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

A.H. Barnett, T. Betcke
2008 Journal of Computational Physics  
The Method of Fundamental Solutions (MFS) is a popular tool to solve Laplace and Helmholtz boundary value problems. Its main drawback is that it often leads to ill-conditioned systems of equations.  ...  In this paper we investigate for the interior Helmholtz problem on analytic domains how the singularities (charge points) of the MFS basis functions have to be chosen such that approximate solutions can  ...  In Section 2 we give rigorous results for the convergence and the numerical stability of the MFS for Helmholtz problems on the unit disc, with analytic boundary data, using charge points on a concentric  ... 
doi:10.1016/j.jcp.2008.04.008 fatcat:cilfz2spwnc5znwxmmxhj54d7a

Numerical investigation on convergence of boundary knot method in the analysis of homogeneous Helmholtz, modified Helmholtz, and convection–diffusion problems

W. Chen, Y.C. Hon
2003 Computer Methods in Applied Mechanics and Engineering  
This paper concerns a numerical study of convergence properties of the boundary knot method (BKM) applied to the solution of 2D and 3D homogeneous Helmholtz, modified Helmholtz, and convection-diffusion  ...  In order to avoid the interference of the DRM, this study focuses on the investigation of the convergence property of the BKM for homogeneous problems.  ...  The first author was then a visiting research fellow to the Mathematics Department of City University of Hong Kong.  ... 
doi:10.1016/s0045-7825(03)00216-0 fatcat:2h24uh4a7jgfdhkwputhuczqva

Nitsche's method for Helmholtz problems with embedded interfaces

Zilong Zou, Wilkins Aquino, Isaac Harari
2016 International Journal for Numerical Methods in Engineering  
We provide analytical results that establish the well-posedness of Helmholtz variational problems and convergence of the corresponding finite element discretizations when Nitsche's method is used to enforce  ...  Doing so, we demonstrate the asymptotic convergence of the proposed method and show that numerical results are in accordance with the theoretical analysis.  ...  The rest of the article is organized as follows. In section 2, we provide the analytical background on the analysis of the Helmholtz variational problem and Nitsche's method.  ... 
doi:10.1002/nme.5369 pmid:28713177 pmcid:PMC5509378 fatcat:4gzvivlk2vdfxlzwrgwnnj45uy

Boundary knot method for 2D and 3D Helmholtz and convection-diffusion problems under complicated geometry

Y. C. Hon, W. Chen
2003 International Journal for Numerical Methods in Engineering  
The stability and convergence of the BKM are numerically illustrated and the completeness issue is also discussed.  ...  The purpose of this paper is to extend the BKM to solve 2D Helmholtz and convection-diffusion problems under rather complicated irregular geometry. The method is also first applied to 3D problems.  ...  ACKNOWLEDGEMENTS The second author is grateful of a visiting research fellowship to Mathematical Department of City University of Hong Kong.  ... 
doi:10.1002/nme.642 fatcat:jbatoghwxzamnif5ncltqkaihy

An alternating iterative MFS algorithm for the Cauchy problem for the modified Helmholtz equation

Liviu Marin
2010 Computational Mechanics  
The iterative MFS algorithm is tested for Cauchy problems for the two-dimensional modified Helmholtz operator to confirm the numerical convergence, stability and accuracy of the method.  ...  The two mixed, well-posed and direct problems corresponding to every iteration of the numerical procedure are solved using the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization  ...  The numerical solution for the Cauchy problem for two-and three-dimensional Helmholtz-type equations by employing the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization  ... 
doi:10.1007/s00466-010-0480-6 fatcat:bluczkntpnbutfaxga4r6cktsm

Boundary knot method based on geodesic distance for anisotropic problems

Bangti Jin, Wen Chen
2006 Journal of Computational Physics  
Numerical results show that the BKM based on the geodesic distance can produce highly accurate solutions of anisotropic problems with a relatively small number of knots.  ...  This paper makes the first attempt to use the geodesic distance with the RBF-based boundary knot method (BKM) to solve 2D and 3D anisotropic Helmholtz-type and convection-diffusion problems.  ...  Acknowledgements We thank the anonymous reviewers of this paper for their very helpful comments and suggestions to significantly improve the academic quality of this paper.  ... 
doi:10.1016/j.jcp.2005.11.032 fatcat:g33j3wxbrrdxjebqm2wrhcbo2e

The method of fundamental solutions for the Cauchy problem associated with two-dimensional Helmholtz-type equations

Liviu Marin, Daniel Lesnic
2005 Computers & structures  
The convergence and the stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain, and decreasing  ...  In this paper, the application of the method of fundamental solutions to the Cauchy problem associated with twodimensional Helmholtz-type equations is investigated.  ...  Acknowledgment Liviu Marin would like to acknowledge the financial support received from the EPSRC.  ... 
doi:10.1016/j.compstruc.2004.10.005 fatcat:no2ct5dk7fbuxef46gr6fu4pha

A meshless method for the numerical solution of the Cauchy problem associated with three-dimensional Helmholtz-type equations

Liviu Marin
2005 Applied Mathematics and Computation  
The convergence, accuracy and stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain, and decreasing  ...  In this paper, the application of the method of fundamental solutions to the Cauchy problem associated with three-dimensional Helmholtz-type equations is investigated.  ...  Method of fundamental solutions The fundamental solutions F H and F MH of the Helmholtz ðL D þ k 2 Þ and the modified Helmholtz ðL D À k 2 Þ equations (1), respectively, in the three-dimensional case are  ... 
doi:10.1016/j.amc.2004.04.052 fatcat:ztbdzmpehbaw5jebprrst5o74e

