Two-scale composite finite element method for Dirichlet problems on complicated domains


In this paper, we define a new class of finite elements for the discretization of problems with Dirichlet boundary conditions. In contrast to standard finite elements, the minimal dimension of the approximation space is independent of the domain geometry and this is especially advantageous for problems on domains with complicated micro-structures. For the proposed finite element method we prove the optimal-order approximation (up to logarithmic terms) and convergence estimates valid also in the cases when the exact solution has a reduced regularity due to re-entering corners of the domain boundary. Numerical experiments confirm the theoretical results and show the potential of our proposed method.

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  1. 1.

    Bank, R.E., Xu, J.: An Algorithm for Coarsening Unstructured Meshes. Numer. Math. 73(1), 1–36 (1996)

    Article  MathSciNet  Google Scholar 

  2. 2.

    Brenner, S., Scott, L.R.: The mathematical theory of finite element methods, Springer, 2002

  3. 3.

    Ciarlet, P.: The finite element method for elliptic problems. North-Holland, 1987

  4. 4.

    Hackbusch, W., Sauter, S.: Composite finite elements for the approximation of PDEs on domains with complicated micro-structures. Numer. Math. 75(4), 447–472 (1997)

    Article  MathSciNet  Google Scholar 

  5. 5.

    Hackbusch, W., Sauter, S.: Composite finite elements for problems containing small geometric details. Part II: Implementation and numerical results. Computing and Visualization in Science 1(1), 15–25 (1997)

    Google Scholar 

  6. 6.

    Kornhuber, R., Yserentant, H.: Multilevel Methods for Elliptic Problems on Domains not Resolved by the Coarse Grid. Contemporary Mathematics 180, 49–60 (1994)

    MATH  MathSciNet  Google Scholar 

  7. 7.

    Rech, M.: Composite finite elements: An adaptive two-scale approach to the non-conforming approximation of Dirichlet problems on complicated domains, PhD thesis, Universität Zürich (planned for 2006)

  8. 8.

    Rech, M., Repin, S., Sauter, S., Smolianski, A.: Composite finite elements for the Dirichlet problem with an a-posteriori controlled, adaptive approximation of the boundary condition (planned for 2006)

  9. 9.

    Repin, S., Sauter, S., Smolianski, A.: A posteriori error estimation for the Dirichlet problem with account of the error in the approximation of boundary conditions. Computing 70, 205–233 (2003)

    MATH  MathSciNet  Google Scholar 

  10. 10.

    Sauter, S., Warnke, R.: Extension operators and approximation on domains containing small geometric details. East-West J. Numer. Math. 7(1), 61–78 (1999)

    MathSciNet  Google Scholar 

  11. 11.

    Stein, E.: Singular integrals and differentiability properties of functions. Princeton University Press, Princeton, 1970

  12. 12.

    Strang, G., Fix. G.J.: An analysis of the finite element method. Prentice-Hall, Englewood Cliffs, New Jersey, 1973

  13. 13.

    Verfürth, R.: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley-Teubner, Chichester, 1996

  14. 14.

    Yserentant, H.: Coarse grid spaces for domains with a complicated boundary. Numerical Algorithms 21, 387–392 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  15. 15.

    Xu, J.: The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids, Computing 56, 215–235 (1996)

    Google Scholar 

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Rech, M., Sauter, S. & Smolianski, A. Two-scale composite finite element method for Dirichlet problems on complicated domains. Numer. Math. 102, 681–708 (2006).

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Mathematics Subject Classification (2000)

  • 35J20
  • 65N15
  • 65N30