# New Stabilization of Hybridizable Discontinuous Galerkin discretization for anisotropic wave equation in Voigt notation

Hélène Barucq¹, Florian Faucher¹ and Ha Pham¹

¹ Makutu, Inria, TotalEnergies, UPPA, CNRS, University of Pau and Pays de l'Adour, F64000 Pau,
[firstname.lastname@inria.fr](mailto:firstname.lastname@inria.fr), https://team.inria.fr/makutu/

**Key Words:** Hybridizable Discontinuous Galerkin method, stabilization, time-harmonic wave.

In this talk, we discuss a Hybridized Discontinuous Galerkin (HDG) method to solve the time-harmonic linear anisotropic elastic equation, in the context of simulating wave propagation in frequency domain in heterogeneous Earth subsurfaces. HDG methods can handle material heterogeneity and nonconforming meshes, to provide accurate and stable performance at tenable computational costs, to exploit efficiently advanced and parallel computing architecture. We follow the original HDG method devised by Cockburn and co-authors [1, 2] for second-order elliptic equation, which was adapted and analyzed for Helmholtz equation in [3]. We propose a formulation which is based on Voigt notation so that the discrete problem includes general anisotropy and facilitates efficient numerical implementation. The associated numerical flux is derived for anisotropy under the assumption of three distinct wave speeds, by extending the hybridization technique of upwind flux in [5] for isotropy. We carry out in-depth numerical investigations to study the effect of our new stabilization operator in the definition of numerical trace for varying degrees of anisotropy. We illustrate the performance of the HDG method on some anisotropic 2D and 3D cases. More details are available in our research report [4].

## REFERENCES

[1] Bernardo Cockburn, Bo Dong, and Johnny Guzmàn. A superconvergent LDG-hybridizable Galerkin method for second-order elliptic problems. Mathematics of Computation, 77(264):1887–1916, 2008.

[2] Bernardo Cockburn, Jayadeep Gopalakrishnan, and Raytcho Lazarov. Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM Journal on Numerical Analysis, 47(2):1319–1365, 2009

[3] Roland Griesmaier and Peter Monk. Error analysis for a hybridizable discontinuous Galerkin method for the Helmholtz equation. Journal of Scientific Computing, 49:291–310, 2011.

[4] Ha Pham, Florian Faucher, and Hélène Barucq. On the implementation of Hybridizable Discontinuous Galerkin discretization for linear anisotropic elastic wave equation: Voigt-notation and stabilization, RR-9533, https://hal.science/hal-04356602, 2023.

[5] S Terrana, Jean-Pierre Vilotte, and L Guillot. A spectral hybridizable discontinuous Galerkin method for elastic-acoustic wave propagation. Geophysical Journal International, 213(1):574–602, 2018.