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A Posteriori Error Estimates for Finite Element Methods for Systems of Nonlinear, Dispersive Equations

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Abstract

The present study regards the numerical approximation of solutions of systems of Korteweg-de Vries type, coupled through their nonlinear terms. In our previous work [9], we constructed conservative and dissipative finite element methods for these systems and presented a priori error estimates for the semidiscrete schemes. In this sequel, we present a posteriori error estimates for the semidiscrete and fully discrete approximations introduced in [9]. The key tool employed to effect our analysis is the dispersive reconstruction developed by Karakashian and Makridakis [20] for related discontinuous Galerkin methods. We conclude by providing a set of numerical experiments designed to validate the a posteriori theory and explore the effectivity of the resulting error indicators.

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Acknowledgements

This work was supported in part by the National Science Foundation under grant DMS-1620288

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Authors and Affiliations

  1. Department of Mathematics, University of Tennessee, Knoxville, TN, USA

    Ohannes A. Karakashian

  2. Advanced Technology Integration Department, Dynetics, Inc., Huntsville, AL, USA

    Michael M. Wise

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  1. Ohannes A. Karakashian
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  2. Michael M. Wise
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Correspondence to Ohannes A. Karakashian.

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Karakashian, O.A., Wise, M.M. A Posteriori Error Estimates for Finite Element Methods for Systems of Nonlinear, Dispersive Equations. Commun. Appl. Math. Comput. 4, 823–854 (2022). https://doi.org/10.1007/s42967-021-00143-4

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  • DOI: https://doi.org/10.1007/s42967-021-00143-4

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