A preprint on HDG method for anisotropy is available online:
Numerical investigation of stabilization in the Hybridizable Discontinuous Galerkin method for linear anisotropic elastic equation, arXiv preprint 2403.02862, pp. 1–34, 2024.
The associated research report (79 pages) is also available on Hal: On the implementation of Hybridizable Discontinuous Galerkin discretization for linear anisotropic elastic wave equation: Voigt-notation and stabilization, Research Report RR-9533, Inria Bordeaux Team Makutu, pp. 1–79, 2024.
Abstract
This work concerns the implementation of the hybridizable discontinuous Galerkin (HDG) method to solve the linear anisotropic elastic equation in the frequency domain. First-order formulation with the compliance tensor and Voigt notation are employed to provide a compact description of the discretized problem and flexibility with highly heterogeneous media. We further focus on the question of optimal choice of stabilization in the definition of HDG numerical traces. For this purpose, we construct a hybridized Godunov-upwind flux for anisotropic elasticity possessing three distinct wavespeeds. This stabilization removes the need to choose scaling factors, contrary to identity and Kelvin-Christoffel based stabilizations which are popular choices in literature. We carry out comparisons among these families for isotropic and anisotropic material, with constant background and highly heterogeneous ones, in two and three dimensions. They establish the optimality of the Godunov stabilization which can be used as a reference choice for generic material and different types of waves.
