The modified quadrature method for classical pseudodifferential equations of negative order on smooth closed curves

Saranen, Jukka; Schroderus, L.

doi:10.1016/0377-0427(94)90322-0

Cited by 10 publications

(2 citation statements)

References 9 publications

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“…There, optimal convergence order in the scale of periodic Sobolev space is established, but only the case of index 0 is treated, a simple matrix form of the method. Then, J. Saranen and L. Schroderus extended some of the results in [19] to the scale of periodic Hölder spaces [20].…”

Section: Introductionmentioning

confidence: 96%

On the collocation methods for singular integral equations with Hilbert kernel

2008

Math. Comp.

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In the present paper, we introduce some singular integral operators, singular quadrature operators and discretization matrices of singular integral equations with Hilbert kernel. These results both improve the classical theory of singular integral equations and develop the theory of singular quadrature with Hilbert kernel. Then by using them a unified framework for various collocation methods of numerical solutions of singular integral equations with Hilbert kernel is given. Under the framework, it is very simple and obvious to obtain the coincidence theorem of collocation methods, then the existence and convergence for constructing approximate solutions are also given based on the coincidence theorem.

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Section: Introductionmentioning

confidence: 96%

On the collocation methods for singular integral equations with Hilbert kernel

2008

Math. Comp.

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“…Moreover, we present maximum norm error estimates in the case of boundary integral operators of integer order. Concerning the error analysis, we utilize the approach relying on the concepts of stability, consistency and convergence known from [6,8] and also from [9,10,11]. Our analysis is different from that of [3, 41 especially because of the consistency estimates, describing the accuracy of approximation when discretizing operators by using product integration.…”

Section: Introductionmentioning

confidence: 99%

Convergence Results for Discrete Trigonometric Collocation Methods with Product Integration in Hölder-Zygmund Spaces

Schroderus¹

1997

Z. Anal. Anwend.

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In this paper convergence results with respect to Hölder-Zygmund norms-including also maximum norm error estimates-are derived for the fully discrete trigonometric collocation method presented earlier by Saranen and Vainikko for solution of boundary integral equations on smooth closed curves. Approximation of the integral operator is based on product integration for which the explicit Fourier representation of the main part is not needed, and still the convergence of arbitrarily high rate for smooth solutions can be achieved. Saranen and Vainikko have given their analysis with respect to Sobolev norms yielding results that do not imply pointwise error estimates of optimal order. In this work the approach is based on the use of Hölder-Zygmund norms, and the optimal order maximum norm estimates are accomplished.

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Two‐Grid Solution of Symm's Integral Equation

Saranen

Vainikko

1996

Mathematische Nachrichten

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We propose two-grid iteration methods for Symm's integral equation discretized by quadrature-collocation or quadrature methods. Asymptotically the optimal order of error estimate is achieved already on the fist iteration, for some modifications on the second iteration. This enables us to introduce some solvers which are of the optimal convergence order and cheap in a practical implementation; the cost varies between 0 (N') and O(N1og N ) arithmetic operations. Numerical experiments confirm the approximation properties of the schemes. doesn't equal 1, see [13], [18]. Introduce the standard parametrization zy(t> = e -l J 2 cos 2nt , z i ( t ) = e-'/'sin 2nt of the circle ro of radius e-l/'. One can decompose equation (1.2) into Integral Equations Appl., 1 (1988), 549-579 485-495

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The modified quadrature method for classical pseudodifferential equations of negative order on smooth closed curves

Cited by 10 publications

References 9 publications

On the collocation methods for singular integral equations with Hilbert kernel

On the collocation methods for singular integral equations with Hilbert kernel

Convergence Results for Discrete Trigonometric Collocation Methods with Product Integration in Hölder-Zygmund Spaces

Two‐Grid Solution of Symm's Integral Equation

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