A hybridizable discontinuous Galerkin method with transmission variables for time-harmonic acoustic problems in heterogeneous media

TL;DR

This work extends hybridizable discontinuous Galerkin (HDG) methods to time-harmonic acoustics in heterogeneous media by introducing CHDG, which uses transmission variables as the hybrid unknown. The CHDG formulation can be written as $(\mathrm{I}-\Pi\mathrm{S})g=b$, and a key result is that, for symmetric fluxes with a positive self-adjoint $\mathcal{A}_F$, the operator $\Pi\mathrm{S}$ is a strict contraction, enabling fixed-point iterations without relaxation. Extensive 2D benchmarks show that CHDG consistently reduces iteration counts for GMRES and CGNR compared to standard HDG, with significant gains in challenging cases such as impedance boundary conditions and near resonances (e.g., Marmousi). The findings highlight the practical impact of CHDG for fast solvers in heterogeneous media and point to promising extensions to 3D and other wave problems. All mathematical notation is presented with explicit delimiters $...$.

Abstract

We consider the finite element solution of time-harmonic wave propagation problems in heterogeneous media with hybridizable discontinuous Galerkin (HDG) methods. In the case of homogeneous media, it has been observed that the iterative solution of the linear system can be accelerated by hybridizing with transmission variables instead of numerical traces, as performed in standard approaches. In this work, we extend the HDG method with transmission variables, which is called the CHDG method, to the heterogeneous case with piecewise constant physical coefficients. In particular, we consider formulations with standard upwind and general symmetric fluxes. The CHDG hybridized system can be written as a fixed-point problem, which can be solved with stationary iterative schemes for a class of symmetric fluxes. The standard HDG and CHDG methods are systematically studied with the different numerical fluxes by considering a series of 2D numerical benchmarks. The convergence of standard iterative schemes is always faster with the extended CHDG method than with the standard HDG methods, with upwind and scalar symmetric fluxes.

Figures (5)

• Figure 1: Real part of the reference solutions $p_{\mathrm{ref}}$ for both benchmark problems with the default parameters corresponding to the first cases of each benchmark in Tables \ref{['tab:homog']} and \ref{['tab:inhomog']}.
• Figure 2: Results for the benchmark problems with homogeneous media. Error history with different iterative schemes and different DG methods. The dashed horizontal lines correspond to the relative numerical errors obtained with a direct solver for the different numerical fluxes.
• Figure 3: Results of the benchmark problems with heterogeneous media. Error history with different iterative schemes and different DG methods. The dashed horizontal lines correspond to the relative numerical errors obtained with a direct solver for the different numerical fluxes. In most graphs, the green lines (CHDG 'Upw') are hidden by the corresponding blue lines (CHDG 'Sym-0').
• Figure 4: Marmousi benchmark problem. Profile of velocity $c(\mathbf{x})$ (a), of density $\rho(\mathbf{x})$(b), mesh (c) and real part of the reference pressure field obtained with a direct solver (d).
• Figure 5: Marmousi benchmark problem. Relative residual (left) and relative error history (right) with different iterative schemes and DG methods. For the relative residual history, we consider for each case both the residual associated with the hybridized system (in $\times$, plain lines) and the one associated with the physical system (in physical norm, dashed lines). The relative error is computed by comparing the numerical physical solution obtained at each iteration with the reference numerical physical solution obtained with a direct solver.

Theorems & Definitions (18)

• Proposition 2.3: Positivity and self-adjointness of the second-order operator
• proof
• Remark 3.2: Case with upwind fluxes
• Theorem 3.4: Well-posedness of the local discrete problem
• proof
• Remark 3.5: Case with upwind fluxes
• Theorem 4.3: Well-posedness of the local discrete problem
• proof
• Remark 4.4: Case with upwind fluxes
• Lemma 4.7