Special boundary approximation methods for laplace equation problems with boundary singularities— Applications to the motz problem

Li, Z. C.; Chan, Ya-Ting; Georgiou, Georgios C.; Xenophontos, Christos

doi:10.1016/j.camwa.2005.01.030

Cited by 16 publications

(19 citation statements)

References 17 publications

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“…The condition number is cond = 1.711 for n = 160 when 1 19 = 3.184 × 10 −19 , 2 19 = 3.652 × 10 −18 . In BAMs (see [11]), the best results are obtained in the classic BAM in the case of the Motz problem when the corresponding maximum errors are 1 = 5.96 × 10 −9 , 2 = 4.45 × 10 −8 , and the condition number of the matrix of the algebraic problem is cond = 0.787 × 10 6 . Some versions of the BAMs give a high accurate results for the leading coefficients j .…”

Section: Numerical Resultsmentioning

confidence: 92%

“…The convergence of the solution with the number of singular functions is exponential. Furthermore, the BAMs give high accurate results for the leading coefficients j , in particular, as shown in [11], for the Motz problem the number of converged significant digits for 1 is 13. The BAMs can be applied in solving the problems of biharmonic equation also (see [13]).…”

Section: Introductionmentioning

confidence: 93%

“…The BAMs can be applied in solving the problems of biharmonic equation also (see [13]). However, as it happens in all versions of BAMs (see [3,11,14]), there is a loss of stability as N increases (i.e. the condition number becomes larger).…”

Section: Introductionmentioning

confidence: 96%

“…the condition number becomes larger). Moreover, as reported in Table 6 in [11] (see also Table 5 in [5]) an error in maximum norm for the approximate solution is of order E−08, and the condition number of the associated matrix of algebraic problem is 0.787E+6. In the case of SFBIM as pointed out in [15] (see also [9]) the stiffness matrix is also ill conditioned, when N is less than the number of Lagrange multipliers N .…”

Section: Introductionmentioning

confidence: 98%

“…In the past few years, Georgiou and co-workers [8][9][10] developed a new version of BAMs (see also [11]), the singular function boundary integral method (SFBIM) for the solution of plane Laplacian problem with boundary singularities. This method is based on the approximation of the 1278 A.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

On solving the cracked‐beam problem by block method

Dosiyev¹,

Buranay²

2007

Commun. Numer. Meth. Engng.

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SUMMARYAn extremely accurate solution is obtained for the cracked-beam problem by one-block version of the block method (BM). The obtained numerical results demonstrate the exponential convergence of the BM with respect to the number of quadrature nodes. A simple and high accurate formula to compute the stress intensity factor is given. The comparisons with other high accurate results in the literature have been carried out.

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Section: Numerical Resultsmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On solving the cracked‐beam problem by block method

Dosiyev¹,

Buranay²

2007

Commun. Numer. Meth. Engng.

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Trefftz, collocation, and other boundary methods—A comparison

Huang

et al. 2006

Numerical Methods Partial

110

114

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In this article we survey the Trefftz method (TM), the collocation method (CM), and the collocation Trefftz method (CTM). We also review the coupling techniques for the interzonal conditions, which include the indirect Trefftz method, the original Trefftz method, the penalty plus hybrid Trefftz method, and the direct Trefftz method. Other boundary methods are also briefly described. Key issues in these algorithms, including the error analysis, are addressed. New numerical results are reported. Comparisons among TMs and other numerical methods are made. It is concluded that the CTM is the simplest algorithm and provides the most accurate solution with the best numerical stability.

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Analysis of the singular function boundary integral method for a biharmonic problem with one boundary singularity

Christodoulou

Elliotis

Georgiou

et al. 2011

Numerical Methods Partial

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In this article, we analyze the singular function boundary integral method (SFBIM) for a two-dimensional biharmonic problem with one boundary singularity, as a model for the Newtonian stick-slip flow problem. In the SFBIM, the leading terms of the local asymptotic solution expansion near the singular point are used to approximate the solution, and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multiplier functions. By means of Green's theorem, the resulting discretized equations are posed and solved on the boundary of the domain, away from the point where the singularity arises. We analyze the convergence of the method and prove that the coefficients in the local asymptotic expansion, also referred to as stress intensity factors, are approximated at an exponential rate as the number of the employed expansion terms is increased. Our theoretical results are illustrated through a numerical experiment.

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Special boundary approximation methods for laplace equation problems with boundary singularities— Applications to the motz problem

Cited by 16 publications

References 17 publications

On solving the cracked‐beam problem by block method

On solving the cracked‐beam problem by block method

Trefftz, collocation, and other boundary methods—A comparison

Analysis of the singular function boundary integral method for a biharmonic problem with one boundary singularity

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