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A posteriori error analysis for finite element methods with projection operators as applied to explicit time integration techniques

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Abstract

We derive a posteriori error estimates for two classes of explicit finite difference schemes for ordinary differential equations. To facilitate the analysis, we derive a systematic reformulation of the finite difference schemes as finite element methods. The a posteriori error estimates quantify various sources of discretization errors, including effects arising from explicit discretization. This provides a way to judge the relative sizes of the contributions, which in turn can be used to guide the choice of various discretization parameters in order to achieve accuracy in an efficient way. We demonstrate the accuracy of the estimate and the behavior of various error contributions in a set of numerical examples.

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Authors and Affiliations

  1. Department of Mathematics, Chemistry, and Physics, Western Texas A&M University, Canyon, TX, 79016, USA

    J. B. Collins

  2. Department of Statistics, Colorado State University, Fort Collins, CO, 80523, USA

    D. Estep

  3. Department of Mathematics, Colorado State University, Fort Collins, CO, 80523, USA

    S. Tavener

Authors
  1. J. B. Collins
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  2. D. Estep
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  3. S. Tavener
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Corresponding author

Correspondence to D. Estep.

Additional information

Communicated by Ralf Hiptmair.

This research is supported in part by the Defense Threat Reduction Agency (HDTRA1-09-1-0036), Department of Energy (DE-FG02-04ER25620, DE-FG02-05ER25699, DE-FC02-07ER54909, DE-SC0001724, DE-SC0005304, INL00120133), Idaho National Laboratory (00069249, 00115474), Lawrence Livermore National Laboratory (B584647, B590495), National Science Foundation (DMS-0107832, DMS-0715135, DGE-0221595003, MSPA-CSE-0434354, ECCS-0700559, DMS-1065046, DMS-1016268, DMS-FRG-1065046), National Institutes of Health (#R01GM096192).

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Collins, J.B., Estep, D. & Tavener, S. A posteriori error analysis for finite element methods with projection operators as applied to explicit time integration techniques. Bit Numer Math 55, 1017–1042 (2015). https://doi.org/10.1007/s10543-014-0534-9

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  • DOI: https://doi.org/10.1007/s10543-014-0534-9

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