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Residual Based Error Estimate for Higher Order Trefftz-Like Trial Functions on Adaptively Refined Polygonal Meshes

  • Steffen WeißerEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 103)

Abstract

The BEM-based Finite Element Method is one of the promising strategies which are applicable for the approximation of boundary value problems on general polygonal and polyhedral meshes. The flexibility with respect to meshes arises from the implicit definition of trial functions in a Trefftz-like manner. These functions are treated locally by means of Boundary Element Methods (BEM). The following presentation deals with the formulation of higher order trial functions and their application in uniform and adaptive mesh refinement strategies. For the adaptive refinement, a residual based error estimate is used on general polygonal meshes for the higher order, conforming trial functions. The first numerical results, in the context of adaptive refinement, show optimal rates of convergence with respect to the number of degrees of freedom.

Keywords

Boundary Element Method Trial Function Adaptive Mesh Domain Decomposition Method Polygonal Mesh 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. Marini, A. Russo, Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23(1), 199–214 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    L. Beirão da Veiga, F. Brezzi, L. Marini, Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51(2), 794–812 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    L. Beirão da Veiga, K. Lipnikov, G. Manzini, Arbitrary-order nodal mimetic discretizations of elliptic problems on polygonal meshes. SIAM J. Numer. Anal. 49(5), 1737–1760 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    D. Copeland, U. Langer, D. Pusch, From the boundary element domain decomposition methods to local Trefftz finite element methods on polyhedral meshes, in Domain Decomposition Methods in Science and Engineering XVIII. ed. by M. Bercovier, M. Gander, R. Kornhuber, O. Widlund. Lecture Notes in Computational Science and Engineering, vol. 70 (Springer, Berlin/Heidelberg, 2009), pp. 315–322Google Scholar
  5. 5.
    D.M. Copeland, Boundary-element-based finite element methods for Helmholtz and Maxwell equations on general polyhedral meshes. Int. J. Appl. Math. Comput. Sci. 5(1), 60–73 (2009)MathSciNetGoogle Scholar
  6. 6.
    W. Dörfler, A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33(3), 1106–1124 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Y. Efendiev, J. Galvis, R. Lazarov, S. Weißer, Mixed FEM for second order elliptic problems on polygonal meshes with BEM-based spaces, in Large-Scale Scientific Computing, ed. by I. Lirkov, S. Margenov, J. Waśniewski. Lecture Notes in Computer Science (Springer, Berlin/Heidelberg, 2014), pp. 331–338Google Scholar
  8. 8.
    R. Hiptmair, A. Moiola, I. Perugia, Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations. Math. Comput. 82(281), 247–268 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    C. Hofreither, L 2 error estimates for a nonstandard finite element method on polyhedral meshes. J. Numer. Math. 19(1), 27–39 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    _, A non-standard finite element method using boundary integral operators, Ph.D. thesis, Johannes Kepler University, Linz, Dec 2012Google Scholar
  11. 11.
    C. Hofreither, U. Langer, C. Pechstein, Analysis of a non-standard finite element method based on boundary integral operators. Electron. Trans. Numer. Anal. 37, 413–436 (2010)zbMATHMathSciNetGoogle Scholar
  12. 12.
    _____________________________, A non-standard finite element method for convection-diffusion-reaction problems on polyhedral meshes. AIP Conf. Proc. 1404(1), 397–404 (2011)Google Scholar
  13. 13.
    P. Joshi, M. Meyer, T. DeRose, B. Green, T. Sanocki, Harmonic coordinates for character articulation. ACM Trans. Graph. 26(3), 71.1–71.9 (2007)CrossRefGoogle Scholar
  14. 14.
    S. Martin, P. Kaufmann, M. Botsch, M. Wicke, M. Gross, Polyhedral finite elements using harmonic basis functions. Comput. Graph. Forum 27(5), 1521–1529 (2008)CrossRefGoogle Scholar
  15. 15.
    W.C.H. McLean, Strongly Elliptic Systems and Boundary Integral Equations (Cambridge University Press, Cambridge, 2000)zbMATHGoogle Scholar
  16. 16.
    C. Pechstein, C. Hofreither, A rigorous error analysis of coupled FEM-BEM problems with arbitrary many subdomains. Lect. Notes Appl. Comput. Mech. 66, 109–132 (2013)CrossRefMathSciNetGoogle Scholar
  17. 17.
    S. Rjasanow, O. Steinbach, The Fast Solution of Boundary Integral Equations. Mathematical and Analytical Techniques with Applications to Engineering (Springer, New York/London, 2007)Google Scholar
  18. 18.
    S. Rjasanow, S. Weißer, Higher order BEM-based FEM on polygonal meshes. SIAM J. Numer. Anal. 50(5), 2357–2378 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    _____, FEM with Trefftz trial functions on polyhedral elements. J. Comput. Appl. Math. 263, 202–217 (2014)Google Scholar
  20. 20.
    A. Tabarraei, N. Sukumar, Application of polygonal finite elements in linear elasticity. Int. J. Comput. Methods 3(4), 503–520 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    S. Weißer, Residual error estimate for BEM-based FEM on polygonal meshes. Numer. Math. 118(4), 765–788 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    ____________________________, Finite Element Methods with local Trefftz trial functions, Ph.D. thesis, Universität des Saarlandes, Saarbrücken, Sept 2012Google Scholar
  23. 23.
    _______________________________________________________________________, Arbitrary order Trefftz-like basis functions on polygonal meshes and realization in BEM-based FEM. Comput. Math. Appl. 67(7), 1390–1406 (2014)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Universität des SaarlandesSaarbrückenGermany

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