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Boundary integral operators for the heat equation

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Abstract

We study the integral operators on the lateral boundary of a space-time cylinder that are given by the boundary values and the normal derivatives of the single and double layer potentials defined with the fundamental solution of the heat equation. For Lipschitz cylinders we show that the 2×2 matrix of these operators defines a bounded and positive definite bilinear form on certain anisotropic Sobolev spaces. By restriction, this implies the positivity of the single layer heat potential and of the normal derivative of the double layer heat potential. Continuity and bijectivity of these operators in a certain range of Sobolev spaces are also shown. As an application, we derive error estimates for various Galerkin methods. An example is the numerical approximation of an eddy current problem which is an interface problem with the heat equation in one domain and the Laplace equation in a second domain. Results of numerical computations for this problem are presented.

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References

  1. M. Abramowitz, I. A. Stegun,Handbook of Mathematical Functions, Dover Publications, New York 1972.

    Google Scholar 

  2. D. N. Arnold, P. J. Noon. Coercivity of the single layer heat potential.J. Comput. Math. 7 (1989) 100–104.

    Google Scholar 

  3. C. A. Brebbia, J. C. F. Telles, L. C. Wrobel.Boundary Element Techniques. Springer-Verlag, Berlin 1984.

    Google Scholar 

  4. C. A. Brebbia, L. A. Wrobel. The solution of parabolic problems using the dual reciprocity boundary element. InT. A. Cruse, editor,Advanced Boundary Element Methods, pages 55–72. Springer-Verlag, Berlin 1988.

    Google Scholar 

  5. R. M. Brown. Layer potentials and boundary value problems for the heat equation on Lipschitz cylinders. Thesis, University of Minnesota 1987.

  6. R. M. Brown. The method of layer potentials for the heat equation in Lipschitz cylinders.Amer. J. Math. 111 (1989) 339–379.

    Google Scholar 

  7. J. Chazarain, A. Piriou.Introduction à la Théorie des Equations aux Dérivées Partielles Linéaires. Gauthier-Villars, Paris 1981.

    Google Scholar 

  8. R. R. Coifman, A. McIntosh, I. Meyer. L'intégrale de Cauchy définit un opérateur borné surL 2 pour les courbes lipschitziennes.Ann. of Math. 116 (1982) 361–387.

    Google Scholar 

  9. M. Costabel.Starke Elliptizität von Randintegraloperatoren erster Art. Habilitationsschrift. THD-Preprint 982, Technische Hochschule Darmstadt 1984.

  10. M. Costabel. Boundary integral operators on Lipschitz domains: Elementary results.SIAM J. Math. Anal. 19 (1988) 613–626.

    Google Scholar 

  11. M. Costabel, V. J. Ervin, E. P. Stephan. Symmetric coupling of finite elements and boundary elements for a parabolic-elliptic interface problem.Quart. Appl. Math. To appear.

  12. M. Costabel, K. Onishi, W. L. Wendland. A boundary element Galerkin method for the Neumann problem of the heat equation. InH. W. Engl, C. W. Groetsch, editors,Inverse and Ill-posed Problems, pages 369–384. Academic Press 1987.

  13. M. Costabel, E. P. Stephan. An improved boundary element Galerkin method for three-dimensional crack problems.Integral Equations Oper. Theory 10 (1987) 467–504.

    Google Scholar 

  14. M. Costabel, E. P. Stephan. Strongly elliptic boundary integral equations for electromagnetic transmission problems.Proc. Royal Soc. Edinburgh 109A (1988) 271–296.

    Google Scholar 

  15. M. Costabel, W. L. Wendland. Strong ellipticity of boundary integral operators.J. Reine Angew. Math. 372 (1986) 39–63.

    Google Scholar 

  16. J. Dieudonné.Eléments d'Analyse, volume 8. Gauthier-Villars, Paris 1978.

    Google Scholar 

  17. E. Fabes. Layer potential methods for boundary value problems on Lipschitz domains. InJ. Král, J. Lukeš, I. Netuka, J. Veselý, editors,Potential Theory-Surveys and Problems, pages 55–80. Springer-Verlag, Berlin-Heidelberg-New York 1988.

    Google Scholar 

  18. E. B. Fabes, N. M. Rivière. Dirichlet and Neumann problems for the heat equation inC 1-cylinders.AMS Proc. Symp. Pure Math. XXXV (1979).

  19. E. B. Fabes, S. Salsa. Estimates of caloric measure and the initial-Dirichlet problem for the heat equation in Lipschitz cylinders.Trans. Amer. Math. Soc. 279 (1983) 635–650.

    Google Scholar 

  20. P. Grisvard.Boundary Value Problems in Non-Smooth Domains. Pitman, London 1985.

    Google Scholar 

  21. G. Grubb.Functional Calculus of Pseudo-Differential Boundary Value Problems. Birkhäuser, Boston 1986.

    Google Scholar 

  22. A. Hammoudi. Equation de la chaleur sur une base polygonale. Thesis, Université de Nantes 1987.

  23. A. Hammoudi. Equation de la chaleur sur une base polygonale.C. R. Acad. Sci. Paris 306 (1988) 165–170.

    Google Scholar 

  24. S. Hariharan, R. C. MacCamy. An integral equation procedure for eddy current problems.J. Comput. Phys. 45 (1982) 80–99.

