arXiv subjects

Pierre-Henri Cocquet

Publications and source records attributed to Pierre-Henri Cocquet.

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

From PDEs constrained optimization to controllability problems via time domain decomposition

This paper focuses on the application of time domain decomposition to solve partial differential equations constrained optimization problems and controllability problems. After clarifying the link between these two types of problems, we show that applying time domain decomposition to both problems leads to the same convergence behavior. Our numerical experiments also confirm these theoretical findings.

math.NA↗

arXiv2020

Optimization of Bathymetry for Long Waves with Small Amplitude

This paper deals with bathymetry-oriented optimization in the case of long waves with small amplitude. Under these two assumptions, the free-surface incompressible Navier-Stokes system can be written as a wave equation where the bathymetry appears as a parameter in the spatial operator. Looking then for time-harmonic fields and writing the bottom topography as a perturbation of a flat bottom, we end up with a heterogeneous Helmholtz equation with impedance boundary condition. In this way, we study some PDE-constrained optimization problem for a Helmholtz equation in heterogeneous media whose coefficients are only bounded with bounded variation. We provide necessary condition for a general cost function to have at least one optimal solution. We also prove the convergence of a finite element approximation of the solution to the considered Helmholtz equation as well as the convergence of discrete optimum toward the continuous ones. We end this paper with some numerical experiments to illustrate the theoretical results and show that some of their assumptions could actually be removed.

math.OC↗