Time discretization of parabolic boundary integral equations

SummaryIn time-dependent boundary integral equations, a boundary element method in space can be coupled with a different type of discretization in time. For the latter a procedure based on linear multistep methods is proposed, which is applicable whenever the Laplace transform of the fundamental solution is known. The stability properties of the method are obtained from those of the underlying multistep method. In the absence of a space discretization, the numerical solution given by the proposed method is identical to that of a semi-discretization in time of the partial differential equation by the underlying multistep method. The theory is presented for the single layer potential equation of the heat equation. Convergence estimates, which are pointwise in time and expressed in terms of the boundary data, are obtained for full discretizations using Galerkin or collocation boundary element methods in space. Numerical examples are included.

[1]  M. S. Agranovič.  BOUNDARY VALUE PROBLEMS FOR SYSTEMS WITH A PARAMETER , 1971 .

[2]  Reinhold Schneider,et al.  Spline approximation methods for multidimensional periodic pseudodifferential equations , 1992 .

[3]  Ernst Hairer,et al.  FAST NUMERICAL SOLUTION OF NONLINEAR VOLTERRA CONVOLUTION EQUATIONS , 1985 .

[4]  Reinhold Schneider,et al.  Stability of a spline collocation method for strongly elliptic multidimensional singular integral equations , 1990 .

[5]  B. Silbermann,et al.  Numerical Analysis for Integral and Related Operator Equations , 1991 .

[6]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[7]  Martin Costabel,et al.  Spline collocation for strongly elliptic equations on the torus , 1992 .

[8]  Martin Costabel,et al.  Boundary integral operators for the heat equation , 1990 .

[9]  Heinz Antes,et al.  Anwendungen der Methode der Randelemente in der Elastodynamik und der Fluiddynamik , 1988 .

[10]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[11]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[12]  C. Lubich,et al.  On convolution quadrature and Hille-Phillips operational calculus , 1992 .

[13]  C. Lubich.  Convolution quadrature and discretized operational calculus. II , 1988 .

[14]  R. Coifman,et al.  Fast wavelet transforms and numerical algorithms I , 1991 .

[15]  Siegfried Prössdorf,et al.  Ein Lokalisierungsprinzip in der Theorie der Spline‐Approximationen und einige Anwendungen , 1984 .

[16]  Witold Pogorzelski,et al.  Integral equations and their applications , 1966 .

[17]  W. Hackbusch,et al.  Integralgleichungen : Theorie und Numerik , 1989 .

[18]  M. Shubin.  Pseudodifferential Operators and Spectral Theory , 1987 .

[19]  W. Hackbusch,et al.  On the fast matrix multiplication in the boundary element method by panel clustering , 1989 .

[20]  S. Prössdorf,et al.  Convergence of Spline Approximation Methods for Periodic Elliptic Pseudodifferential Equations , 1992 .

[21]  Wolfgang L. Wendland,et al.  On the asymptotic convergence of collocation methods with spline functions of even degree , 1985 .

[22]  H. Brezis.  Analyse fonctionnelle : théorie et applications , 1983 .

[23]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .