Research

I develop and analyze numerical methods for partial differential equations, with an emphasis on the interaction between discretization and iterative solution. My current work focuses on high-frequency wave propagation, including Maxwell discretizations, domain decomposition, and phase space methods for scattering.

Discretizations and solvers for electromagnetic waves

Hybridizable discontinuous Galerkin methods

With Théophile Chaumont-Frelet and Axel Modave, I extended the CHDG method from Helmholtz to the three-dimensional time-harmonic Maxwell equations with constant coefficients. The method uses incoming and outgoing transmission variables on element faces. Eliminating the fields inside each element reduces the global problem to incoming interface unknowns. We prove well-posedness of the local problems and contractivity of the associated fixed-point iteration.

The paper develops a nodal implementation and compares CGNR and GMRES solver strategies through three-dimensional benchmarks, including a large-scale experiment using a parallel C++ code. I am preparing CHDG.jl for public release on GitHub.

Three-dimensional electromagnetic field computed with the CHDG method.

Three-dimensional electromagnetic field computed with the CHDG method.

Related publication: A Hybridizable Discontinuous Galerkin Method with Transmission Variables for Time-Harmonic Electromagnetic Problems, SIAM Journal on Scientific Computing (2026) · HAL version.

Domain decomposition for heterogeneous and anisotropic media

In joint work with Marcella Bonazzoli, Patrick Ciarlet Jr., and Axel Modave at POEMS, I studied a two-level additive Schwarz preconditioner for the time-harmonic Maxwell equations with spatially varying, anisotropic material coefficients.

We derive convergence estimates for preconditioned GMRES that explicitly account for material contrast and anisotropy. The analysis covers proportional conductivity and a restricted class of deviations from it. Numerical experiments at fixed frequency show that a finer coarse mesh can largely compensate for the deterioration in convergence caused by increasing anisotropy or heterogeneity. They also show successful solver behavior for conductivity choices beyond the sufficient conditions covered by the analysis.

Field visualizations from the heterogeneous, anisotropic Maxwell experiments.

Field visualizations from the heterogeneous, anisotropic Maxwell experiments.

Phase space methods for high-frequency waves

At LJLL, I work on phase space methods within PSINumScat (Phase-space-inspired Numerical Methods for High Frequency Wave Scattering), an ERC Synergy project. The project uses high-frequency analysis to develop numerical methods for acoustic and electromagnetic wave scattering.

Earlier work: a posteriori error estimation and adaptivity

During my PhD, I studied how a posteriori error estimates can guide computational effort in nonlinear and nonsmooth elliptic problems. The estimators separate errors due to regularization, linearization, and discretization, making it possible to guide mesh refinement and choose stopping criteria for each stage of the computation.

With François Févotte and Martin Vohralík, I developed adaptive regularization strategies based on this principle, including applications to Richards’ equation for flow in porous media. Joint work with A. Harnist, K. Mitra, and Martin Vohralík establishes robust augmented-energy estimates for Lipschitz and strongly monotone elliptic problems.

First stage of the perched-water simulation

Second stage of the perched-water simulation

Third stage of the perched-water simulation

Evolution of perched water in a Richards equation simulation.

The computational side of this work includes EquilibratedFlux.jl and joint work with François Févotte on a thread-parallel implementation of equilibrated flux reconstruction in Julia.