Convergence analysis of hybrid high-order methods for the wave equation

Erik Burman (1) , Omar Duran (2, 3) , Alexandre Ern (2, 3) , Morgane Steins (2, 3, 4)

1 UCL London - Department of Mathematics
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
3 SERENA - Simulation for the Environment: Reliable and Efficient Numerical Algorithms
4 DYN - Laboratoire d'études de DYNamique

We prove error estimates for the wave equation semi-discretized in space by the hybrid high-order (HHO) method. These estimates lead to optimal convergence rates for smooth solutions. We consider first the second-order formulation in time, for which we establish $H^1$ and $L^2$-error estimates, and the first-order formulation, for which we establish $H^1$-error estimates. For both formulations, the space semi-discrete HHO scheme has close links with hybridizable discontinuous Galerkin schemes from the literature. Numerical experiments using either the Newmark scheme or diagonally-implicit Runge-Kutta schemes for the time discretization illustrate the theoretical findings and show that the proposed numerical schemes can be used to simulate accurately the propagation of elastic waves in heterogeneous media.

Mots clés: Elastodynamics; Error analysis; Wave equation; Hybrid high-order methods

Domaines: Analyse numérique [math.NA]

Erik Burman, Omar Duran, Alexandre Ern, Morgane Steins. Convergence analysis of hybrid high-order methods for the wave equation. Journal of Scientific Computing, 2021, 87 (91), ⟨10.1007/s10915-021-01492-1⟩. ⟨hal-02922720v3⟩