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The norm of a discretized gradient in $$\varvec{H({{\mathrm{div}}})^*}$$ for a posteriori finite element error analysis | Numerische Mathematik Skip to main content
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The norm of a discretized gradient in \(\varvec{H({{\mathrm{div}}})^*}\) for a posteriori finite element error analysis

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Abstract

This paper characterizes the norm of the residual of mixed schemes in their natural functional framework with fluxes or stresses in \(H({{\mathrm{div}}})\) and displacements in \(L^2\). Under some natural conditions on an associated Fortin interpolation operator, reliable and efficient error estimates are introduced that circumvent the duality technique and so do not suffer from reduced elliptic regularity for non-convex domains. For the Laplace, Stokes, and Lamé equations, this generalizes known estimators to non-convex domains and introduces new a posteriori error estimators.

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Authors and Affiliations

  1. Institut für Mathematik, Humboldt Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Carsten Carstensen

  2. Institut für Numerische Simulation, Universität Bonn, Wegelerstr. 6, 53115, Bonn, Germany

    Daniel Peterseim

  3. Fachbereich Mathematik, Universität Salzburg, Hellbrunnerstr. 34, 5020, Salzburg, Austria

    Andreas Schröder

Authors
  1. Carsten Carstensen
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  2. Daniel Peterseim
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  3. Andreas Schröder
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Corresponding author

Correspondence to Daniel Peterseim.

Additional information

Dedicated to Professor Volker Mehrmann on the occasion of his 60th birthday.

The authors gratefully acknowledge support by the Deutsche Forschungsgemeinschaft in the Priority Program 1748 “Reliable simulation techniques in solid mechanics. Development of non-standard discretization methods, mechanical and mathematical analysis” under the projects “Foundation and application of generalized mixed FEM towards nonlinear problems in solid mechanics”, “Adaptive isogeometric modeling of propagating strong discontinuities in heterogeneous materials” and “High-order immersed-boundary methods in solid mechanics for structures generated by additive processes”.

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Carstensen, C., Peterseim, D. & Schröder, A. The norm of a discretized gradient in \(\varvec{H({{\mathrm{div}}})^*}\) for a posteriori finite element error analysis. Numer. Math. 132, 519–539 (2016). https://doi.org/10.1007/s00211-015-0728-3

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