Pre-Publicación 2026-20
Gabriel N. Gatica, Salim Meddahi:
A hybridizable discontinuous Galerkin method for the elasto--acoustic transmission problem
Abstract:
We study a first-order velocity--stress--pressure formulation of the linear elasto-acoustic transmission problem, which couples elastodynamics in a solid subdomain with acoustic wave propagation in a compressible inviscid fluid. Unified Hilbert spaces are constructed to encode the transmission conditions within the operator domain. Casting the system as an abstract Cauchy problem, we verify the Lumer-Phillips conditions for the resulting block skew-symmetric operator and thereby establish existence, uniqueness, and energy stability of strong solutions via a contractive $C_0$-semigroup. We then develop a Hybridizable Discontinuous Galerkin (HDG) discretization of the semi-discrete system. A key feature of the method is a tailored penalization jump that permits free tangential slip on the acoustic side of the interface, consistent with the inviscid fluid model, while preserving the symmetry of the stabilization form. We prove consistency, stability, and optimal $hp$-convergence in the energy norm. Numerical experiments validate the theoretical predictions and confirm the robust performance of the method in both well-conditioned and stiff parameter settings relevant to practical applications, indicating in particular that the scheme is free of volumetric locking in the nearly incompressible regime.
