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Discontinuous Galerkin \(hp\)-BEM with quasi-uniform meshes

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Abstract

We present and analyze a discontinuous variant of the \(hp\)-version of the boundary element Galerkin method with quasi-uniform meshes. The model problem is that of the hypersingular integral operator on an (open or closed) polyhedral surface. We prove a quasi-optimal error estimate and conclude convergence orders which are quasi-optimal for the \(h\)-version with arbitrary degree and almost quasi-optimal for the \(p\)-version. Numerical results underline the theory.

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References

  1. Ainsworth, M., McLean, W., Tran, T.: The conditioning of boundary element equations on locally refined meshes and preconditioning by diagonal scaling. SIAM J. Numer. Anal. 36, 1901–1932 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bespalov, A., Heuer, N.: The \(p\)-version of the boundary element method for hypersingular operators on piecewise plane open surfaces. Numer. Math. 100, 185–209 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Buffa, A., Costabel, M., Sheen, D.: On traces for H(curl, \(\Omega \)) in Lipschitz domains. J. Math. Anal. Appl. 276, 845–867 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chouly, F., Heuer, N.: A Nitsche-based domain decomposition method for hypersingular integral equations. Numer. Math. 121, 705–729 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Dauge, M.: Elliptic boundary value problems on corner domains. In: Lecture Notes in Mathematics, vol. 1341, Springer, Berlin, (1988)

  8. Ervin, V.J., Heuer, N., Stephan, E.P.: On the \(h\)-\(p\) version of the boundary element method for Symm’s integral equation on polygons. Comput. Methods Appl. Mech. Eng. 110, 25–38 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gatica, G.N., Healey, M., Heuer, N.: The boundary element method with Lagrangian multipliers Numer. Methods Partial Differ. Eq. 25, 1303–1319 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Healey, M., Heuer, N.: Mortar boundary elements. SIAM J. Numer. Anal. 48, 1395–1418 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Heuer, N.: Additive Schwarz method for the \(p\)-version of the boundary element method for the single layer potential operator on a plane screen. Numer. Math. 88, 485–511 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Heuer, N.: On the equivalence of fractional-order Sobolev semi-norms. arXiv: 1211.0340, (2012)

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Heuer, N., Sayas, F.-J.: Crouzeix-Raviart boundary elements. Numer. Math. 112, 381–401 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Lions, J.-L., Magenes, E.: Non-homogeneous boundary value problems and applications I. Springer, New York (1972)

    Book  MATH  Google Scholar 

  16. Nédélec, J.-C.: Integral equations with nonintegrable kernels. Integral Equ. Oper. Theory 5, 562–572 (1982)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. von Petersdorff, T.: Randwertprobleme der Elastizitätstheorie für Polyeder - Singularitäten und Approximation mit Randelementmethoden, PhD thesis, Technische Hochschule Darmstadt, Germany (1989)

  19. von Petersdorff, T., Stephan, E.P.: Regularity of mixed boundary value problems in \({\rm I\!R}^3\) and boundary element methods on graded meshes. Math. Methods Appl. Sci. 12, 229–249 (1990)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Macul, Santiago, Chile

    Norbert Heuer

  2. Departamento de Matemáticas, Facultad de Ciencias, Universidad de Oviedo, Calvo Sotelo s/n, Oviedo, Spain

    Salim Meddahi

Authors
  1. Norbert Heuer
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  2. Salim Meddahi
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Correspondence to Norbert Heuer.

Additional information

Supported by FONDECYT project 1110324, CONICYT project Anillo ACT1118 (ANANUM) and by Ministery of Education of Spain through project MTM2010-18427.

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Heuer, N., Meddahi, S. Discontinuous Galerkin \(hp\)-BEM with quasi-uniform meshes. Numer. Math. 125, 679–703 (2013). https://doi.org/10.1007/s00211-013-0547-3

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  • DOI: https://doi.org/10.1007/s00211-013-0547-3

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