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Advanced BEM for BVPs in Polygonal/Polyhedral Domains: h- and p-Versions

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Advanced Boundary Element Methods

Part of the book series: Springer Series in Computational Mathematics ((SSCM,volume 52))

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Abstract

This chapter presents, h−, p −BEM on graded meshes and hp −BEM on quasiuniform meshes for the numerical treatment of boundary value problems in polygonal and polyhedral domains. For ease of presentation we also introduce here the hp −version on geometrically graded meshes (for details and proofs see Chap. 8). For the solutions of Dirichlet and Neumann problems we present decompositions into a sum of special singularity terms (describing their edge and corner behaviors) and in regular parts (see Theorem 7.3, Theorem 7.12 for two-dimensions and Theorem 7.7, Theorem 7.16 for three dimensions). These regularity results by von Petersdorff, Stephan [425] are based on the seminal works of Dauge [141] and Kondratiev [270]. Chapter 7 is organized as follows: The results for the single layer integral equation covering the Dirichlet problem are presented in Sect. 7.1 ; those for the hypersingular integral equation covering the Neumann problem in Sect. 7.2. Then in Sect. 7.3 the proofs for the results for the integral equations on curves are given, whereas in Sect. 7.4 the results for the integral equations on surfaces. We present approximation results for solutions of the integral equations on graded meshes in 2D and 3D from the PhD thesis by von Petersdorff [423], see also [426]. Also in detail we investigate the hp −version of BEM on quasi uniform meshes on polygons based on the paper by Suri and Stephan [405]. For the p-version BEM with quasi uniform meshes on polyhedra we refer to [51, 52, 374].

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References

  1. I. Babuška, M.R. Dorr, Error estimates for the combined h and p versions of the finite element method. Numer. Math. 37, 257–277 (1981)

    Article  MathSciNet  Google Scholar 

  2. I. Babuška, B.Q. Guo, E.P. Stephan, The h-p version of the boundary element method with geometric mesh on polygonal domains. Comput. Methods Appl. Mech. Eng. 80, 319–325 (1990)

    Article  MathSciNet  Google Scholar 

  3. I. Babuška, B.Q. Guo, E.P. Stephan, On the exponential convergence of the h-p version for boundary element Galerkin methods on polygons. Math. Methods Appl. Sci. 12, 413–427 (1990)

    Article  MathSciNet  Google Scholar 

  4. I. Babuška, B.Q. Guo, M. Suri, Implementation of nonhomogeneous Dirichlet boundary conditions in the p-version of the finite element method. Impact Comput. Sci. Eng. 1, 36–63 (1989)

    Article  Google Scholar 

  5. I. Babuška, M. Suri, The h-p version of the finite element method with quasi-uniform meshes. RAIRO Modél. Math. Anal. Numér. 21, 199–238 (1987)

    Article  MathSciNet  Google Scholar 

  6. I. Babuška, M. Suri, The optimal convergence rate of the p-version of the finite element method. SIAM J. Numer. Anal. 24, 750–776 (1987)

    Article  MathSciNet  Google Scholar 

  7. I. Babuška, M. Suri, The treatment of nonhomogeneous Dirichlet boundary conditions by the p-version of the finite element method. Numer. Math. 55, 97–121 (1989)

    Article  MathSciNet  Google Scholar 

  8. A. Bespalov, N. Heuer, The p-version of the boundary element method for a three-dimensional crack problem. J. Integr. Equ. Appl. 17, 243–258 (2005)

    Article  MathSciNet  Google Scholar 

  9. A. Bespalov, N. Heuer, The p-version of the boundary element method for weakly singular operators on piecewise plane open surfaces. Numer. Math. 106, 69–97 (2007)

    Article  MathSciNet  Google Scholar 

  10. A. Bespalov, N. Heuer, The hp-version of the boundary element method with quasi-uniform meshes in three dimensions. M2AN Math. Model. Numer. Anal. 42, 821–849 (2008)

    Google Scholar 

  11. C. Carstensen, M. Maischak, D. Praetorius, E.P. Stephan, Residual-based a posteriori error estimate for hypersingular equation on surfaces. Numer. Math. 97, 397–425 (2004)

    Article  MathSciNet  Google Scholar 

  12. M. Costabel, E.P. Stephan, Curvature terms in the asymptotic expansions for solutions of boundary integral equations on curved polygons. J. Integr. Equ. 5, 353–371 (1983)

    MathSciNet  MATH  Google Scholar 

  13. M. Costabel, E.P. Stephan, Boundary Integral Equation for Mixed Boundary Value Problem in Polygonal Domains and Galerkin Approximation. Mathematical Models and Methods in Mechanics. Banach Center Publications, vol. 15 (PWN - Polish Scientific Publishers, 1985), pp. 175–251

    Google Scholar 

  14. M. Dauge, Elliptic Boundary Value Problems on Corner Domains. Smoothness and Asymptotics of Solutions. Lecture Notes in Mathematics, vol. 1341 (Springer, Berlin, 1988)

    Google Scholar 

  15. B.Q. Guo, N. Heuer, The optimal rate of convergence of the p-version of the boundary element method in two dimensions. Numer. Math. 98, 499–538 (2004)

