My work lies at the interface of numerical analysis and scientific computing I develop robust and scalable algorithms for large-scale PDEs.
Relevant papers/preprints: [Rapaport, Chaumont-Frelet, Modave, HAL preprint 2025]
The CHDG method—originally developed for acoustics—had not been extended to full 3D Maxwell equations. My work provides the first analysis and implementation of CHDG for the time-harmonic Maxwell equations.
First CHDG formulation for Maxwell in 3D: A stable hybridizable discretization for high-frequency electromagnetic fields, avoiding the complexity of Nédélec elements while enforcing tangential continuity through a hybrid trace variable.
Well-posed local elimination problems: I prove that the elementwise static-condensation systems defining the local fields are uniquely solvable
Provable contraction of the hybrid solver: The global trace problem admits a contracting fixed-point iteration (in the spirit of UWVF), with convergence estimates valid in the high-frequency regime.
Large-scale numerical validation: Parallel implementations show good scalability, including simulations with ≈13 million degrees of freedom using high-order discretizations.
Relevant papers/preprints: [Févotte, Rappaport, Vohralík, Comp. Geo. 2024, Févotte, Rappaport, Vohralík, CMAME, 2024, Harnist, Mitra, Rappaport, Vohralík, HAL preprint 2023]
A recurring theme in my work is the development of guaranteed a posteriori error estimators and adaptive algorithms for nonlinear or nonsmooth PDEs. These estimators provide quantitative control of discretization, linearization, and model regularization errors, and they form the basis of fully automated numerical methods.
Equilibrated flux estimators for nonsmooth elliptic problems: Construction of guaranteed upper bounds for the energy error, with estimators that distinguish the contributions of regularization, linearization, and discretization. This leads to adaptive stopping criteria for each component and robust convergence guarantees.
Adaptive regularization for nonlinear and degenerate PDEs: Introduction of smoothing strategies that render nonsmooth nonlinearities amenable to Newton linearization. The amount of regularization is controlled by error estimators on the energy difference, ensuring that regularization is only applied where necessary.
Richards’ equation and porous media flow: A framework for replacing degenerate constitutive relations by smooth counterparts, combined with a posteriori estimators to drive adaptive regularization and solver decisions. Numerical results on benchmark problems demonstrate improved robustness for these challenging multiphysics models.
Across these problems, the overarching goal is to design adaptive algorithms that autonomously adjust regularization parameters, Newton stopping tolerances, and mesh resolution in response to certified error estimators.
