arXiv subjects

Pierre Marchand

Publications and source records attributed to Pierre Marchand.

At least 19 recordsLinked to original sources

arXiv2026

Multi-domain FEM-BEM coupling with several impenetrable obstacles

In a recent paper we have analyzed a new formulation of the coupling of finite and boundary element methods (FEM-BEM) for Helmholtz problems, involving several heterogeneous bounded subdomains and one homogeneous unbounded subdomain. This formulation, called Generalized Optimized Schwarz Method (GOSM), is substructured, that is, its unknowns are associated with the subdomains interfaces. To derive the GOSM, the first step was to prove that a solution to the Helmholtz problem satisfies a specific multi-domain variational formulation, which involves one operator for each subdomain, each operator being independent of the others. In the present contribution, we design a variational formulation of that type for a more general geometrical and material configuration: several heterogeneous bounded subdomains, impenetrable obstacles and homogeneous subdomains are allowed. Note that, like in our recent paper, we assume that only one subdomain is unbounded, and that its boundary is bounded. The domain partition can have cross-points, that is, points where at least three subdomains are adjacent. We also prove that a solution to the initial Helmholtz problem can be recovered from a solution to the multi-domain variational formulation. This shows that the GOSM is a general and flexible framework to model acoustic wave propagation, as it can handle both multi-domain FEM-BEM coupling and (weakly imposed) boundary conditions on several obstacles.

math.NA↗

arXiv2026

Chemical Complexity in the Early Stages of Star Formation in the SKAO Era

About 350 molecules have been identified in the interstellar medium (ISM), including complex molecules relevant to prebiotic chemistry. A remarkable level of molecular diversity has been observed from the earliest stages of star formation, providing the initial chemical inventory inherited by planetary systems. Radio observations have played a pivotal role in these discoveries, starting with the identification of the first polyatomic molecule, $\text{NH}_3$ (Cheung et al. 1968). (Sub-)millimeter observations have revealed complex organic molecules of prebiotic relevance, including formamide ($\text{NH}_2\text{CHO}$), glycolaldehyde ($\text{CH}_2\text{OHCHO}$), and even urea ($(\text{NH}_2)_2\text{CO}$), and hydroxylamine ($\text{NH}_2\text{OH}$), which are possible precursors of RNA nucleotides (Ceccarelli et al. 2023; Jim\'enez-Serra et al. 2020). However, in dense protostellar regions, dust opacity hampers the detection of molecular emission. Additionally, large molecules and those containing heavy atoms, which have rotational transitions at lower frequencies, often remain inaccessible to current instruments. The Square Kilometre Array Observatory (SKAO) will provide an unprecedented combination of sensitivity and angular resolution at radio wavelengths. This will allow for the detection of prebiotic species and offer new insights into the chemical pathways that shape emerging planetary systems (Jim\'enez-Serra et al. 2022). This chapter details the scientific questions and advancements that the SKAO, and more specifically, SKA-Mid equipped with the Band 5 receivers, will pursue in the field of astrochemistry, focusing on the chemical complexity in both high-mass and solar-type star-forming regions.

astro-ph.SR↗

arXiv2026

Discrete FEM-BEM coupling with the Generalized Optimized Schwarz Method

The present contribution aims at developing a non-overlapping Domain Decomposition (DD) approach to the solution of acoustic wave propagation boundary value problems based on the Helmholtz equation, on both bounded and unbounded domains. This DD solver, called Generalized Optimized Schwarz Method (GOSM), is a substructuring method, that is, the unknowns of an iteration are associated with the subdomains interfaces. We extend the analysis presented in a previous paper of one of the author to a fully discrete setting. We do not consider only a specific set of boundary conditions, but a whole class including, e.g., Dirichlet, Neumann, and Robin conditions. Our analysis will also cover interface conditions corresponding to a Finite Element Method - Boundary Element Method (FEM-BEM) coupling. In particular, we shall focus on three classical FEM-BEM couplings, namely the Costabel, Johnson-N\'ed\'elec and Bielak-MacCamy couplings. As a remarkable outcome, the present contribution yields well-posed substructured formulations of these classical FEM-BEM couplings for wavenumbers different from classical spurious resonances. We also establish an explicit relation between the dimensions of the kernels of the initial variational formulation, the local problems and the substructured formulation. That relation especially holds for any wavenumber for the substructured formulation of Costabel FEM-BEM coupling, which allows us to prove that the latter formulation is well-posed even at spurious resonances. Besides, we introduce a systematically geometrically convergent iterative method for the Costabel FEM-BEM coupling, with estimates on the convergence speed.

