Error estimates for Galerkin finite element methods for the Camassa–Holm equation | Numerische Mathematik
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Error estimates for Galerkin finite element methods for the Camassa–Holm equation

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Abstract

We consider the Camassa–Holm (CH) equation, a nonlinear dispersive wave equation that models one-way propagation of long waves of moderately small amplitude. We discretize in space the periodic initial-value problem for CH (written in its original and in system form), using the standard Galerkin finite element method with smooth splines on a uniform mesh, and prove optimal-order \(L^{2}\)-error estimates for the semidiscrete approximation. Using the fourth-order accurate, explicit, “classical” Runge–Kutta scheme for time-stepping, we construct a highly accurate, stable, fully discrete scheme that we employ in numerical experiments to approximate solutions of CH, mainly smooth travelling waves and nonsmooth solitons of the ‘peakon’ type.

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Acknowledgements

V.A.D. and D.E.M. acknowledge travel support by grant MTM2014-54710 of the Ministerió de Economia y Competividad, Spain. D.E.M. was supported by the Marsden Fund administered by the Royal Society of New Zealand with Contract Number VUW1418.

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Authors and Affiliations

  1. Mathematics Department, National and Kapodistrian University of Athens, 15784, Zographou, Greece

    D. C. Antonopoulos & V. A. Dougalis

  2. Institute of Applied and Computational Mathematics, FORTH, 70013, Heraklion, Greece

    D. C. Antonopoulos & V. A. Dougalis

  3. School of Mathematics and Statistics, Victoria University of Wellington, Wellington, 6140, New Zealand

    D. E. Mitsotakis

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  1. D. C. Antonopoulos
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  2. V. A. Dougalis
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  3. D. E. Mitsotakis
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Correspondence to V. A. Dougalis.

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Antonopoulos, D.C., Dougalis, V.A. & Mitsotakis, D.E. Error estimates for Galerkin finite element methods for the Camassa–Holm equation. Numer. Math. 142, 833–862 (2019). https://doi.org/10.1007/s00211-019-01045-7

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