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International Conference on Large-Scale Scientific Computing

LSSC 2013: Large-Scale Scientific Computing pp 331–338Cite as

Mixed FEM for Second Order Elliptic Problems on Polygonal Meshes with BEM-Based Spaces

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 8353)

Abstract

We present a Boundary Element Method (BEM)-based FEM for mixed formulations of second order elliptic problems in two dimensions. The challenge, we would like to address, is a proper construction of \(\mathbf H(\mathrm {div})\)–conforming vector valued trial functions on arbitrary polygonal partitions of the domain. The proposed construction generates trial functions on polygonal elements which inherit some of the properties of the unknown solution. In the numerical realization, the relevant local problems are treated by means of boundary integral formulations. We test the accuracy of the method on two model problems.

Keywords

  • Mixed formulation
  • BEM-based FEM
  • Polygonal mesh

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References

  1. Beirão da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A.: Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23(1), 199–214 (2013)

    CrossRef  MATH  MathSciNet  Google Scholar 

  2. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer Series in Computational Mathematics, vol. 15. Springer, New York (1991)

    MATH  Google Scholar 

  3. Brezzi, F., Lipnikov, K., Shashkov, M.: Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal. 43(5), 1872–1896 (2005)

    CrossRef  MATH  MathSciNet  Google Scholar 

  4. Chen, Z., Hou, T.Y.: A mixed multiscale finite element method for elliptic problems with oscillating coefficients. Math. Comp. 72(242), 541–576 (2003)

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. Copeland D., Langer U., Pusch D.: From the boundary element domain decomposition methods to local Trefftz finite element methods on polyhedral meshes. In: Bercovier M., Gander M., Kornhuber R., Widlund O. (eds) Domain Decomposition Methods in Science and Engineering XVIII. Lecture Notes in Computational Science and Engineering (LNCSE), pp. 315–322. Springer, Heidelberg (2009)

    Google Scholar 

  6. Efendiev Y., Galvis J., Lazarov R.D., Weißer S.: Mixed formulation of BEM-based FEM for second order elliptic problems on general polygonal meshes (in progress)

    Google Scholar 

  7. Efendiev, Y., Hou, T.Y.: Multiscale Finite Element Methods. Theory and Applications. Surveys and Tutorials in the Appl. Math. Sci. Springer, New York (2009)

    MATH  Google Scholar 

  8. Kuznetsov, Y., Repin, S.: New mixed finite element method on polygonal and polyhedral meshes. Russian J. Numer. Anal. Math. Model. 18(3), 261–278 (2003)

    CrossRef  MATH  MathSciNet  Google Scholar 

  9. Rjasanow, S., Weißer, S.: Developments in BEM-based finite element methods on polygonal and polyhedral meshes. In: Eberhardsteiner, J., Böhm, H.J., Rammerstorfer, F.G. (eds.) ECCOMAS 2012, e-Book Full Papers, pp. 1421–1431. Vienna University of Technology, Austria (2012)

    Google Scholar 

  10. Rjasanow, S., Weißer, S.: Higher order BEM-based FEM on polygonal meshes. SIAM J. Numer. Anal. 50(5), 2357–2378 (2012)

    CrossRef  MATH  MathSciNet  Google Scholar 

  11. Steinbach, O.: Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements. Springer, New York (2008)

    CrossRef  Google Scholar 

  12. Weißer, S.: Residual error estimate for BEM-based FEM on polygonal meshes. Numer. Math. 118(4), 765–788 (2011)

    CrossRef  MATH  MathSciNet  Google Scholar 

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Acknowledgments

The research of Y. Efendiev, J. Galvis, and R. Lazarov has been supported in parts by award KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST). R. Lazarov is also supported in part by the award made by NSF DMS-1016525. Y. Efendiev would like to acknowledge a partial support from NSF (724704, 0811180, 0934837) and DOE.

Author information

Authors and Affiliations

  1. Texas A and M University, College Station, TX, 77843, USA

    Yalchin Efendiev & Raytcho Lazarov

  2. Matematicas, Universidad Nacional de Colombia, Carrera 45 No 26-85 - Edificio Uriel Gutierréz, Bogotá D.C., Colombia

    Juan Galvis

  3. Saarland University, Saarbrücken, Germany

    Steffen Weißer

Authors
  1. Yalchin Efendiev
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  2. Juan Galvis
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  3. Raytcho Lazarov
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  4. Steffen Weißer
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Corresponding author

Correspondence to Steffen Weißer .

Editor information

Editors and Affiliations

  1. Inst. of Info. & Comm. Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Lirkov

  2. Bulgarian Academy of Sciences, Sofia, Bulgaria

    Svetozar Margenov

  3. Dept. of Informatics and Mathematical Modell, Technical University of Denmark, Kongens Lyngby, Denmark

    Jerzy Waśniewski

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Efendiev, Y., Galvis, J., Lazarov, R., Weißer, S. (2014). Mixed FEM for Second Order Elliptic Problems on Polygonal Meshes with BEM-Based Spaces. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds) Large-Scale Scientific Computing. LSSC 2013. Lecture Notes in Computer Science(), vol 8353. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43880-0_37

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  • DOI: https://doi.org/10.1007/978-3-662-43880-0_37

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-43879-4

  • Online ISBN: 978-3-662-43880-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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