some simple numerical results to test the estimates. We expect that similar techniques can be used to prove error estimates for the UWVF applied to Maxwell's equations and elasticity.Mathematics Subject Classification. 65N15, 65N30, 35J05.

show abstract

Section: Introductionmentioning

confidence: 99%

Error estimates for the Ultra Weak Variational Formulation of the Helmholtz equation

Buffa¹,

Monk²

2008

ESAIM: M2AN

106

View full text Add to dashboard Cite

show abstract

“…The discrete version of this statement is that, with the MFS, there exists a sequence of charge points and strengths whose field converges to u s . If Γ F is suitably chosen one can show that there are such sequences that give rapid exponential convergence in certain norms (see [4], and Section 5.1). Unlike boundary integral methods which enable a second kind formulation, (4.1) is necessarily first kind.…”

Section: The Methods Of Fundamentalmentioning

confidence: 99%

“…This better captures solution behavior than standard polynomial bases, and is closely related to the method of particular solutions or collocation methods [23]. Typical choices of basis functions are plane waves [26], Fourier-Bessel expansions [14,27], and fundamental solutions of the Helmholtz equation [6,16,4].…”

mentioning

confidence: 99%

An Exponentially Convergent Nonpolynomial Finite Element Method for Time-Harmonic Scattering from Polygons

Barnett¹,

Betcke²

2010

SIAM J. Sci. Comput.

Self Cite

View full text Add to dashboard Cite

Abstract. In recent years nonpolynomial finite element methods have received increasing attention for the efficient solution of wave problems. As with their close cousin the method of particular solutions, high efficiency comes from using solutions to the Helmholtz equation as basis functions. We present and analyze such a method for the scattering of two-dimensional scalar waves from a polygonal domain that achieves exponential convergence purely by increasing the number of basis functions in each element. Key ingredients are the use of basis functions that capture the singularities at corners, and representing the scattered field towards infinity by a combination of fundamental solutions. The solution is obtained by minimizing a least-squares functional, which we discretize in such a way that a matrix least-squares problem is obtained. We give computable exponential bounds on the rate of convergence of the least-squares functional that are in very good agreement with the observed numerical convergence. Challenging numerical examples, including a nonconvex polygon with several corner singularities, and a cavity domain, are solved to around 10 digits of accuracy with a few seconds of CPU time. The examples are implemented concisely with MPSpack, a MATLAB toolbox for wave computations with nonpolynomial basis functions, developed by the authors. A code example is included.

show abstract

“…The approximate representation of the MFS solution for this problem is written as [3,[15][16][17] u(…”

Section: Formulation Of Singular Boundary Methodsmentioning

confidence: 99%

A method of fundamental solution without fictitious boundary

Chen¹,

Wang²

2009

Mesh Reduction Methods

View full text Add to dashboard Cite

This paper proposes a novel meshless boundary method called the singular boundary method (SBM). This method is mathematically simple, easy-to-program, and truly meshless. Like the method of fundamental solution (MFS), the SBM employs the singular fundamental solution of the governing equation of interest as the interpolation basis function. However, unlike the MFS, the source and collocation points of the SBM coincide on the physical boundary without the requirement of fictitious boundary. In order to avoid the singularity at origin, this method proposes an inverse interpolation technique to evaluate the singular diagonal elements of the interpolation matrix. This study tests the SBM successfully to three benchmark problems, which shows that the method has rapid convergent rate and is numerically stable.

show abstract

Stability and convergence of the method of fundamental solutions for Helmholtz problems on analytic domains

Cited by 159 publications

References 29 publications

Error estimates for the Ultra Weak Variational Formulation of the Helmholtz equation

Error estimates for the Ultra Weak Variational Formulation of the Helmholtz equation

An Exponentially Convergent Nonpolynomial Finite Element Method for Time-Harmonic Scattering from Polygons

A method of fundamental solution without fictitious boundary

Contact Info

Product

Resources

About