    Article  MathSciNet  Google Scholar 

  16. B.Q. Guo, N. Heuer, The optimal convergence of the h-p version of the boundary element method with quasiuniform meshes for elliptic problems on polygonal domains. Adv. Comput. Math. 24, 353–374 (2006)

    Article  MathSciNet  Google Scholar 

  17. N. Heuer, M. Maischak, E.P. Stephan, Exponential convergence of the hp-version for the boundary element method on open surfaces. Numer. Math. 83, 641–666 (1999)

    Article  MathSciNet  Google Scholar 

  18. N. Heuer, E.P. Stephan, The hp-version of the boundary element method on polygons. J. Integr. Equ. Appl. 8, 173–212 (1996)

    Article  MathSciNet  Google Scholar 

  19. V.A. Kondratiev, Boundary value problems for elliptic equations in domains with conical and angular points. Trans. Moscow Math. Soc. 16, 227–313 (1967)

    MathSciNet  Google Scholar 

  20. V.A. Kozlov, V.G. Maz’ya, J. Rossmann, Elliptic Boundary Value Problems in Domains with Point Singularities. Mathematical Surveys and Monographs, vol. 52 (American Mathematical Society, Providence, RI, 1997)

    Google Scholar 

  21. V.G. Maz’ya, B.A. Plamenevskiı̆, The coefficients in the asymptotic expansion of the solutions of elliptic boundary value problems near an edge. Dokl. Akad. Nauk SSSR 229, 33–36 (1976)

    Google Scholar 

  22. V.G. Maz’ya, B.A. Plamenevskiı̆, The coefficients in the asymptotics of solutions of elliptic boundary value problems with conical points. Math. Nachr. 76, 29–60 (1977)

    Google Scholar 

  23. S.A. Nazarov, Asymptotic behavior of the solution of the Dirichlet problem in an angular domain with a periodically changing boundary Mat. Zametki 49, 86–96 (1991)

    MathSciNet  Google Scholar 

  24. S.A. Nazarov, B.A. Plamenevskiı̆, Elliptic Problems in Domains with Piecewise Smooth Boundaries. De Gruyter Expositions in Mathematics, vol. 13 (Walter de Gruyter & Co., Berlin, 1994)

    Google Scholar 

  25. Ch. Schwab, M. Suri, The optimal p-version approximation of singularities on polyhedra in the boundary element method. SIAM J. Numer. Anal. 33, 729–759 (1996)

    Article  MathSciNet  Google Scholar 

  26. E.P. Stephan, A boundary integral equation method for three-dimensional crack problems in elasticity. Math. Methods Appl. Sci. 8, 609–623 (1986)

    Article  MathSciNet  Google Scholar 

  27. E.P. Stephan, Boundary integral equations for screen problems in R 3. Integr. Equ. Oper. Theory 10, 236–257 (1987)

    Article  MathSciNet  Google Scholar 

  28. E.P. Stephan, M. Suri, On the convergence of the p-version of the boundary element Galerkin method. Math. Comput. 52, 31–48 (1989)

    MathSciNet  MATH  Google Scholar 

  29. E.P. Stephan, M. Suri, The h-p version of the boundary element method on polygonal domains with quasiuniform meshes. RAIRO Modél. Math. Anal. Numér. 25, 783–807 (1991)

    Article  MathSciNet  Google Scholar 

  30. E.P. Stephan, W.L. Wendland, An augmented Galerkin procedure for the boundary integral method applied to two-dimensional screen and crack problems. Appl. Anal. 18, 183–219 (1984)

    Article  MathSciNet  Google Scholar 

  31. E.P. Stephan, W.L. Wendland, An augmented Galerkin procedure for the boundary integral method applied to mixed boundary value problems. Appl. Numer. Math. 1, 121–143 (1985)

    Article  MathSciNet  Google Scholar 

  32. M.E. Taylor, Pseudodifferential Operators. Princeton Mathematical Series, vol. 34 (Princeton University Press, Princeton, NJ, 1981)

    Google Scholar 

  33. T. von Petersdorff, Randwertprobleme der Elastizitätstheorie für Polyeder - Singularitäten und Approximation mit Randelementmethoden, Ph.D. thesis, TU Darmstadt, 1989

    Google Scholar 

  34. T. von Petersdorff, E.P. Stephan, Decompositions in edge and corner singularities for the solution of the Dirichlet problem of the Laplacian in a polyhedron. Math. Nachr. 149, 71–103 (1990)

    Article  MathSciNet  Google Scholar 

  35. T. von Petersdorff, E.P. Stephan, Regularity of mixed boundary value problems in R 3 and boundary element methods on graded meshes. Math. Methods Appl. Sci. 12, 229–249 (1990)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Fakultät für Luft- und Raumfahrttechnik, Universität der Bundeswehr München, Neubiberg/München, Germany

    Joachim Gwinner

  2. Institut für Angewandte Mathematik, Leibniz Universität Hannover, Hannover, Germany

    Ernst Peter Stephan

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  1. Joachim Gwinner
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  2. Ernst Peter Stephan
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Gwinner, J., Stephan, E.P. (2018). Advanced BEM for BVPs in Polygonal/Polyhedral Domains: h- and p-Versions. In: Advanced Boundary Element Methods. Springer Series in Computational Mathematics, vol 52. Springer, Cham. https://doi.org/10.1007/978-3-319-92001-6_7

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