A meshless, integration-free, and boundary-only RBF technique

W. Chen, M. Tanaka
2002 Computers and Mathematics with Applications  
In particular, due to the use of nonsingular general solutions rather than singular fundamental solutions, the BKM is different from the method of fundamental solution in that the former does not require  ...  Based on the radial basis function (RBF), nonsingular general solution, and dual reciprocity method (DRM), this paper presents an inherently meshless, integration-free, boundaryonly RBF collocation technique  ...  NONSINGULAR GENERAL SOLUTION AND BOUNDARY KNOT METHOD One may think that the placement of source points outside domain in the MFS is to avoid the singularities of fundamental solutions.  ... 
doi:10.1016/s0898-1221(01)00293-0 fatcat:2xm54piqfndgjdcpddbwwkobua

Investigation of regularized techniques for boundary knot method

Fuzhang Wang, Wen Chen, Xinrong Jiang
2009 International Journal for Numerical Methods in Biomedical Engineering  
We consider three regularization methods and two approaches for the determination of the regularization parameter.  ...  Our numerical experiments show that Tikhonov regularization in conjunction with generalized cross-validation approach outperforms the other regularization techniques in the BKM solution of Helmholtz and  ...  ACKNOWLEDGEMENTS We are very grateful of Prof. C.S. Chen for helpful suggestions to improve academic quality and readability of this paper.  ... 
doi:10.1002/cnm.1275 fatcat:oikqui2nurho3oh6spw4jnxbny

BOUNDARY PARTICLE METHOD FOR INVERSE CAUCHY PROBLEMS OF INHOMOGENEOUS HELMHOLTZ EQUATIONS

Wen Chen, Zhuojia Fu
2009 Journal of Marine Science and Technology  
This paper investigates the boundary particle method (BPM) coupled with truncated singular value decomposition (TSVD) regularization technique on the solution of inverse Cauchy problems of inhomogeneous  ...  In this study, numerical experiments demonstrate that the BPM in conjunction with the TSVD is highly accurate, computationally efficient and stable for inverse Cauchy problems.  ...  Section III describes the boundary particle method for the Cauchy problem associated with inhomogeneous Helmholtz equations, followed by the Section IV to numerically examine the efficiency and stability  ... 
doi:10.51400/2709-6998.1952 fatcat:xe7s4zwtmjbylmfbk7mp4z6d6q

Singular boundary method for modified Helmholtz equations

Wen Chen, Jin-Yang Zhang, Zhuo-Jia Fu
2014 Engineering analysis with boundary elements  
This study makes the first attempt to apply a recent strong-form boundary collocation method using the singular fundamental solutions, namely the singular boundary method (SBM), to 2D and 3D modified Helmholtz  ...  Numerical demonstrations show the feasibility and accuracy of the present SBM in some benchmark problems.  ...  Acknowledgments The authors thank the anonymous reviewers of this article for their very helpful comments and suggestions to significantly improve the academic quality of this article.  ... 
doi:10.1016/j.enganabound.2014.02.007 fatcat:juyttnixqbajdccta4m7bktwva

A Novel Technique for Estimating the Numerical Error in Solving the Helmholtz Equation

Kue-Hong Chen, Cheng-Tsung Chen
2020 Computers Materials & Continua  
In this study, we applied a defined auxiliary problem in a novel error estimation technique to estimate the numerical error in the method of fundamental solutions (MFS) for solving the Helmholtz equation  ...  The defined auxiliary problem is substituted for the real problem, and its analytical solution is generated using the complementary solution set of the governing equation.  ...  N for Case 3 Field solution for Case 3; (a) analytical solution and (b) method of fundamental solutions (MFS) and the analytical solution to this problem is expressed as [ ] ( ) 2 ( ) 2 ,  ... 
doi:10.32604/cmc.2020.08864 fatcat:nktdizqtbvhy3jdrhln4lb7hu4

A new investigation into regularization techniques for the method of fundamental solutions

Ji Lin, Wen Chen, Fuzhang Wang
2011 Mathematics and Computers in Simulation  
This study examines different regularization approaches to investigate the solution stability of the method of fundamental solutions (MFS).  ...  Meanwhile, we have investigated the relationship among the condition number, the effective condition number, and the MFS solution accuracy.  ...  Acknowledgements The work described in this paper was supported by National  ... 
doi:10.1016/j.matcom.2010.10.030 fatcat:zcorzrkqmveefirmpcxkuhs6au

A Simple, Accurate and Semi-Analytical Meshless Method for Solving Laplace and Helmholtz Equations in Complex Two-Dimensional Geometries

Xingxing Yue, Buwen Jiang, Xiaoxuan Xue, Chao Yang
2022 Mathematics  
The LVBE-MCM is a semi-analytical and domain-type meshless collocation method that is based on the fundamental solution of the governing equation, which is different from the traditional virtual boundary  ...  Numerical experiments, including irregular and doubly connected domains, demonstrate that the LVBE-MCM is accurate, stable, and convergent for solving both Laplace and Helmholtz equations.  ...  Conflicts of Interest: The authors declare no conflict of interest.  ... 
doi:10.3390/math10050833 fatcat:bb6ro2fodvg3nerotu734nxwue
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