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Boundary integral approximation of a heat-diffusion problem in time-harmonic regime

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In this paper we propose and analyse numerical methods for the approximation of the solution of Helmholtz transmission problems in the half plane. The problems we deal with arise from the study of some models in photothermal science. The solutions to the problem are represented as single layer potentials and an equivalent system of boundary integral equations is derived. We then give abstract necessary and sufficient conditions for convergence of Petrov–Galerkin discretizations of the boundary integral system and show for three different cases that these conditions are satisfied. We extend the results to other situations not related to thermal science and to non-smooth interfaces. Finally, we propose a simple full discretization of a Petrov–Galerkin scheme with periodic spline spaces and show some numerical experiments.

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References

  1. A.J. Ångström, Ann. Physik. Lpz. 114 (1861).

  2. D.P. Almond and P.M. Patel, Phototermal Science and Techniques (Chapman and Hall, London, 1996).

    Google Scholar 

  3. D.N. Arnold, A spline-trigonometric Galerkin method and an exponentially convergent boundary integral method, Math. Comput. 41 (1983) 383–397.

    Article  MATH  Google Scholar 

  4. S.E. Bialkowski, Photothermal Spectroscopy Methods for Chemical Analysis (Wiley, New York, 1996).

    Google Scholar 

  5. G. Chen and J. Zhou, Boundary Element Methods (Academic, London, 1992).

    MATH  Google Scholar 

  6. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd edition (Springer, Berlin Heidelberg New York, 1998).

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. M. Costabel and E. Stephan, A direct boundary integral equation method for transmission problems, J. Math. Anal. Appl. 106 (1985) 367–413.

    Article  MATH  MathSciNet  Google Scholar 

  9. M. Crouzeix and F.J. Sayas, Asymptotic expansions of the error of spline Galerkin boundary element methods, Numer. Math. 78 (1998) 523–547.

    Article  MATH  MathSciNet  Google Scholar 

  10. V. Domínguez and F.J. Sayas, An asymptotic series approach to qualocation methods, J. Integral Equations Appl. 15 (2003) 113–151.

    Article  MATH  MathSciNet  Google Scholar 

  11. M. Feistauer, J. Felcman and M. Lukáčová-Medvid'ová, On the convergence of a combined finite volume-finite element method for nonlinear convection-diffusion problems, Num. Methods Part. Diff. Eq. 13 (1997) 163–190.

    Article  MATH  Google Scholar 

  12. G.C. Hsiao, P. Kopp and W.L. Wendland, A Galerkin collocation method for some integral equations of the first kind, Computing 25 (1980) 89–130.

    Article  MATH  MathSciNet  Google Scholar 

  13. G.C. Hsiao, P. Kopp and W.L. Wendland, Some applications of a Galerkin-collocation method for boundary integral equations of the first kind, Math. Methods Appl. Sci. 6 (1984) 280–325.

    Article  MATH  MathSciNet  Google Scholar 

  14. T. Hohage and F.J. Sayas, Numerical solution of a heat diffusion problem by boundary integral equation methods using the Laplace transform, to appear in Numer. Math.

  15. A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems (Springer, Berlin Heidelberg New York, 1996).

    MATH  Google Scholar 

  16. R.E. Kleinman and P.A. Martin, On single integral equations for the transmission problem of acoustics, SIAM J. Appl. Math 48 (1988) 307–325.

    Article  MathSciNet  Google Scholar 

  17. R. Kress, Linear Integral Equations, 2nd edition (Springer, Berlin Heidelberg New York, 1999).

    MATH  Google Scholar 

  18. R. Kress and G.F. Roach, Transmission problems for the Helmholtz equation, J. Math. Phys. 19 (1978) 1433–1437.

    Article  MATH  MathSciNet  Google Scholar 

  19. A. Mandelis, Photoacoustic and Thermal Wave Phenomena in Semiconductors (North-Holland, New York, 1987).

    Google Scholar 

  20. A. Mandelis, Diffusion waves and their uses, Physics Today 53 (2000) 29–34.

    Article  Google Scholar 

  21. W. McLean, Strongly Elliptic Systems and Boundary Integral Equations (Cambridge University Press, Cambridge, 2000).

    MATH  Google Scholar 

  22. S. Meddahi and F.J. Sayas, Analysis of a new BEM-FEM coupling for two dimensional fluid-solid iteraction, Num. Methods Part. Diff. Eq. 21 (2005) 1017–1154

    Article  MATH  MathSciNet  Google Scholar 

  23. S. Prössdorf and B. Silbermann, Numerical Analysis for Integral and Related Operator Equations (Akademie-Verlag, Berlin, 1991).

    Google Scholar 

  24. M.L. Rapún, Numerical Methods for the Study of the Scattering of Thermal Waves, Ph.D. Thesis, University of Zaragoza, 2004 (in Spanish).

  25. J. Saranen and G. Vainikko, Periodic Integral and Pseudodifferential Equations with Numerical Approximation (Springer, Berlin Heidelberg New York, 2002).

    MATH  Google Scholar 

  26. O. Steinbach, On a generalized \(L^2\) projection and some related stability estimates in Sobolev spaces, Numer. Math. 90 (2002) 775–786.

    Article  MATH  MathSciNet  Google Scholar 

  27. O. Steinbach and W.L. Wendland, On C. Neumann's method for second-order elliptic systems in domains with non-smooth boundaries, J. Math. Anal. Appl. 262 (2001) 733–748.

    Article  MATH  MathSciNet  Google Scholar 

  28. D. Sheen, I.H. Sloan and V. Thomée, A parallel method for time-discretization of parabolic problems based on contour integral representation and quadrature, Math. Comp. 69 (2000) 177–195.

    Article  MATH  MathSciNet  Google Scholar 

  29. D. Sheen, I.H. Sloan and V. Thomée, A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature, IMA J. Numer. Anal. 23 (2003) 269–299.

    Article  MATH  MathSciNet  Google Scholar 

  30. J.M. Terrón, A. Sánchez-Lavega and A. Salazar, Multiple scattering of thermal waves by a coated subsurface cylindrical inclusion, J. Appl. Phys. 89 (2001) 5696–5702.

    Article  Google Scholar 

  31. J.M. Terrón, A. Salazar and A. Sánchez-Lavega, General solution for the thermal wave scattering in fiber composites, J. Appl. Phys. 91 (2002) 1087–1098.

    Article  Google Scholar 

  32. R.H. Torres and G.V. Welland, The Helmholtz equation and transmission problems with Lipschitz interfaces, Indiana Univ. Math. J. 42 (1993) 1457–1485.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  34. J. Xu and L. Zikatanov, Some observations on Babuška and Brezzi theories, Numer. Math. 94 (2003) 195–202.

    Article  MATH  MathSciNet  Google Scholar 

  35. A. Yodh and B. Chance, Spectroscopy and imaging with diffusing light, Physics Today 3 (1995) 34–40.

    Article  Google Scholar 

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Correspondence to María-Luisa Rapún.

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Rapún, ML., Sayas, FJ. Boundary integral approximation of a heat-diffusion problem in time-harmonic regime. Numer Algor 41, 127–160 (2006). https://doi.org/10.1007/s11075-005-9002-6

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  • DOI: https://doi.org/10.1007/s11075-005-9002-6

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