The Approximate Solution of Elliptic Boundary-Value Problems by Fundamental Solutions

Mathon, Rudolf; Johnston, Roy L.

doi:10.1137/0714043

Cited by 330 publications

(122 citation statements)

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“…Kupradze also suggested the MFS for time-dependent problems [31]; in particular, for the solution of the heat equation. The MFS was first investigated, as a numerical technique, by Mathon and Johnston [39] in 1977. In their work, the coefficients in (1.7) were chosen to minimize the L 2 −distance of the approximate solution from the boundary data of (1.5).…”

Section: Introductionmentioning

confidence: 99%

Applicability and applications of the method of fundamental solutions

Smyrlis

2009

Math. Comp.

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Abstract. In the present work, we investigate the applicability of the method of fundamental solutions for the solution of boundary value problems of elliptic partial differential equations and elliptic systems. More specifically, we study whether linear combinations of fundamental solutions can approximate the solutions of the boundary value problems under consideration. In our study, the singularities of the fundamental solutions lie on a prescribed pseudo-boundary -the boundary of a domain which embraces the domain of the problem under consideration. We extend previous density results of Kupradze and Aleksidze, and of Bogomolny, to more general domains and partial differential operators, and with respect to more appropriate norms. Our domains may possess holes and their boundaries are only required to satisfy a rather weak boundary requirement, namely the segment condition. Our density results are with respect to the norms of the spaces C (Ω). Analogous density results are obtainable with respect to Hölder norms. We have studied approximation by fundamental solutions of the Laplacian, biharmonic and m−harmonic and modified Helmholtz and poly-Helmholtz operators. In the case of elliptic systems, we obtain analogous density results for the Cauchy-Navier operator as well as for an operator which arises in the linear theory of thermo-elasticity. We also study alternative formulations of the method of fundamental solutions in cases when linear combinations of fundamental solutions of the equations under consideration are not dense in the solution space. Finally, we show that linear combinations of fundamental solutions of operators of order m ≥ 4, with singularities lying on a prescribed pseudo-boundary, are not in general dense in the corresponding solution space.

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Section: Introductionmentioning

confidence: 99%

Applicability and applications of the method of fundamental solutions

Smyrlis

2009

Math. Comp.

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“…Location of the surface dU in Kupradze-Alexidze's approach is quite arbitrary. Mathon and Johnston [8] in an analogous context proposed using nonlinear optimization algorithms for the optimal disposition of the surface d U or, more precisely, points x. e dU. As mentioned in [8], serious computational problems can arise when different points merge in the process of optimization.…”

Section: Idclmentioning

confidence: 99%

Untitled

1995

Quart. Appl. Math.

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Abstract. The Direct Boundary Integral Equation Method (BIEM), which leads to second-kind singular integral equations for all types of commonly used boundaryvalue problems of the theory of elasticity, is formulated. The exposition is applied to anisotropic bodies with arbitrary elastic anisotropy.1. Introduction. The "direct" in connection with modern terminology [1] are called such formulations of the boundary integral equation method that lead to equations with unknown densities, located on the boundary surface and having real geometric or physical meaning. In contrast to that in indirect approaches, unknown densities do not have any physical or geometric meaning. The advantage of the direct formulation is based on the possibility of determining unknown surface densities just in the process of resolution, and quite often these data are of main interest.The first direct BIEM formulations in the theory of elasticity are, to all appearances, due to Kupradze and Alexidze [2,3]. Their approach was based on the Somigliana identity written for the complement Q_ = R"\fl, where fi is an open region under consideration:

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“…Mathon and Johnston [13] first showed numerical results obtained by the MFS. The papers [11], [14] discuss some mathematical theories on the MFS.…”

Section: Introductionmentioning

confidence: 99%

Method of fundamental solutions with optimal regularization techniques for the Cauchy problem of the Laplace equation with singular points

Shigeta

Young

2009

Journal of Computational Physics

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The purpose of this study is to propose a high-accuracy and fast numerical method for the Cauchy problem of the Laplace equation. Our problem is directly discretized by the method of fundamental solutions (MFS). The Tikhonov regularization method stabilizes a numerical solution of the problem for given Cauchy data with high noises. The accuracy of the numerical solution depends on a regularization parameter of the Tikhonov regularization technique and some parameters of MFS. The L-curve determines a suitable regularization parameter for obtaining an accurate solution. Numerical experiments show that such a suitable regularization parameter coincides with the optimal one. Moreover, a better choice of the parameters of MFS is numerically observed. It is noteworthy that a problem whose solution has singular points can successfully be solved. It is concluded that the numerical method proposed in this paper is effective for a problem with an irregular domain, singular points, and the Cauchy data with high noises.

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The Approximate Solution of Elliptic Boundary-Value Problems by Fundamental Solutions

Cited by 330 publications

References 7 publications

Applicability and applications of the method of fundamental solutions

Applicability and applications of the method of fundamental solutions

Untitled

Method of fundamental solutions with optimal regularization techniques for the Cauchy problem of the Laplace equation with singular points

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