A meshless polyharmonic-type boundary interpolation method for solving boundary integral equations

Int. J. Comput. Methods

2013

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A special regularization method based on higher-order partial differential equations is presented. Instead of using the fundamental solution of the original partial differential operator with source points located outside of the domain, the original second-order partial differential equation is approximated by a higher-order one, the fundamental solution of which is continuous at the origin. This allows the use of the method of fundamental solutions (MFS) for the approximate problem. Due to the continuity of the modified operator, the source points and the boundary collocation points are allowed to coincide, which makes the solution process simpler. This regularization technique is generalized to various problems and combined with the extremely efficient quadtreebased multigrid methods. Approximation theorems and numerical experiences are also presented.

Section: Gáspármentioning

confidence: 94%

Section: Gáspármentioning

confidence: 99%

Regularization Techniques for the Method of Fundamental Solutions

Int. J. Comput. Methods

2013

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“…The above mentioned numerical difficulties can be decreased by applying the direct multi-elliptic interpolation [7][8][9].…”

Section: Direct Multi-elliptic Interpolation For Creating Particular mentioning

confidence: 99%

Multi-level meshless methods based on direct multi-elliptic interpolation

Journal of Computational and Applied Mathematics

2009

a r t i c l e i n f o MSC: 65N55 65N99 Keywords: Meshless methods Radial basis functions Multi-elliptic interpolation Multi-level methods a b s t r a c tA short overview on the direct multi-elliptic interpolation and the related meshless methods for solving partial differential equations is given. A new technique is proposed which produces a biharmonic interpolation along the boundary and solves the original problem inside the domain. An error estimation is also derived. To implement the method, quadtree-based multi-level methods are used. The approach avoids the use of large, dense and ill-conditioned matrices and significantly reduces the computational cost.

“…Thus, the use of large ill-conditioned matrices can be completely avoided and the computational cost can be significantly reduced. For details, see Gáspár [8].…”

Section: Regularization By Higher Order Problemsmentioning

confidence: 99%

“…To overcome this difficulty, a desingularization method is used in general (Young, Chen and Lee [16], Šarler [15], Chen and Wang [5]; see also Gu, Chen and Zhang [12], Gu, Chen and He [11]). In its simplest form, consider the auxiliary Dirichlet problem ∆ = 0 in Ω Γ = 1 (8) which has the unique solution ≡ 1. Expressing in the same form,…”

Section: Regularization By Truncationmentioning

confidence: 99%

Some variants of the method of fundamental solutions: regularization using radial and nearly radial basis functions

2013

Open Mathematics

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The method of fundamental solutions and some versions applied to mixed boundary value problems are considered. Several strategies are outlined to avoid the problems due to the singularity of the fundamental solutions: the use of higher order fundamental solutions, and the use of nearly fundamental solutions and special fundamental solutions concentrated on lines instead of points. The errors of the approximations as well as the problem of ill-conditioned matrices are illustrated via numerical examples.