On stability of approximation methods for the Muskhelishvili equation

Didenko, Victor D.; Silbermann, Bernd

doi:10.1016/s0377-0427(02)00393-x

Cited by 9 publications

(20 citation statements)

References 21 publications

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“…For the reader's convenience, we also recall some results of [7,8] concerning the operator A Γ,k ∈ B 2 (Γ) and adapt them to the situation considered here.…”

Section: φ(T)mentioning

confidence: 99%

“…Further, to study invertibility in B Γ /J Γ one can use a localizing principle. Recall that localizing principles are very efficient, and they have been successfully used in approximation methods for singular integral equations with conjugation [5], Mellin integral equations with conjugation [9], the Muskhelishvili and Sherman-Lauricella equations on piecewise smooth curves [4,7]. For the convenience of the reader, we reformulate the Allan's local principle for real C * -algebras here.…”

Section: Theorem 34 Let γ Be a Simple Closed Piecewise Smooth Contomentioning

confidence: 99%

“…It turns out now that stability depends on certain operators associated with the corner points of Γ and on certain specific parameters of the corresponding approximation method. Nevertheless, for the Galerkin method using piecewise constant splines, necessary and sufficient stability conditions have been obtained [7]. Moreover, in [7] an approximation method based on the rectangular quadrature rule has also been studied.…”

mentioning

confidence: 99%

“…Nevertheless, for the Galerkin method using piecewise constant splines, necessary and sufficient stability conditions have been obtained [7]. Moreover, in [7] an approximation method based on the rectangular quadrature rule has also been studied. The last two methods mentioned have both advantages and drawbacks.…”

mentioning

confidence: 99%

“…In the process, special grids refined in neighbourhoods of the corner points are used. We establish necessary and sufficient conditions of the stability of the method, and it is worth mentioning that the corresponding operators responsible for the stability of the spline Galerkin method from [7] have a simpler structure. Moreover, it is easily seen that those operators are always Fredholm.…”

mentioning

confidence: 99%

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On the Stability of the Nyström Method for the Muskhelishvili Equation on Contours with Corners

Didenko¹,

Helsing²

2013

SIAM J. Numer. Anal.

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Abstract. The stability of the Nyström method for the Muskhelishvili equation on piecewise smooth simple contours Γ is studied. It is shown that in the space L 2 the method is stable if and only if certain operators Aτ j from an algebra of Toeplitz operators are invertible. The operators Aτ j depend on the parameters of the equation considered, on the opening angles θ j of the corner points τ j ∈ Γ and on parameters of the approximation method mentioned. Numerical experiments show that there are opening angles where the operators Aτ j are non-invertible. Therefore, for contours with such corners the method under consideration is not stable. Otherwise, the method is always stable. Numerical examples show an excellent convergence of the method.

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“…For the reader's convenience, we also recall some results of [7,8] concerning the operator A Γ,k ∈ B 2 (Γ) and adapt them to the situation considered here.…”

Section: φ(T)mentioning

confidence: 99%

Section: Theorem 34 Let γ Be a Simple Closed Piecewise Smooth Contomentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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On the Stability of the Nyström Method for the Muskhelishvili Equation on Contours with Corners

Didenko¹,

Helsing²

2013

SIAM J. Numer. Anal.

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Approximate solution of boundary integral equations for biharmonic problems in non‐smooth domains

Didenko

Helsing

2013

Proc Appl Math and Mech

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This paper deals with approximate solutions to integral equations arising in boundary value problems for the biharmonic equation in simply connected piecewise smooth domains. The approximation method considered demonstrates excellent convergence even in the case of boundary conditions discontinuous at corner points. In an application we obtain very accurate approximations for some characteristics of two-dimensional Stokes flow in non-smooth domains.

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Mellin convolution operators in Bessel potential spaces

Didenko

Duduchava

2016

Journal of Mathematical Analysis and Applications

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Mellin convolution equations acting in Bessel potential spaces are considered. The study is based upon two results. The first one concerns the interaction of Mellin convolutions and Bessel potential operators (BPOs). In contrast to the Fourier convolutions, BPOs and Mellin convolutions do not commute and we derive an explicit formula for the corresponding commutator in the case of Mellin convolutions with meromorphic symbols. These results are used in the lifting of the Mellin convolution operators acting on Bessel potential spaces up to operators on Lebesgue spaces. The operators arising belong to an algebra generated by Mellin and Fourier convolutions acting on L p -spaces. Fredholm conditions and index formulae for such operators have been obtained earlier by R. Duduchava and are employed here. Note that the results of the present work find numerous applications in boundary value problems for partial differential equations, in particular, for equations in domains with angular points.• In general, Mellin convolution operators are not bounded in neither Besov nor Bessel potential spaces. Therefore, in order to study equations (2) in the spaces of interest, one has to find a subclass of multipliers with the boundedness property.

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On stability of approximation methods for the Muskhelishvili equation

Cited by 9 publications

References 21 publications

On the Stability of the Nyström Method for the Muskhelishvili Equation on Contours with Corners

On the Stability of the Nyström Method for the Muskhelishvili Equation on Contours with Corners

Approximate solution of boundary integral equations for biharmonic problems in non‐smooth domains

Mellin convolution operators in Bessel potential spaces

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