An equilibrated method of fundamental solutions to choose the best source points for the Laplace equation.

*(English)*Zbl 1352.65637Summary: For the method of fundamental solutions (MFS), a trial solution is expressed as a linear combination of fundamental solutions. However, the accuracy of MFS is heavily dependent on the distribution of source points. Two distributions of source points are frequently adopted: one on a circle with a radius \(R\), and another along an offset \(D\) to the boundary, where \(R\) and \(D\) are problem dependent constants. In the present paper, we propose a new method to choose the best source points, by using the MFS with multiple lengths \(R_k\) for the distribution of source points, which are solved from an uncoupled system of nonlinear algebraic equations. Based on the concept of equilibrated matrix, the multiple-length \(R_k\) is fully determined by the collocated points and a parameter \(R\) or \(D\), such that the condition number of the multiple-length MFS (MLMFS) can be reduced smaller than that of the original MFS. This new technique significantly improves the accuracy of the numerical solution in several orders than the MFS with the distribution of source points using \(R\) or \(D\). Some numerical tests for the Laplace equation confirm that the MLMFS has a good efficiency and accuracy, and the computational cost is rather cheap.

##### MSC:

65N80 | Fundamental solutions, Green’s function methods, etc. for boundary value problems involving PDEs |

##### Keywords:

Laplace equation; collocation method; method of fundamental solutions (MFS); multiple-length MFS (MLMFS)
PDF
BibTeX
XML
Cite

\textit{C.-S. Liu}, Eng. Anal. Bound. Elem. 36, No. 8, 1235--1245 (2012; Zbl 1352.65637)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Fairweather, G.; Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv comput math, 9, 69-95, (1998) · Zbl 0922.65074 |

[2] | Saavedra, I.; Power, H., Multipole fast algorithm for the least-squares approach of the method of fundamental solutions for three-dimensional harmonic problems, Numer methods partial differential equation, 19, 828-845, (2003) · Zbl 1038.65127 |

[3] | Bogomolny, A., Fundamental solutions method for elliptic boundary value problems, SIAM J numer anal, 22, 644-669, (1985) · Zbl 0579.65121 |

[4] | Smyrlis, Y.S.; Karageorghis, A., Some aspects of the method of fundamental solutions for certain harmonic problems, J sci comput, 16, 341-371, (2001) · Zbl 0995.65116 |

[5] | Cho, H.A.; Golberg, M.A.; Muleshkov, A.S.; Li, X., Trefftz methods for time-dependent partial differential equations, CMC: comput mater contin, 1, 1-37, (2004) |

[6] | Hon, Y.C.; Wei, T., The method of fundamental solution for solving multidimensional inverse heat conduction problems, CMES: comput model eng sci, 7, 119-132, (2005) · Zbl 1114.80004 |

[7] | Young, D.L.; Chen, K.H.; Lee, C.W., Novel meshless method for solving the potential problems with arbitrary domain, J comput phys, 209, 290-321, (2005) · Zbl 1073.65139 |

[8] | Young, D.L.; Ruan, J.W., Method of fundamental solutions for scattering problems of electromagnetic waves, CMES: comput model eng sci, 7, 223-232, (2005) · Zbl 1106.78008 |

[9] | Young, D.L.; Tsai, C.C.; Lin, Y.C.; Chen, C.S., The method of fundamental solutions for eigenfrequencies of plate vibrations, CMC: comput mater contin, 4, 1-10, (2006) |

[10] | Chen CS, Karageorghis A, Smyrlis YS., editors. The method of fundamental solutions—a meshless method. Dynamic Publishers Inc.; 2008. |

[11] | Liu, C.S., The method of fundamental solutions for solving the backward heat conduction problem with conditioning by a new post-conditioner, Numer heat transfer B: fundam, 60, 57-72, (2011) |

[12] | Karageorghis, A.; Lesnic, D.; Marin, L., A survey of applications of the MFS to inverse problems, Inv prob sci eng, 19, 309-336, (2011) · Zbl 1220.65157 |

[13] | Golberg, M.A.; Chen, C.S., Discrete projection methods for integral equations, (1996), Computational Mechanics Publications Southampton |

[14] | Chen, C.S.; Cho, H.A.; Golberg, M.A., Some comments on the ill-conditioning of the method of fundamental solutions, Eng anal boundary elem, 30, 405-410, (2006) · Zbl 1187.65136 |

[15] | Smyrlis, Y.S.; Karageorghis, A., Numerical analysis of the MFS for certain harmonic problems, M2AN math model numer anal, 38, 495-517, (2004) · Zbl 1079.65108 |

[16] | Alves, C.J.S.; Leitão, V.M.A., Crack analysis using an enriched MFS domain decomposition technique, Eng anal boundary elem, 30, 160-166, (2006) · Zbl 1195.74148 |

[17] | Tsai, C.C.; Lin, Y.C.; Young, D.L.; Atluri, S.N., Investigations on the accuracy and condition number for the method of fundamental solutions, CMES: comput model eng sci, 16, 103-114, (2006) |

