International Conference on Spectral and High Order Methods
12th - 16th July 2021 | Vienna, Austria
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plenary 3: Alexandre Ern
Hybrid high-order methods for the wave equation
1University College London; 2University of Bergen; 3Ecole des Ponts and INRIA, France
We consider hybrid high-order (HHO) methods for the wave equation. 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 and discuss their design and analysis when using unfitted meshes to approximate interface problems where a curved interface separates subdomains with different material properties. 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 are shown and, in particular, illustrate the benefits of using high-order approximations in space.
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