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A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems

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Abstract

The effect of numerical quadrature in finite element methods for solving quasilinear elliptic problems of nonmonotone type is studied. Under similar assumption on the quadrature formula as for linear problems, optimal error estimates in the L 2 and the H 1 norms are proved. The numerical solution obtained from the finite element method with quadrature formula is shown to be unique for a sufficiently fine mesh. The analysis is valid for both simplicial and rectangular finite elements of arbitrary order. Numerical experiments corroborate the theoretical convergence rates.

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Author notes

  1. Gilles Vilmart

    Present address: École Normale Supérieure de Cachan, Antenne de Bretagne, av. Robert Schuman, 35170, Bruz, France

Authors and Affiliations

  1. Section de Mathématiques, École Polytechnique Fédérale de Lausanne, Station 8, 1015, Lausanne, Switzerland

    Assyr Abdulle & Gilles Vilmart

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  1. Assyr Abdulle
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  2. Gilles Vilmart
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Correspondence to Assyr Abdulle.

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Abdulle, A., Vilmart, G. A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems. Numer. Math. 121, 397–431 (2012). https://doi.org/10.1007/s00211-011-0438-4

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  • DOI: https://doi.org/10.1007/s00211-011-0438-4

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