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Description
The Hybrid high-order (HHO) method was originally devised for diffusion and elasticity problems [1, 2], and the realm of applications has been considerably extended since then. In this work, we consider the version of the HHO method that uses as discrete unknowns cell- and face-based polynomials of degree (k + 1) (cells) and k (faces) order with 0 ≤ k, yielding for steady problems optimal convergence of order (k + 1) in the energy norm. Firstly, we address the time second-order form of the wave equation, and we devise, analyze, and evaluate numerically an HHO scheme for the space discretization combined with a Newmark-like time-marching scheme. Secondly, for the first-order form, diagonally implicit and explicit Runge–Kutta time-marching schemes combined with the HHO method are considered, and we highlight the link with the hybridizable discontinuous Galerkin (HDG) methods [3]. Finally, we present numerical experiments recovering optimal convergence rates for smooth solutions and exhibiting robust performances in the case of contrasted media. Further insight into our results can be found in [4, 5].
REFERENCES
[1] D.A. Di Pietro, A. Ern, and S. Lemaire. An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators. Computational Methods in Applied Mathematics. 14 (2014) 461-472.
[2] D.A. Di Pietro and A. Ern, A hybrid high-order locking-free method for linear elasticity on general meshes. Comput Methods Appl. Mech. Eng. 283 (2015) 1–21.
[3] M. Stanglmeier, N.C. Nguyen, J. Peraire, and B. Cockburn. An explicit hybridizable discontinuous galerkin method for the acoustic wave equation.
Computer Methods in Applied Mechanics and Engineering, 300:748–769, March 2016.
[4] E. Burman, O. Duran, and A. Ern. Hybrid high-order methods for the acoustic wave equation in the time domain. working paper or preprint, hal-02922702, Aug 2020.
[5] E. Burman, O. Duran, A. Ern, and M. Steins. Convergence analysis of hybrid high-order methods for the wave equation. working paper or preprint, hal-02922720, Aug 2020.
