arXiv subjects

Antoine Tonnoir

Publications and source records attributed to Antoine Tonnoir.

3 recordsLinked to original sources

arXiv2026

Asymptotic dispersion correction for the isotropic elastic Helmholtz equation discretized with a MAC scheme

The numerical simulation of time-harmonic wave propagation in elastic media plays an important role in applications such as geophysics and non-destructive testing. Accurate discretization of the elastic Helmholtz equation at high frequencies is challenging due to numerical dispersion and pollution effects. In this work, we develop an asymptotic dispersion correction for a Marker-And-Cell (MAC) discretization of the isotropic elastic Helmholtz equation. We characterize the discrete dispersion relation of the scheme and determine the leading-order term in the dispersion error in both two and three spatial dimensions. Based on this analysis, we derive a correction that is asymptotically optimal in the limit of vanishing mesh size. The proposed approach improves the agreement between the discrete and continuous wave propagation properties while preserving the structure of the underlying discretization. We also establish a connection between the factorization of the dispersion relation and the structure of the grad-div operator symbol, providing additional insight into the algebraic structure of the elastic problem. Numerical experiments finally demonstrate a substantial reduction of relative errors and confirm the effectiveness of the proposed correction. We further provide numerical evidence that the corrected discretization improves the convergence behavior of multigrid solvers.

math.NA↗

arXiv2026

Modal analysis of a domain decomposition method for Maxwell's equations in a waveguide

Time-harmonic wave propagation problems, especially those governed by Maxwell's equations, pose significant computational challenges due to the non-self-adjoint nature of the operators and the large, non-Hermitian linear systems resulting from discretization. Domain decomposition methods, particularly one-level Schwarz methods, offer a promising framework to tackle these challenges, with recent advancements showing the potential for weak scalability under certain conditions. In this paper, we analyze the weak scalability of one-level Schwarz methods for Maxwell's equations in strip-wise domain decompositions, focusing on waveguides with general cross sections and different types of transmission conditions such as impedance or perfectly matched layers (PMLs). By combining techniques from the limiting spectrum analysis of Toeplitz matrices and the modal decomposition of Maxwell's solutions, we provide a novel theoretical framework that extends previous work to more complex geometries and transmission conditions. Numerical experiments confirm that the limiting spectrum effectively predicts practical behavior even with a modest number of subdomains. Furthermore, we demonstrate that the one-level Schwarz method can achieve robustness with respect to the wave number under specific domain decomposition parameters, offering new insights into its applicability for large-scale electromagnetic wave problems.

math.NA↗

arXiv2024

Stochastic porous media equation with Robin boundary conditions, gravity-driven infiltration and multiplicative noise

We aim at studying a novel mathematical model associated to a physical phenomenon of infiltration in an homogeneous porous medium. The particularities of our system are connected to the presence of a gravitational acceleration term proportional to the level of saturation, and of a Brownian multiplicative perturbation. Furthermore, the boundary conditions intervene in a Robin manner with the distinction of the behavior along the inflow and outflow respectively. We provide qualitative results of well-posedness, the investigation being conducted through a functional approach.

math.AP↗