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

doi:10.1007/s42967-021-00143-4

Abstract

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.

Key words: Finite element methods, Discontinuous Galerkin methods, Korteweg-de Vries equation, A posteriori error estimates, Conservation laws, Nonlinear equations, Dispersive equations

CLC Number:

Ohannes A. Karakashian, Michael M. Wise. A Posteriori Error Estimates for Finite Element Methods for Systems of Nonlinear, Dispersive Equations[J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 823-854.

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