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Error estimates for the combinedh andp versions of the finite element method

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Summary

In theh-version of the finite element method, convergence is achieved by refining the mesh while keeping the degree of the elements fixed. On the other hand, thep-version keeps the mesh fixed and increases the degree of the elements. In this paper, we prove estimates showing the simultaneous dependence of the order of approximation on both the element degrees and the mesh. In addition, it is shown that a proper design of the mesh and distribution of element degrees lead to a better than polynomial rate of convergence with respect to the number of degrees of freedom, even in the presence of corner singularities. Numerical results comparing theh-version,p-version, and combinedh-p-version for a one dimensional problem are presented.

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References

  1. Babuška, I., Kellogg, R.B., Pitkäranta, J.: Direct and inverse error estimates for finite elements with mesh refinement. Numer. Math.33, 447–471 (1979)

    Google Scholar 

  2. Babuška, I., Szabo, B.A., Katz, I.N.: Thep-version of the finite element method. Report WU/CCM-79/1, Center for Computational Mechanics, Washington University, SINUM (1981)

  3. Bergh, J., Löfstrom, J.: Interpolation spaces. Berlin-Heidelberg-New York: Springer 1976

    Google Scholar 

  4. Ciarlet, P.: The finite element method for elliptic problems. Amsterdam: North-Holland 1978

    Google Scholar 

  5. DeVore, R., Scherer, K.: Variable knot, variable degree spline approximation tox β. In: Quantitative approximation, Proceedings of the Bonn Conference. New York: Academic Press 1979

    Google Scholar 

  6. Grisvard, P.: Boundary value problems in non-smooth domains. Lecture Notes 19, University of Maryland, 1980

  7. Kondrat'ev, V.A.: Boundary problems for elliptic equations with conical or angular points. Trans. Moscow Math Soc.16, 227–313 (1967)

    Google Scholar 

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

    Google Scholar 

  9. Widlund, O.: On best error bounds for approximation by piecewise polynomial functions. Numer. Math.27, 327–338, 1977

    Google Scholar 

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Authors and Affiliations

  1. Institute for Physical Science and Technology, University of Maryland, 20742, College Park, Maryland, USA

    Ivo Babuška

  2. Naval Surface Weapons Center, 20910, Silver Spring, Maryland, USA

    Milo R. Dorr

Authors
  1. Ivo Babuška
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  2. Milo R. Dorr
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Babuška, I., Dorr, M.R. Error estimates for the combinedh andp versions of the finite element method. Numer. Math. 37, 257–277 (1981). https://doi.org/10.1007/BF01398256

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  • DOI: https://doi.org/10.1007/BF01398256

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