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On a posteriori error estimation using distances between numerical solutions and angles between truncation errors

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  • Alekseev, A.K.
  • Bondarev, A.E.

Abstract

The geometric properties of the ensemble of numerical solutions obtained by the algorithms of different inner structure are addressed from the prospects for a posteriori error estimation. The numerical results are presented for the two-dimensional inviscid supersonic flows, containing shock waves. The truncation errors are computed using a postprocessor, the approximation errors are calculated by the subtraction of the numerical and the analytic solutions. The angles between the approximation errors are found to be far from zero that enables a posteriori estimation of the error norm. The correlation of the angles between the approximation errors and the corresponding angles between the computable truncation errors is observed in numerical tests and discussed from the viewpoint of the measure concentration effect and the algorithmic randomness. The analysis of the truncation errors’ geometry and the distances between solutions enables the estimation of the approximation error norm on the ensemble of numerical solutions obtained by the independent algorithms.

Suggested Citation

  • Alekseev, A.K. & Bondarev, A.E., 2021. "On a posteriori error estimation using distances between numerical solutions and angles between truncation errors," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 892-904.
    • Handle: RePEc:eee:matcom:v:190:y:2021:i:c:p:892-904
    DOI: 10.1016/j.matcom.2021.06.014
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    References listed on IDEAS

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    1. Segeth, Karel, 2010. "A review of some a posteriori error estimates for adaptive finite element methods," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 80(8), pages 1589-1600.
    2. Repin, Sergey I., 1999. "A unified approach to a posteriori error estimation based on duality error majorants," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 50(1), pages 305-321.
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