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A posteriori error analysis for higher order dissipative methods for evolution problems

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Abstract

We prove a posteriori error estimates for time discretizations by the discontinuous Galerkin method dG(q) and the corresponding implicit Runge- Kutta-Radau method IRK-R(q) of arbitrary order q≥0 for both linear and nonlinear evolution problems of the form \(u^{\prime} + \mathfrak{F}(u) = f\), with γ2-angle bounded operator \(\mathfrak{F}\). The key ingredient is a novel higher order reconstruction \(\widehat{U}\) of the discrete solution U, which restores continuity and leads to the differential equation \(\widehat{U}^{\prime}+\Pi\mathfrak{F}(U)=F\) for a suitable interpolation operator Π and piecewise polynomial approximation F of f. We discuss applications to linear PDE, such as the convection-diffusion equation (γ ≥ 1/2) and the wave equation (formally γ =  ∞), and nonlinear PDE corresponding to subgradient operators (γ =  1), such as the p-Laplacian, as well as Lipschitz operators (γ ≥ 1/2). We also derive conditional a posteriori error estimates for the time-dependent minimal surface problem.

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Crete, 71409, Heraklion-Crete, Greece

    Charalambos Makridakis

  2. Institute of Applied and Computational Mathematics, FORTH, 71110, Heraklion - Crete, Greece

    Charalambos Makridakis

  3. Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD, 20742, USA

    Ricardo H. Nochetto

Authors
  1. Charalambos Makridakis
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  2. Ricardo H. Nochetto
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Correspondence to Charalambos Makridakis.

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Partially supported by the European Union RTN-network HYKE, HPRN-CT-2002-00282, and the EU Marie Curie Development Host Site, HPMD-CT-2001-00121.

Partially supported by NSF Grants DMS-9971450 and DMS-0204670 and the General Research Board of the University of Maryland.

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Makridakis, C., Nochetto, R.H. A posteriori error analysis for higher order dissipative methods for evolution problems. Numer. Math. 104, 489–514 (2006). https://doi.org/10.1007/s00211-006-0013-6

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