math.NA↗

arXiv2025

Spurious resonances for substructured FEM-BEM coupling

We are interested in time-harmonic acoustic scattering by an impenetrable obstacle in a medium where the wavenumber is constant in an exterior unbounded subdomain and is possibly heterogeneous in a bounded subdomain. The associated Helmholtz boundary value problem can be solved by coupling the Finite Element Method (FEM) in the heterogeneous subdomain with the Boundary Element Method (BEM) in the homogeneous subdomain. Recently, we designed and analyzed a new substructured FEM-BEM formulation, called Generalized Optimized Schwarz Method (GOSM). Unfortunately, it is well known that, even when the initial boundary value problem is well-posed, the variational formulation of classical FEM-BEM couplings can be ill-posed for certain wavenumbers, called spurious resonances. In this paper, we focus on the Johnson-N\'ed\'elec and Costabel couplings and show that the GOSM derived from both is not immune to that issue. In particular, we give an explicit expression of the kernel of the local operator associated with the interface between the FEM and BEM subdomains. That kernel and the one of classical FEM-BEM couplings are simultaneously non-trivial.

math.NA↗

arXiv2025

Preconditioning of GMRES for Helmholtz problems with quasimodes

Finite element methods are effective for Helmholtz problems involving complex geometries and heterogeneous media. However, the resulting linear systems are often large, indefinite, and challenging for iterative solvers, particularly at high wave numbers or near resonant conditions. We derive a GMRES convergence bound that incorporates the nonlinear behavior of the relative residual and relates convergence to harmonic Ritz values. This perspective reveals how small eigenvalues associated with quasimodes can hinder convergence, and when they cease to have an effect. These phenomena occur in domain decomposition, and we illustrate them through numerical experiments. We also combine domain decomposition methods with deflation techniques using (approximate) eigenvectors tailored to resonant regimes. Their impact on GMRES performance is evaluated.

math.NA↗

arXiv2025

The evolution of complex organic molecules during star formation

Complex organic molecules (COMs) are thought to be the precursors of pre-biotic molecules and are observed in many protostellar sources. For this paper we studied the formation of COMs during star formation and their evolution in the midplane of the circumstellar disk up to the end of the Class I stage. We used the Analytical Protostellar Environment (APE) code to perform analytical simulations of star formation and the Nautilus code to model the chemical evolution. Most COMs mainly form during the collapse or in the disk, except the lightest (CH3CCH, C3H6, CH3OH, CH3CHO, CH3OCH3, C2H5OH, CH3CN, CH3NC, C2H3CN, and CH3SH), which are significantly inherited by the disk from the prestellar phase. Over the first 150 kyr of the disk, the abundances of several COMs in the midplane vary negligibly (e.g., CH3CCH, CH3OH, and CH3CN), while others experience a variation of one order of magnitude (e.g., C2H3CHO HOCH2CHO, and CH3COCH2OH). Changing physical conditions also have an impact on the abundance profiles of COMs in the disk, and their inheritance. For example, increasing the temperature of the molecular cloud from 10 K to 15 K significantly promotes the formation of COMs in the prestellar phase, notably c-C2H4O and N-bearing species. Conversely, increasing the cloud mass from 2 Msol to 5 Msol only has a minor effect on the disk abundances in the early stages.

astro-ph.GA↗

arXiv2025

Convergence rates of curved boundary element methods for the 3D Laplace and Helmholtz equations

We establish improved convergence rates for curved boundary element methods applied to the three-dimensional (3D) Laplace and Helmholtz equations with smooth geometry and data. Our analysis relies on a precise analysis of the consistency errors introduced by the perturbed bilinear and sesquilinear forms. We illustrate our results with numerical experiments in 3D based on basis functions and curved triangular elements up to order four.

math.NA↗

arXiv2025

Convergence analysis of GMRES applied to Helmholtz problems near resonances

The finite element solution of Helmholtz problems near resonant or quasi-resonant frequencies poses significant challenges, as iterative solvers typically suffer from severely degraded convergence. We analyze the convergence behavior of GMRES applied to linear systems arising from such configurations. Theoretical convergence estimates are derived based on harmonic Ritz values, highlighting their proximity to small eigenvalues as a key determining factor. We further examine deflation strategies and their interplay with preconditioning techniques, using the Complex Shifted Laplacian preconditioner as a case study. Numerical experiments on resonant and quasi-resonant test cases validate the theoretical framework and demonstrate the effectiveness of deflation strategies. This study provides new insights and practical guidance for analyzing and improving iterative solvers for time-harmonic problems near resonances.

math.NA↗

arXiv2025

APE: An analytical protostellar environment to provide physical conditions to chemical models and synthetic observations

