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Nyström method for BEM of the heat equation with moving boundaries

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Abstract

A direct boundary integral equation method for the heat equation based on Nyström discretization is proposed and analyzed. For problems with moving geometries, a weakly and strongly singular Green’s integral equation is formulated. Here the hypersingular integral operator, i.e., the normal trace of the double-layer potential, must be understood as a Hadamard finite part integral. The thermal layer potentials are regarded as generalized Abel integral operators in time and discretized with a singularity-corrected trapezoidal rule. The spatial discretization is a standard quadrature rule for smooth surface integrals. The discretized systems lead to an explicit time stepping scheme and is effective for solving the Dirichlet and Neumann boundary value problems based on both the weakly and/or strongly singular integral equations.

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References

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

    MathSciNet  MATH  Google Scholar 

  2. Atkinson, K.E.: The numerical solution of integral equations of the second kind. Cambridge University Press, Cambridge (1997)

    Book  Google Scholar 

  3. Brattkus, K., Meiron, D.: Numerical simulations of unsteady crystal growth. SIAM J. Appl. Math. 52, 1303–1320 (1992)

    Article  MathSciNet  Google Scholar 

  4. Costabel, M.: Boundary integral operators for the heat equation. Integr. Equ. Oper. Theory 13(4), 498–552 (1990)

    Article  MathSciNet  Google Scholar 

  5. Costabel, M.: Time-dependent problems with the boundary integral equation method. In: Stein, E., de Borst, R., Hughes, T. (eds.) Encyclopedia of computational mathematics. Wiley, New York (2004)

  6. Dohr, S.O., Steinbach, M., Of, M.G., Zapletal, J.: A parallel solver for a preconditioned space-time boundary element method for the heat equation. Technical report, arXiv (2018)

  7. Greengard, L., Lin, P.: Spectral approximation of the free-space heat kernel. Appl. Comput. Harmonic Anal. 9, 83–97 (1999)

    Article  MathSciNet  Google Scholar 

  8. Greengard, L., Strain, J.: A fast algorithm for the evaluation of heat potentials. Comm. Pure Appl. Math. XLIII, 949–963 (1990)

    Article  MathSciNet  Google Scholar 

  9. Harbrecht, H., Tausch, J.: A fast sparse grid based space–time boundary element method for the nonstationary heat equation. Numer. Math. 140(1), 239–264 (2018)

    Article  MathSciNet  Google Scholar 

  10. Kress, R.: Linear integral equations, volume 82 of applied mathematical sciences. Springer, Berlin (1989)

    Book  Google Scholar 

  11. Lubich, C., Schneider, R.: Time discretization of parabolic boundary integral equations. Numer. Math., 63(1) (1992)

    Article  MathSciNet  Google Scholar 

  12. Mason, N., Tausch, J.: Quadrature for parabolic Galerkin BEM with moving surfaces. Comput. Math. Appl. 77(1), 1–14 (2019)

    Article  MathSciNet  Google Scholar 

  13. Messner, M., Schanz, M., Tausch, J.: An efficient Galerkin boundary element method for the transient heat equation. SIAM J. Sci. Comput. 258(1), A1554–A1576 (2015)

    Article  MathSciNet  Google Scholar 

  14. Power, H., Ibanez, M. T.: An efficient direct bem numerical scheme for phase change problems using fourier series. Comput. Methods Appl. Mech. Engrg. 191, 2371–2402 (2002)

    Article  Google Scholar 

  15. Schanz, M., Antes, H.: Application of operational quadrature methods in time domain boundary element methods. Meccanica 32(3), 179–186 (1997)

    Article  Google Scholar 

  16. Tao, L., Yong, H.: A generalization of discrete Gronwall inequality and its application to weakly singular Volterra integral equation of the second kind. J. Math. Anal. Appl. 282, 56–62 (2003)

    Article  MathSciNet  Google Scholar 

  17. Tausch, J.: A fast method for solving the heat equation by layer potentials. J. Comput. Phys. 224, 956–969 (2007)

    Article  MathSciNet  Google Scholar 

  18. Tausch, J.: Nyström discretization of parabolic boundary integral equations. Appl. Numer. Math. 59(11), 2843–2856 (2009)

    Article  MathSciNet  Google Scholar 

  19. Tausch, J.: The generalized Euler-Maclaurin formula for the numerical solution of Abel-type integral equations. J. Integral Eqns. Appl. 22(1), 115–140 (2010)

    Article  MathSciNet  Google Scholar 

  20. Wang, J., Greengard, L.: Hybrid asymptotic/numerical methods for the evaluation of layer heat potentials in two dimensions. Adv. Comput. Math. 45(2), 847–867 (2019)

    Article  MathSciNet  Google Scholar 

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Funding

This material is based upon the work supported by the National Science Foundation under grant DMS-1115931.

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Authors and Affiliations

  1. Southern Methodist University, Dallas, TX, 75275, USA

    Johannes Tausch

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  1. Johannes Tausch
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Correspondence to Johannes Tausch.

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Communicated by: Leslie Greengard

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Tausch, J. Nyström method for BEM of the heat equation with moving boundaries. Adv Comput Math 45, 2953–2968 (2019). https://doi.org/10.1007/s10444-019-09720-x

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  • DOI: https://doi.org/10.1007/s10444-019-09720-x

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