Spline collocation for convolutional parabolic boundary integral equations

Costabel, Martin; Saranen, Jukka

doi:10.1007/s002110050004

Cited by 9 publications

(16 citation statements)

References 23 publications

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“…This latter condition arises in a natural way by imposing the vanishing initial condition. Concerning the assumption which relates the order of the operator to the degree of the splines we have not been able to obtain here exactly the same results as in [8]. However, our new condition is not too restrictive in the usual applications; for example for the single layer heat operator it just puts the piecewise constant splines in the spatial variable out of consideration.…”

Section: Introductionmentioning

confidence: 54%

“…In this work we extend the spline collocation results of [8] to the case of smooth boundary curves. The analysis is carried out by applying a localization method.…”

Section: Introductionmentioning

confidence: 98%

“…However, our new condition is not too restrictive in the usual applications; for example for the single layer heat operator it just puts the piecewise constant splines in the spatial variable out of consideration. In the first paper [8] we have considered also an improved approximation result which is obtained when using splines of even degree with respect to the spatial variable. We expect that this result remains valid also for the present case of a general smooth boundary but we do not consider this question in the present paper.…”

Section: Introductionmentioning

confidence: 98%

“…In [8] we proved stability and optimal order error estimates for case of circular domains. The analysis was mainly based on Fourier technique where the operator is given as a parabolic pseudodifferential operator with a convolutional principal part.…”

Section: Introductionmentioning

confidence: 98%

“…In this paper we continue our study [8] concerning the spline collocation solution of the parabolic boundary integral equations in the plane. In [8] we proved stability and optimal order error estimates for case of circular domains.…”

Section: Introductionmentioning

confidence: 99%

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The spline collocation method for parabolic boundary integral equations on smooth curves

Costabel

Saranen

2003

Numerische Mathematik

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We consider the spline collocation method for a class of parabolic pseudodifferential operators. We show optimal order convergence results in a large scale of anisotropic Sobolev spaces. The results cover the classical boundary integral equations for the heat equation in the general case where the spatial domain has a smooth boundary in the plane. Our proof is based on a localization technique for which we use our recent results proved for parabolic pseudodifferential operators. For the localization we need also some special spline approximation results in anisotropic Sobolev spaces.

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Section: Introductionmentioning

confidence: 54%

“…In this work we extend the spline collocation results of [8] to the case of smooth boundary curves. The analysis is carried out by applying a localization method.…”

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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The spline collocation method for parabolic boundary integral equations on smooth curves

Costabel

Saranen

2003

Numerische Mathematik

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Time‐Dependent Problems with the Boundary Integral Equation Method

Costabel¹

2004

Encyclopedia of Computational Mechanics

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Time-dependent problems that are modeled by initial-boundary value problems for parabolic or hyperbolic partial differential equations can be treated with the boundary integral equation method. The ideal situation is when the right-hand side in the partial differential equation and the initial conditions vanish, the data are given only on the boundary of the domain, the equation has constant coefficients, and the domain does not depend on time. In this situation, the transformation of the problem to a boundary integral equation follows the same well-known lines as for the case of stationary or time-harmonic problems modeled by elliptic boundary value problems. The same main advantages of the reduction to the boundary prevail: Reduction of the dimension by one, and reduction of an unbounded exterior domain to a bounded boundary.There are, however, specific difficulties due to the additional time dimension: Apart from the practical problems of increased complexity related to the higher dimension, there can appear new stability problems. In the stationary case, one often has unconditional stability for reasonable approximation methods, and this stability is closely related to variational formulations based on the ellipticity of the underlying boundary value problem. In the timedependent case, instabilities have been observed in practice, but due to the absence of ellipticity, the stability analysis is more difficult and fewer theoretical results are available.In this article, the mathematical principles governing the construction of boundary integral equation methods for time-dependent problems are presented. We describe some of the main algorithms that are used in practice and have been analyzed in the mathematical literature.

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Time‐Dependent Problems with the Boundary Integral Equation Method

Costabel

2017

Encyclopedia of Computational Mechanics Second Edition

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Time‐dependent problems that are modeled by initial‐boundary value problems for parabolic or hyperbolic partial differential equations can be treated with the boundary integral equation (BIE) method. The ideal situation is when the right‐hand side in the partial differential equation and the initial conditions vanish, the data are given only on the boundary of the domain, the equation is linear with constant coefficients, and the domain does not depend on time. In this situation, the transformation of the problem to a BIE follows the same well‐known lines as for the case of stationary or time‐harmonic problems modeled by elliptic boundary value problems. The same main advantages of the reduction to the boundary prevail: reduction of the dimension by one, and reduction of an unbounded exterior domain to a bounded boundary. There are, however, specific difficulties due to the additional time dimension. Apart from the practical problems of increased complexity related to the higher dimension, there can appear new stability problems. In the stationary case, one often has unconditional stability for reasonable approximation methods, and this stability is closely related to variational formulations based on the ellipticity of the underlying boundary value problem. In the time‐dependent case, instabilities have been observed in practice, but due to the absence of ellipticity, the stability analysis is more difficult and fewer theoretical results are available. In this chapter, the mathematical principles governing the construction of BIE methods for time‐dependent problems are presented. We describe some of the main numerical algorithms that are used in practice and have been analyzed in the mathematical literature.

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Spline collocation for convolutional parabolic boundary integral equations

Cited by 9 publications

References 23 publications

The spline collocation method for parabolic boundary integral equations on smooth curves

The spline collocation method for parabolic boundary integral equations on smooth curves

Time‐Dependent Problems with the Boundary Integral Equation Method

Time‐Dependent Problems with the Boundary Integral Equation Method

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