Chemical modeling and synthetic observations are powerful methods to interpret observations, both requiring a knowledge of the physical conditions. In this paper, we present the Analytical Protostellar Environment (APE) code, which aims at making chemical simulations and synthetic observations accessible. APE contains a physical model of protostellar evolution (including the central object, the envelope, the protoplanetary disk and the outflow) as well as interfaces to publicly available codes to perform chemical simulations, radiative transfer calculations, and synthetic interferometry imaging. APE produces density and temperature maps of protostellar systems. The code can also follow individual particles throughout their journey in a collapsing core. APE includes a treatment of the dust grain size-distribution to compute opacities self-consistently for subsequent radiative transfer. We show an example of application of APE by computing chemical abundance maps of CO, CN, CS, H2CO, and CH3OH in a Class I protostellar system. We also performed synthetic ALMA observations of their molecular emission assuming an edge-on source inclination. The moment 0 maps of CO, CS, and H2CO display an X-shaped emission similar to what is observed toward the Class I source IRAS 04302+2247.

astro-ph.SR↗

arXiv2024

Protoplanetary disk size under non-ideal magnetohydrodynamics: A general formalism with inclined magnetic field

Many mechanisms have been proposed to alleviate the magnetic catastrophe, which prevents the Keplerian disk from forming inside a collapsing magnetized core. Such propositions include inclined field and non-ideal magnetohydrodynamics effects, and have been supported with numerical experiments. Models have been formulated for typical disk sizes when a field threads the rotating disk, parallel to the rotation axis, while observations at the core scales do not seem to show evident correlation between the directions of angular momentum and the magnetic field. In the present study, we propose a new model that considers both vertical and horizontal fields and discuss their effects on the protoplanetary disk size.

astro-ph.EP↗

arXiv2023

Fast methods for tracking grain coagulation and ionization. III. Protostellar collapse with non-ideal MHD

Dust grains influence many aspects of star formation, including planet formation, opacities for radiative transfer, chemistry, and the magnetic field via Ohmic, Hall, and ambipolar diffusion. The size distribution of the dust grains is the primary characteristic influencing all these aspects. Grain size increases by coagulation throughout the star formation process. We describe here numerical simulations of protostellar collapse using methods described in earlier papers of this series. We compute the evolution of the grain size distribution from coagulation and the non-ideal magnetohydrodynamics effects self-consistently and at low numerical cost. We find that the coagulation efficiency is mostly affected by the time spent in high-density regions. Starting from sub-micron radii, grain sizes reach more than 100 {\mu}m in an inner protoplanetary disk that is only 1000 years old. We also show that the growth of grains significantly affects the resistivities, and indirectly the dynamics and angular momentum of the disk.

astro-ph.SR↗

arXiv2022

Protostellar collapse simulations in spherical geometry with dust coagulation and fragmentation

We model the coagulation and fragmentation of dust grains during the protostellar collapse with our newly developed shark code. It solves the gas-dust hydrodynamics in a spherical geometry and the coagulation/fragmentation equation. It also computes the ionization state of the cloud and the Ohmic, ambipolar and Hall resistivities. We find that the dust size distribution evolves significantly during the collapse, large grain formation being controlled by the turbulent differential velocity. When turbulence is included, only ambipolar diffusion remains efficient at removing the small grains from the distribution, brownian motion is only efficient as a standalone process. The macroscopic gas-dust drift is negligible for grain growth and only dynamically significant near the first Larson core. At high density, we find that the coagulated distribution is unaffected by the initial choice of dust distribution. Strong magnetic fields are found to enhance the small grains depletion, causing an important increase of the ambipolar diffusion. This hints that the magnetic field strength could be regulated by the small grain population during the protostellar collapse. Fragmentation could be effective for bare silicates, but its modeling relies on the choice of ill-constrained parameters. It is also found to be negligible for icy grains. When fragmentation occurs, it strongly affects the magnetic resistivities profiles. Dust coagulation is a critical process that needs to be fully taken into account during the protostellar collapse. The onset and feedback of fragmentation remains uncertain and its modeling should be further investigated.

astro-ph.GA↗

arXiv2022

The scattering phase: seen at last

The scattering phase, defined as $ \log \det S ( \lambda ) / 2\pi i $ where $ S ( \lambda ) $ is the (unitary) scattering matrix, is the analogue of the counting function for eigenvalues when dealing with exterior domains and is closely related to Krein's spectral shift function. We revisit classical results on asymptotics of the scattering phase and point out that it is never monotone in the case of strong trapping of waves. Perhaps more importantly, we provide the first numerical calculations of scattering phases for non-radial scatterers. They show that the asymptotic Weyl law is accurate even at low frequencies and reveal effects of trapping such as lack of monotonicity. This is achieved by using the recent high level multiphysics finite element software FreeFEM.

math.SP↗

arXiv2022

Fast methods to track grain coagulation and ionization. II. Extension to thermal ionization

