Adaptive error estimation technique of the Trefftz method for solving the over-specified boundary value problem

Chen, K. H.; Chen, C. T.; Lee, Jaw Fang

doi:10.1016/j.enganabound.2009.02.001

Cited by 4 publications

(4 citation statements)

References 17 publications

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“…x approaches to the source point j s , Equations (4) and (5) will become singular. Equations (4) and 5for the interior problems need to be regularized by using special treatment of the desingularization of subtracting and adding-back technique [6] as follows…”

Section: Adding-back Technique Of Regularized Meshless Methodsmentioning

confidence: 99%

“…However, because of the controversial artificial boundary (off-set boundary), which shift a distance from the real boundary, the MFS has not become a dominant numerical method. Notwithstanding its gain in singularity free, the influence matrix becomes ill-posed matrix [4] ; its result is very unstable since the condition number for the influence matrix becomes very large.…”

Section: Introductionmentioning

confidence: 99%

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A modified version of regularized meshless method for three dimensional potential problem

et al. 2017

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In this study, three-dimensional potential problem is solved using a novel meshless method. Due to the singularity of the kernel functions, the diagonal terms of the influence matrices in the method of fundamental solutions (MFS) are unobtainable. A proposed approach in the literature of the past decade, namely regularized meshless method (RMM), is proposed to overcome such difficulties by using the proposed desingularization (subtracting and adding-back) technique. The main difficulty for the coincidence of the source and collocation points then disappears. However, the disadvantage of RMM is the order of precision of obtained solution is lower than other numerical methods. In this study, we present a novel technique to promote the order of precision of diagonal term; therefore, we can obtain a more precise solution. Finally, we introduce a typical 3-D numerical example to illustrate the technique. The numerical result is compared with those obtained by the RMM; and a more precise result is obtained.

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Section: Adding-back Technique Of Regularized Meshless Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A modified version of regularized meshless method for three dimensional potential problem

et al. 2017

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“…In [9]: u Ã ¼ xy, In [16]: u Ã ¼ expð0:5xÞsinð0:5yÞ and u Ã ¼ x þ y, In [17]: u Ã ¼ x 3 À3xy 2 þ expð2yÞsinð2xÞÀexpðxÞcosðyÞ, In [19]: u Ã ¼ 10yÀ9, and others in [1][2][3][4][13][14][15], the tested Cauchy problems with globally harmonic solutions may also have numerically redundant Cauchy data.…”

Section: Redundancy In Cauchy Datamentioning

confidence: 99%

On numerical experiments for Cauchy problems of elliptic operators

Yang

Ling

2011

Engineering Analysis with Boundary Elements

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“…Another approaches are those based on the method of fundamental solutions (Young et al, 2008;Yan et al, 2008;Shidfar et al, 2009;Huang et al, 2007;Huang and Wang, 1999) or radial basis functions (Cheng and Cabral, 2005;Hon and Wu, 2000). Similar situation arises when some boundary is over-specified (Chen et al, 2009). The authors use the Trefftz method together with regularization and adaptive error estimation technique in order to obtain the optimal regularization parameter.…”

Section: Introductionmentioning

confidence: 99%

Numerical reconstruction of 2D potential fields from discrete measurements

Irša

Galybin²

2011

International Journal of Numerical Methods for Heat &Amp; Fluid Flow

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Purpose -The purpose of this paper is to consider reconstructions of potential 2D fields from discrete measurements. Two potential processes are addressed, steady flow and heat conduction. In the first case, the flow speed and streamlines are determined from the discrete data on flow directions, in the second case, the temperature and flux are recovered from temperature measurements at discrete points. Design/methodology/approach -The method employs the Trefftz element principle and the collocation. The domain is seen as a combination of elements, where the solution is sought as a linear holomorphic function a priori satisfying the governing equations. Continuity of piecewise holomorphic functions is imposed at collocation points located on the element interfaces. These form the first group of equations. The second group of equations is formed by addressing the measured data, therefore the matrix coefficients may reflect experimental errors. In the case of fluid flow, all equations are homogeneous, therefore one normalising equation is added, which provides existence of a non-trivial solution. The system is over-determined; it is solved by the least squares method. Findings -For the heat flow problem, the determination of heat flux is unique, while for the fluid flow, the determined streamlines are unique and the determination of speed contains one free multiplicative positive constant. Several examples are presented to illustrate the methods and investigate their efficiency and sensitivity to noisy data.Research limitations/implications -The approach can be applied to other 2D potential problems. Originality/value -The paper studies two novel formulations of the reconstruction problem for 2D potential fields. It is shown that the suggested numerical method is able to deal directly with discrete experimental data.

show abstract

Adaptive error estimation technique of the Trefftz method for solving the over-specified boundary value problem

Cited by 4 publications

References 17 publications

A modified version of regularized meshless method for three dimensional potential problem

A modified version of regularized meshless method for three dimensional potential problem

On numerical experiments for Cauchy problems of elliptic operators

Numerical reconstruction of 2D potential fields from discrete measurements

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