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Mixed methods on quadrilaterals and hexahedra

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Abstract

We describe a new family of discrete spaces suitable for use with mixed methods on certain quadrilateral and hexahedral meshes. The new spaces are natural in the sense of differential geometry, so all the usual mixed method theory, including the hybrid formulation, carries over to these new elements with proofs unchanged. Because transforming general quadrilaterals into squares introduces nonlinearity and because mixed methods involve the divergence operator, the new spaces are more complicated than either the corresponding Raviart-Thomas spaces for rectangles or corresponding finite element spaces for quadrilaterals. The new spaces are also limited to meshes obtained from a rectangular mesh through the application of a single global bilinear transformation. Despite this limitation, the new elements may be useful in certain topologically regular problems, where initially rectangular grids are deformed to match features of the physical region. They also illustrate the difficulties introduced into the theory of mixed methods by nonlinear transformations.

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References

  1. T. Arbogast, C. Dawson and P.T. Keenan, Mixed finite element methods as finite difference methods for solving elliptic equations on triangular elements, Tech. Report 93-53, Dept. of Comput. and Applied Math., Rice University (1993).

  2. T. Arbogast, M.F. Wheeler and I. Yotov, Mixed finite element methods on general geometry, Tech. Report, Dept. of Computational and Applied Mathematics, Rice University, in preparation.

  3. F. Brezzi and M. Fortin,Mixed and Hybrid Finite Element Methods (Springer, 1991).

  4. B. Char et al.,Maple V Language Reference Manual (Wiley-Interscience, 1991).

  5. L. Cowsar and M.F. Wheeler, Parallel domain decomposition method for mixed finite element methods for elliptic partial differential equations, Tech. Report 90-37, Dept. of Mathematical Sciences, Rice University (1990).

  6. P. Dirac,General Theory of Relativity (Wiley-Interscience, 1975).

  7. L. Eisenhart,Riemannian Geometry (Princeton University Press, 1966).

  8. R. Glowinski and M.F. Wheeler, Domain decomposition and mixed finite element methods for elliptic problems, Tech. Report 87-11, Dept. of Mathematical Sciences, Rice University (1987).

  9. R. Lashof and A. Liulevicius,Topology and Geometry of Locally Euclidean Spaces, Lecture Notes in Mathematics (University of Chicago, Department of Mathematics, 1973).

  10. P.A. Raviart and J. M. Thomas, A mixed finite element method for second order elliptic problems, in:Mathematical Aspects of the Finite Element Method, eds. G. I. and E. Magenes (Springer, 1977).

  11. J.M. Thomas, Sur l'analyse numérique des méthodes d'éléments finis hybrides et mixtes, PhD thesis, Université Pierre et Marie Curie, Paris (1977).

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Keenan, P.T. Mixed methods on quadrilaterals and hexahedra. Numer Algor 7, 269–293 (1994). https://doi.org/10.1007/BF02140687

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  • DOI: https://doi.org/10.1007/BF02140687

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