Thermal ionization is a critical process at temperatures T > 10 3 K, particularly during star formation. An increase in ionization leads to a decrease in nonideal magnetohydrodynamics (MHD) resistivities, which has a significant impact on protoplanetary disks and protostar formation. We developed an extension of the fast computational ionization method presented in our recent paper to include thermal ionization. The model can be used to inexpensively calculate the density of ions and electrons and the electric charge of each size of grains for an arbitrary size distribution. This tool should be particularly useful for the self-consistent calculation of nonideal MHD resistivities in multidimensional simulations, especially of protostellar collapse and protoplanetary disks.

astro-ph.SR↗

arXiv2021

High-frequency estimates on boundary integral operators for the Helmholtz exterior Neumann problem

We study a commonly-used second-kind boundary-integral equation for solving the Helmholtz exterior Neumann problem at high frequency, where, writing $\Gamma$ for the boundary of the obstacle, the relevant integral operators map $L^2(\Gamma)$ to itself. We prove new frequency-explicit bounds on the norms of both the integral operator and its inverse. The bounds on the norm are valid for piecewise-smooth $\Gamma$ and are sharp up to factors of $\log k$ (where $k$ is the wavenumber), and the bounds on the norm of the inverse are valid for smooth $\Gamma$ and are observed to be sharp at least when $\Gamma$ is smooth with strictly-positive curvature. Together, these results give bounds on the condition number of the operator on $L^2(\Gamma)$; this is the first time $L^2(\Gamma)$ condition-number bounds have been proved for this operator for obstacles other than balls.

math.AP↗

arXiv2021

Universal protoplanetary disk size under complete non-ideal magnetohydrodynamics: The interplay between ion-neutral friction, Hall effect, and the Ohmic dissipation

The role of non-ideal magnetohydrodynamics has been proven critical during the formation of the protoplanetary disk, particularly in regulating its size. We provide a simple model to predict the disk size under the interplay among the ambipolar diffusion, the Hall effect, and the Ohmic dissipation. The model predicts a small disk size of around 20 AU, that depends only sub-linearly on disk parameters, for a wide range of initial conditions of sub-Solar mass and moderate magnetization. It is able to explain phenomena manifested in existing numerical simulations, including the bimodal disk behavior under parallel and anti-parallel alignment between the rotation and magnetic field. In the parallel configuration, the disk size decreases and eventually disappears. In the anti-parallel configuration, and the disk has an outer partition (or pseudo-disk) that is flat, shrinking , and short-lived, as well as a inner partition that grows slowly with mass and is long-lived. Even with significant initial magnetization, the vertical field in the disk can only dominate at the early stage when the mass is low, and the toroidal field eventually dominates in all disks.

astro-ph.EP↗

arXiv2021

Fast methods for tracking grain coagulation and ionization. I. Analytic derivation

Dust grains play a major role in many astrophysical contexts. They affect the chemical, magnetic, dynamical, and optical properties of their environment, from galaxies down to the interstellar medium, star-forming regions, and protoplanetary disks. Their coagulation leads to shifts in their size distribution and ultimately to the formation of planets. However, although the coagulation process is reasonably uncomplicated to numerically implement by itself, it is difficult to couple it with multidimensional hydrodynamics numerical simulations because of its high computational cost. We propose here a simple method for tracking the coagulation of grains at far lower cost. Given an initial grain size distribution, the state of the distribution at time t is solely determined by the value of a single variable integrated along the trajectory, independently of the specific path taken by the grains. Although this method cannot account for processes other than coagulation, it is mathematically exact, fast, inexpensive, and can be used to evaluate the effect of grain coagulation in most astrophysical contexts. It is applicable to all coagulation kernels in which local physical conditions and grain properties can be separated. We also describe another method for calculating the average electric charge of grains and the density of ions and electrons in environments that are shielded from radiation fields, given the density and temperature of the gas, the cosmic-ray ionization rate, and the average mass of the ions. The equations we provide are fast to integrate numerically and can be used in multidimensional numerical simulations to self-consistently calculate on the fly the local resistivities that are required to model nonideal magnetohydrodynamics.

astro-ph.GA↗

arXiv2021

Applying GMRES to the Helmholtz equation with strong trapping: how does the number of iterations depend on the frequency?

We consider GMRES applied to discretisations of the high-frequency Helmholtz equation with strong trapping; recall that in this situation the problem is exponentially ill-conditioned through an increasing sequence of frequencies. Under certain assumptions about the distribution of the eigenvalues, we prove upper bounds on how the number of GMRES iterations grows with the frequency. Our main focus is on boundary-integral-equation formulations of the exterior Dirichlet and Neumann obstacle problems in 2- and 3-d; for these problems, we investigate numerically the sharpness (in terms of dependence on frequency) of both our bounds and various quantities entering our bounds. This paper is therefore the first comprehensive study of the frequency-dependence of the number of GMRES iterations for Helmholtz boundary-integral equations under trapping.

math.NA↗