[18] | Young, D.L.; Chen, K.H.; Chen, J.T.; Kao, J.H., A modified method of fundamental solutions with source on the boundary for solving Laplace equations with circular and arbitrary domains, CMES: comput model eng sci, 19, 197-222, (2007) · Zbl 1184.65116 |

[19] | Ramachandran, P.A., Method of fundamental solutions: singular value decomposition analysis, Commun numer methods eng, 18, 789-801, (2002) · Zbl 1016.65095 |

[20] | Liu, C.S., Improving the ill-conditioning of the method of fundamental solutions for 2D Laplace equation, CMES: comput model eng sci, 28, 77-93, (2008) · Zbl 1232.65170 |

[21] | Liu, C.S., A modified Trefftz method for two-dimensional Laplace equation considering the Domain’s characteristic length, CMES: comput model eng sci, 21, 53-65, (2007) · Zbl 1232.65157 |

[22] | Liu, C.S., An effectively modified direct Trefftz method for 2D potential problems considering the Domain’s characteristic length, Eng anal boundary elem, 31, 983-993, (2007) · Zbl 1259.65183 |

[23] | Liu, C.S., A highly accurate collocation Trefftz method for solving the Laplace equation in the doubly connected domains, Numer methods partial differential equation, 24, 179-192, (2008) · Zbl 1130.65114 |

[24] | Shigeta, T.; Young, D.L., Mathematical and numerical studies on meshless methods for exterior unbounded domain problems, J comput phys, 230, 6900-6915, (2011) · Zbl 1252.65202 |

[25] | Christiansen, S., Condition number of matrices derived from two classes of integral equations, Math methods appl sci, 3, 364-392, (1981) · Zbl 0485.65089 |

[26] | Katsurada, M., A mathematical study of the charge simulation method II, J fac sci univ Tokyo sec 1A, 36, 135-162, (1989) · Zbl 0681.65081 |

[27] | Drombosky, T.W.; Meyer, A.L.; Ling, L., Applicability of the method of fundamental solutions, Eng anal boundary elem, 33, 637-643, (2009) · Zbl 1244.65220 |

[28] | Mathon, R.; Johnston, R.L., The approximate solution of elliptic boundary value problems by fundamental solutions, SIAM J numer anal, 14, 638-650, (1977) · Zbl 0368.65058 |

[29] | Johnston, R.L.; Fairweather, G., The method of fundamental solutions for problems in potential flow, Appl math model, 8, 265-270, (1984) · Zbl 0546.76021 |

[30] | Wang, J.G.; Ahmed, M.T.; Leavers, J.D., Nonlinear least squares optimization applied to the method of fundamental solutions for eddy current problems, IEEE trans magnet, 26, 2385-2387, (1990) |

[31] | Katsurada, M.; Okamoto, H., The collocation points of the fundamental solution method for the potential problem, Comput math appl, 31, 123-137, (1996) · Zbl 0852.65101 |

[32] | Alves, C.J.S., On the choice of source points in the method of fundamental solutions, Eng anal boundary elem, 33, 1348-1361, (2009) · Zbl 1244.65216 |

[33] | Ling, L.; Opfer, R.; Schaback, R., Results on meshless collocation techniques, Eng anal boundary elem, 30, 247-253, (2006) · Zbl 1195.65177 |

[34] | Schaback, R., Adaptive numerical solution of MFS systems, (), 1-27 |

[35] | Kupradze, V.D.; Aleksidze, M.A., The method of functional equations for the approximate solution of certain boundary value problems, USSR comput math math phys, 4, 82-126, (1964) · Zbl 0154.17604 |

[36] | Liu, C.S.; Atluri, S.N., An iterative method using an optimal descent vector, for solving an ill-conditioned system bx={\bfb}, better and faster than the conjugate gradient method, CMES: comput model eng sci, 80, 275-298, (2011) |

[37] | Bauer, F.L., Optimally scaled matrices, Numer math, 5, 73-87, (1963) · Zbl 0107.10501 |

[38] | van der Sluis, A., Condition numbers and equilibration of matrices, Numer math, 14, 14-23, (1969) · Zbl 0182.48906 |

[39] | Gautsch, W., Optimally scaled and optimally conditioned Vandermonde and Vandermonde-like matrices, BIT numer math, 51, 103-125, (2011) · Zbl 1214.65021 |

[40] | Liu, C.S.; Hong, H.K.; Atluri, S.N., Novel algorithms based on the conjugate gradient method for inverting ill-conditioned matrices, and a new regularization method to solve ill-posed linear systems, CMES: comput model eng sci, 60, 279-308, (2010) · Zbl 1231.65071 |

[41] | Motz, H., The treatment of singularities in relaxation methods, Q appl math, 4, 371-377, (1946) · Zbl 0030.40002 |

[42] | Li, Z.C., Combined methods for elliptic equations with singularities, interfaces and infinities, (1998), Kluwer Academic Publishers Netherlands · Zbl 0909.65079 |

[43] | Liu, C.S., A highly accurate solver for the mixed-boundary potential problem and singular problem in arbitrary plane domain, CMES: comput model eng sci, 20, 111-122, (2007) |

[44] | Li, Z.C., The method of fundamental solutions for the Laplace equation with mixed boundary problems, (), 29-50 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.