Hybrid high-order (HHO) methods attach discrete unknowns to the mesh cells and faces. Their design is based on a local gradient reconstruction and a local stabilization operator weakly enforcing the matching of the trace of the cell unknowns to the face unknowns. HHO methods can be embedded into the framework of Hybridizable discontinuous Galerkin (HDG) methods, and they are closely related to Weak Galerkin (WG) and to nonconforming Virtual Element methods (ncVEM). HHO methods support polytopal meshes, lead to optimal error estimates, are locally conservative, and are computationally effective owing to the possibility of locally eliminating the cell unknowns.
In this talk, we first gently introduce the main ideas of HHO methods on the Poison model problem. Then, we consider the application of HHO methods for the space semi-discretization of the wave equation. We address both the second-order formulation in time of the wave equation and its reformulation as a first-order system, leading respectively to the use of Newmark and Runge-Kutta schemes for the time discretization. Numerical examples of wave propagation problems through heterogeneous media illustrate the benefits of using high-order approximations in space. The handling of curved interfaces will also be (briefly) discussed [4].
References:
[1] D. Di Pietro, A. Ern, and S. Lemaire, Comp. Methods Appl. Math., 14(4):461-472 (2014).
[2] M. Ciuttin, A. Ern, and N. Pignet, Hybrid high-order methods. A primer with applications to solid mechanics, SpringerBriefs in Mathematics (Springer, 2021).
[3] E. Burman, O. Duran and A. Ern, Commun. Appl. Math. Comput., 4(2):597-633 (2022).
[4] E. Burman, M. Cicuttin, G. Delay and A. Ern, SIAM J. Sci. Comput., 43(2):A859-A882 (2021).