    Google Scholar 

  25. F. K. Hebeker, G. C. Hsiao. On a boundary integral equation approach to a nonstationary problem of isothermal viscous compressible flows. To appear.

  26. D. S. Jerison, C. E. Kenig. The Dirichlet problem in nonsmooth domains.Ann. of Math. 113 (1981) 367–382.

    Google Scholar 

  27. D. S. Jerison, C. E. Kenig. The Neumann problem on Lipschitz domains.Bull. Amer. Math. Soc. 4 (1981) 203–207.

    Google Scholar 

  28. D. S. Jerison, C. E. Kenig. Boundary value problems on Lipschitz domains. InW. Littmann, editor,Studies in Partial Differential Equations. MAA Studies in Mathematics 23, pages 1–68. Math. Assoc. of America, Washington, D. C. 1982.

    Google Scholar 

  29. J. L. Lions.Equations Differentielles Opérationelles et Problèmes aux Limites. Springer-Verlag, Berlin 1961.

    Google Scholar 

  30. J. L. Lions, E. Magenes.Nonhomogeneous Boundary Value Problems and Applications, volume 1. Springer-Verlag, Berlin 1972.

    Google Scholar 

  31. J. L. Lions, E. Magenes.Nonhomogeneous Boundary Value Problems and Applications, volume 2. Springer-Verlag, Berlin 1972.

    Google Scholar 

  32. R. C. MacCamy, M. Suri. A time-dependent interface problem for two-dimensional eddy currents.Quart. Appl. Math. 44 (1987) 675–690.

    Google Scholar 

  33. J. Nečas.Les Méthodes Directes en Théorie des Equations Elliptiques. Masson, Paris 1967.

    Google Scholar 

  34. J.-C. Nédélec. Integral equations with non integrable kernels.Integral Equations Oper. Theory 5 (1982) 562–572.

    Google Scholar 

  35. J.-C. Nédélec. Equations intégrales associées aux problèmes aux limites elliptiques dans des domaines de ℝ3. InR. Dautray, J.-L. Lions, editors,Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques, chapter XI–XIII. Masson, Paris 1988.

    Google Scholar 

  36. P. J. Noon. The single layer heat potential and Galerkin boundary element methods for the heat equation. Thesis, University of Maryland 1988.

  37. K. Onishi. Galerkin method for boundary integral equations in transient heat conduction. InC. A. Brebbia, W. L. Wendland, G. Kuhn, editors,Boundary Elements IX, volume 3, pages 231–248. Springer-Verlag, Berlin 1987

    Google Scholar 

  38. T. V. Petersdorff. Boundary integral equations for mixed Dirichlet, Neumann and transmission problems.Math. Meth. Appl. Sci. 11 (1989) 185–213.

    Google Scholar 

  39. T. V. Petersdorff, E. P. Stephan. Improved boundary element methods for mixed boundary value problems in ℝ3. To appear.

  40. A. Piriou. Une classe d'opérateurs pseudo-différentiels du type de Volterra.Ann. Inst. Fourier Grenoble 20,1 (1970) 77–94.

    Google Scholar 

  41. A. Piriou. Problèmes aux limites généraux pour des opérateurs differentiels paraboliques dans un domaine borné.Ann. Inst. Fourier Grenoble 21,1 (1971) 59–78.

    Google Scholar 

  42. W. Pogorzelski.Integral Equations and their Applications. Pergamon Press, Oxford 1966.

    Google Scholar 

  43. S. Rempel, B.-W. Schulze.Index Theory for Elliptic Boundary Value Problems. Akademie-Verlag, Berlin 1983.

    Google Scholar 

  44. F. Sgallari. A weak formulation of boundary integral equations for time dependent parabolic problems.Appl. Math. Modelling 9 (1985) 295–301.

    Google Scholar 

  45. E. P. Stephan. Boundary integral equations for screen problems in ℝ3 Integral Equations Oper. Theory 10 (1987) 236–257.

    Google Scholar 

  46. G. Verchota. Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains.J. Funct. Anal. 59 (1984) 572–611.

    Google Scholar 

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Authors and Affiliations

  1. Technische Hochschule Darmstadt, Schloßgartenstraße 7, D-6100, Darmstadt, W. Germany

    Martin Costabel

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  1. Martin Costabel
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Parts of this work were done while the author had visiting positions at the Carnegie Mellon University, Pittsburgh, USA, and at the Université de Nantes, France, or was supported by the DFG-Forschergruppe KO 634/32-1.

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Costabel, M. Boundary integral operators for the heat equation. Integr equ oper theory 13, 498–552 (1990). https://doi.org/10.1007/BF01210400

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