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Multisymplecticity of Hybridizable Discontinuous Galerkin Methods

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Abstract

In this paper, we prove necessary and sufficient conditions for a hybridizable discontinuous Galerkin method to satisfy a multisymplectic conservation law, when applied to a canonical Hamiltonian system of partial differential equations. We show that these conditions are satisfied by the “hybridized” versions of several of the most commonly used finite element methods, including mixed, nonconforming, and discontinuous Galerkin methods. (Interestingly, for the continuous Galerkin method in dimension greater than one, we show that multisymplecticity only holds in a weaker sense.) Consequently, these general-purpose finite element methods may be used for structure-preserving discretization (or semidiscretization) of canonical Hamiltonian systems of ODEs or PDEs. This establishes multisymplecticity for a large class of arbitrarily high-order methods on unstructured meshes.

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  1. This result holds since e lies in an affine hyperplane in \( \mathbb {R}^m \), so \( x \cdot \mathbf {n} \) is constant on e. It follows that the degree-\(( r + 1 )\) elements of the Raviart–Thomas space nevertheless have degree-r normal traces.

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Acknowledgements

This research was supported in part by the Marsden Fund of the Royal Society of New Zealand and by the Simons Foundation (Award #279968 to Ari Stern). We also wish to thank the anonymous referees for their helpful comments and suggestions.

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Authors and Affiliations

  1. School of Fundamental Sciences, Massey University, Palmerston North, 5301, New Zealand

    Robert I. McLachlan

  2. Department of Mathematics and Statistics, Washington University in St. Louis, One Brookings Drive, Campus Box 1146, Saint Louis, MO, 63130, USA

    Ari Stern

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  1. Robert I. McLachlan
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Correspondence to Ari Stern.

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Communicated by Arieh Iserles.

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McLachlan, R.I., Stern, A. Multisymplecticity of Hybridizable Discontinuous Galerkin Methods. Found Comput Math 20, 35–69 (2020). https://doi.org/10.1007/s10208-019-09415-1

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