Postdoctoral researcher CERMICS, CNRS, Inria, ENPC, Institut Polytechnique de Paris
SUBMITTED PAPERS
S.P., G. Gabard, T. Chaumont-Frelet, A. Modave (2026) A HDG method with transmission variables for time-harmonic wave propagation problems with constant coefficients Submitted for publication
Iterative finite element solvers for time-harmonic wave problems are notoriously slow to converge, owing to fundamental properties of these problems. We present a variant of the hybridizable discontinuous Galerkin (HDG) method that is better suited to fast iterative solution. Unlike the standard hybridization strategy, which eliminates physical unknowns by introducing an auxiliary numerical flux on element faces, our approach instead introduces a transmission variable on those faces. For Helmholtz problems, this reformulation, known as CHDG, has been shown to significantly accelerate the convergence of iterative schemes relative to standard HDG. The present work extends CHDG to a general framework covering wave propagation problems with constant coefficients, capable of handling diverse wave types in a unified manner. We prove that the resulting hybridized system is well-posed and amenable to fixed-point iteration. As a practical application, we apply the method to the time-harmonic linearized Euler equations with a uniform subsonic mean flow. The method is illustrated through two-dimensional numerical benchmarks involving both sound and vorticity waves, with a systematic comparison of the convergence behaviour of several iterative schemes across a range of configurations.
S.P., G. Gabard, T. Chaumont-Frelet, A. Modave (2025) A hybridizable discontinuous Galerkin method with transmission variables for time-harmonic acoustic problems in heterogeneous media Journal of Computational Physics
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.
S.P. (2025) Discontinuous Galerkin finite element methods with transmission variables for time-harmonic wave propagation
The goal of this thesis is to develop wave-specific discontinuous Galerkin (DG) methods for wave-propagation phenomena. Linear systems arising from finite element discretization techniques are often difficult to solve due to ill-conditioning issues related to intrinsic properties of time-harmonic problems. DG methods are very versatile and offer a wide framework for formulations that allow the use of fast iterative algorithms as well as non-standard basis functions. In addition, part of this thesis is motivated by the intuition that embedding the oscillating nature of the physical problem into the discretization method should lead to an improvement of the properties of the associated linear system. We consider a variant of the hybridizable discontinuous Galerkin (HDG) method with upwind numerical fluxes. Unlike the standard approach, the hybridization process is achieved using transmission variables whose definitions are driven by the physics of the problem. The resulting method, called the CHDG method, delivers an overall improvement of the speed of convergence of iterative procedures when compared to standard DG and HDG methods. We extend the theory of CHDG, which was initially studied for acoustic problems, to general time-harmonic problems with numerical tests focusing on aeroacoustics in homogeneous media. Moreover, inspired by recent studies on domain decomposition methods, we target another extension of CHDG that includes high-order transmission variables with a focus for the theory and the numerical tests on acoustic for both homogeneous and heterogeneous media. Lastly, with the idea of combining wave-tailored interface treatments with a wave-based basis method, we investigate a Trefftz DG method with propagative plane-wave basis functions applied to Helmholtz problems complemented by high-order absorbing boundary conditions. The numerical methods are studied by using two-dimensional numerical benchmarks implemented in dedicated MATLAB codes.
S.P. (2023) A Trefftz discontinuous Galerkin method with absorbing boundary conditions for the numerical simulation of the Helmholtz problem
The numerical simulation of Helmholtz problems in the time-harmonic acoustic framework set in unbounded domains requires a mesh based approach in bounded numerical domains. Therefore, we present a Trefftz method, which is a particular Discontinuous Galerkin method, applied to the problem complemented by Absorbing Boundary Conditions. Trefftz methods have been largely studied in the framework of Helmholtz equation and have turned out to be suitable. In particular, the use of propagative plane-waves is effective in approximating the solution. Nevertheless, their use is still limited by conditioning issues that arise whenever the approximate space becomes too large. The combination of the wave-based Trefftz method and the relative Ultra Weak Variational Formulation with Absorbing Boundary Conditions represents a novelty that we aim at studying and investigating in the present work. It leads to a new formulation of the problem whose numerical results are exploited to evaluate the efficiency of the approach.