On two methods for elimination of non-unique solutions of an integral equation with logarithmic kernel/<sup>†</sup>

Christiansen, S⊘ren

doi:10.1080/00036818208839372

Cited by 49 publications

(18 citation statements)

References 17 publications

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“…The results show how the condition number depends on the radius of the circle or the aspect ratio of the ellipse. Comparable results are given in [3]. In [4] and [5] the Laplace equation on a circle with Dirichlet boundary conditions is again investigated.…”

Section: Introductionmentioning

confidence: 90%

The condition number of the BEM-matrix arising from Laplace's equation

Dijkstra

Mattheij

2007

EJBE

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We investigate the condition number of the matrices that appear in the boundary element method. In particular we consider the Laplace equation with mixed boundary conditions. For Dirichlet boundary conditions, the condition number of the system matrix increases linearly with the number of boundary elements. We extend the research and search for a relation between the condition number and the number of elements in the case of mixed boundary conditions. In the case of a circular domain, we derive an estimate for the condition number of the system matrix. This matrix consists of two blocks, each block originating from a well-conditioned matrix. We show that the block matrix is also wellconditioned.

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Section: Introductionmentioning

confidence: 90%

The condition number of the BEM-matrix arising from Laplace's equation

Dijkstra

Mattheij

2007

EJBE

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“…has a non-trivial solution for 1 (s), where B d is the boundary of degenerate scale using the fundamental solution U(s, x) = ln(r) [13,[15][16][17][32][33][34]. For determining the degenerate scale systematically from one trial of a normal scale, we provide a flowchart as shown in Figure 2(a) and the numerical results as shown in Table I.…”

Section: Dual Boundary Integral Formulation and Dual Bem For Torsion mentioning

confidence: 99%

“…Burton and Miller [3] solved the problem by combining singular and hypersingular equations with an imaginary constant. Chen et al [4] 235 methods to eliminate degenerate scale, scaling method and restriction method were discussed by Christiansen [33]. He also investigated the condition number of the influence matrix of the fictitious BEM and null-field approach [34].…”

Section: Introductionmentioning

confidence: 99%

Degenerate scale problem when solving Laplace's equation by BEM and its treatment

Chen

Lin

Chen

2004

Numerical Meth Engineering

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SUMMARYIn this paper, Laplace problems are solved by using the dual boundary element method (BEM). It is found that a degenerate scale problem occurs if the conventional BEM is used. In this case, the influence matrix is rank deficient and numerical results become unstable. Both the circular and elliptical bars are studied analytically in the continuous system. In the discrete system, the Fredholm alternative theorem in conjunction with the SVD (Singular Value Decomposition) updating documents is employed to sort out the spurious mode which causes the numerical instability. Three regularization techniques, method of adding a rigid body mode, hypersingular formulation and CHEEF (Combined Helmholtz Exterior integral Equation Formulation) concept, are employed to deal with the rankdeficiency problem. The addition of a rigid body term, c, in the fundamental solution is proved to shift the original degenerate scale to a new degenerate scale by a factor e −c . The torsion rigidities are obtained and compared with analytical solutions. Numerical examples including elliptical, square and triangular bars were demonstrated to show the failure of conventional BEM in case of the degenerate scale. After employing the three regularization techniques, the accuracy of the proposed approaches is achieved.

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“…The eigenvalues Xp have been discussed in the work of F. R. de Hoog [ 10] and S. Christiansen [7]. However, the eigenvalues A given in (4) In the following it is shown that co is an upper bound for \B |0, and that this bound becomes increasingly sharp as n -> co. Theorem 2 tells us that <y can be viewed essentially as the value of \B when n is sufficiently large.…”

Section: Collocation On a Circlementioning

confidence: 90%

The Collocation Method for First-Kind Boundary Integral Equations on Polygonal Regions

Yao¹

1990

Mathematics of Computation

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Abstract.In this paper the collocation method for first-kind boundary integral equations, by using piecewise constant trial functions with uniform mesh, is shown to be equivalent to a projection method for second-kind Fredholm equations. In a certain sense this projection is an interpolation projection. By introducing this technique of analysis, we particularly consider the case of polygonal boundaries. We give asymptotic error estimates in L2 norm on the boundaries, and some superconvergence results for the single layer potential.

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On two methods for elimination of non-unique solutions of an integral equation with logarithmic kernel/^†

Cited by 49 publications

References 17 publications

The condition number of the BEM-matrix arising from Laplace's equation

The condition number of the BEM-matrix arising from Laplace's equation

Degenerate scale problem when solving Laplace's equation by BEM and its treatment

The Collocation Method for First-Kind Boundary Integral Equations on Polygonal Regions

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On two methods for elimination of non-unique solutions of an integral equation with logarithmic kernel/†

Cited by 49 publications

References 17 publications

The condition number of the BEM-matrix arising from Laplace's equation

The condition number of the BEM-matrix arising from Laplace's equation

Degenerate scale problem when solving Laplace's equation by BEM and its treatment

The Collocation Method for First-Kind Boundary Integral Equations on Polygonal Regions

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On two methods for elimination of non-unique solutions of an integral equation with logarithmic kernel/^†