Intrinsic finite element methods for the computation of fluxes for Poisson’s equation

Abstract

In this paper we consider an intrinsic approach for the direct computation of the fluxes for problems in potential theory. We develop a general method for the derivation of intrinsic conforming and non-conforming finite element spaces and appropriate lifting operators for the evaluation of the right-hand side from abstract theoretical principles related to the second Strang Lemma. This intrinsic finite element method is analyzed and convergence with optimal order is proved.

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Notes

  1. 1.

    Here, we use the observation that for a polynomial \(q\in \mathbb {P}_{p}\left( \omega \right) \), \(\omega \subset \Omega \) with positive measure, it holds either \(\left. q\right| _{\omega }=0\) or \(\mathrm{supp}\,q=\omega \). In our application we choose \(q=\mathbf {e} _{1}+\mathbf {e}_{2}\) and apply the argument simplex by simplex.

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Ciarlet, P.G., Ciarlet, P., Sauter, S.A. et al. Intrinsic finite element methods for the computation of fluxes for Poisson’s equation. Numer. Math. 132, 433–462 (2016). https://doi.org/10.1007/s00211-015-0730-9

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Keywords

  • Elliptic boundary value problems
  • Conforming and non-conforming finite element spaces
  • Intrinsic formulation

Mathematics Subject Classification

  • 65N30