arXiv subjects

Xavier Claeys

Publications and source records attributed to Xavier Claeys.

At least 19 recordsLinked to original sources

arXiv2026

Coarse space preconditioning for Generalized Optimized Schwarz Methods. Part I: continuous case

The Generalized Optimized Schwarz Method (GOSM) originally proposed in [Claeys, 2021] is a variant of Depr\'es algorithm, a domain decomposition strategy for the solution of harmonic wave propagation problems. It imposes transmission conditions through interfaces by means of a non-local exchange operator. Conducting our analysis at the continuous level, in an infinite dimensional setting, we propose a coarse space construction for the preconditioning of the GOSM formulation, and provide estimates for the convergence of GMRes applied to the preconditioned equation.

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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.

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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.

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arXiv2025

Hierarchical matrix approximability of inverse of convection dominated finite element matrices

Several researchers have developed a rich toolbox of matrix compression techniques that exploit structure and redundancy in large matrices. Classical methods such as the block low-rank format and the Fast Multipole Method make it possible to manipulate very large systems by representing them in a reduced form. Among the most sophisticated tools in this area are hierarchical matrices (H-matrices), which exploit local properties of the underlying kernel or operator to approximate matrix blocks by low-rank factors, organized in a recursive hierarchy. H-matrices offer a flexible and scalable framework, yielding nearly linear complexity in both storage and computation. Hierarchical matrix techniques, originally developed for boundary integral equations, have recently been applied to matrices stemming from the discretization of advection-dominated problems. However, their effectiveness is limited by the loss of coercivity induced by convection phenomena, where traditional methods fail. Initial work by Le Borne addressed this by modifying the admissibility criterion for structured grids with constant convection, but challenges remain for more general grids and advection fields. In this work, we propose a novel partitioning strategy based on "convection tubes", clusters aligned with the convection vector field. This method does not require a structured grid or constant convection, overcoming the limitations of previous approaches. We present both theoretical analyses and numerical experiments, that demonstrate the efficiency and robustness of our method for convection-dominated PDEs on unstructured grids. The approach builds on a P\'eclet-robust Caccioppoli inequality, crucial for handling convection-dominated problems.

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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.

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arXiv2024

Accelerating non-local exchange in generalized optimized Schwarz methods

The generalized optimised Schwarz method proposed in [Claeys & Parolin, 2022] is a variant of the Despr\'es algorithm for solving harmonic wave problems where transmission conditions are enforced by means of a non-local exchange operator. We introduce and analyse an acceleration technique that significantly reduces the cost of applying this exchange operator without deteriorating the precision and convergence speed of the overall domain decomposition algorithm.

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arXiv2023

Boundary Element Methods for the Laplace Hypersingular Integral Equation on Multiscreens: a two-level Substructuring Preconditioner

We present a preconditioning method for the linear systems arising from the boundary element discretization of the Laplace hypersingular equation on a $2$-dimensional triangulated surface $\Gamma$ in $\mathbb{R}^3$. We allow $\Gamma$ to belong to a large class of geometries that we call polygonal multiscreens, which can be non-manifold. After introducing a new, simple conforming Galerkin discretization, we analyze a substructuring domain-decomposition preconditioner based on ideas originally developed for the Finite Element Method. The surface $\Gamma$ is subdivided into non-overlapping regions, and the application of the preconditioner is obtained via the solution of the hypersingular equation on each patch, plus a coarse subspace correction. We prove that the condition number of the preconditioned linear system grows poly-logarithmically with $H/h$, the ratio of the coarse mesh and fine mesh size, and our numerical results indicate that this bound is sharp. This domain-decomposition algorithm therefore guarantees significant speedups for iterative solvers, even when a large number of subdomains is used.

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arXiv2023

Multi-domain FEM-BEM coupling for acoustic scattering

We model time-harmonic acoustic scattering by an object composed of piece-wise homogeneous parts and an arbitrarily heterogeneous part. We propose and analyze new formulations that couple, adopting a Costabel-type approach, boundary integral equations for the homogeneous subdomains with volume variational formulations for the heterogeneous subdomain. This is an extension of the Costabel FEM-BEM coupling to a multi-domain configuration, with cross-points allowed, i.e. points where three or more subdomains are adjacent. While generally just the exterior unbounded subdomain is treated with the BEM, here we wish to exploit the advantages of BEM whenever it is applicable, that is, for all the homogeneous parts of the scattering object. Our formulation is based on the multi-trace formalism, which initially was introduced for acoustic scattering by piece-wise homogeneous objects. Instead, here we allow the wavenumber to vary arbitrarily in a part of the domain. We prove that the bilinear form associated with the proposed formulation satisfies a G{\aa}rding coercivity inequality, which ensures stability of the variational problem if it is uniquely solvable. We identify conditions for injectivity and construct modified versions immune to spurious resonances.

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arXiv2023

Non-local optimized Schwarz method with physical boundaries

We extend the theoretical framework of non-local optimized Schwarz methods as introduced in [Claeys,2021], considering an Helmholtz equation posed in a bounded cavity supplemented with a variety of conditions modeling material boundaries. The problem is reformulated equivalently as an equation posed on the skeleton of a non-overlapping partition of the computational domain, involving an operator of the form "identity + contraction". The analysis covers the possibility of resonance phenomena where the Helmholtz problem is not uniquely solvable. In case of unique solvability, the skeleton formulation is proved coercive, and an explicit bound for the coercivity constant is provided in terms of the inf-sup constant of the primary Helmholtz boundary value problem.

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arXiv2022

Fractured Meshes

This work introduces ``generalized meshes", a type of meshes suited for the discretization of partial differential equations in non-regular geometries. Generalized meshes extend regular simplicial meshes by allowing for overlapping elements and more flexible adjacency relations. They can have several distinct ``generalized" vertices (or edges, faces) that occupy the same geometric position. These generalized facets are the natural degrees of freedom for classical conforming spaces of discrete differential forms appearing in finite and boundary element applications. Special attention is devoted to the representation of fractured domains and their boundaries. An algorithm is proposed to construct the so-called {\em virtually inflated mesh}, which correspond to a ``two-sided" mesh of a fracture. Discrete $d$-differential forms on the virtually inflated mesh are characterized as the trace space of discrete $d$-differential forms in the surrounding volume.

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arXiv2022

How does the partition of unity influence SORAS preconditioner?

We investigate the influence of the choice of the partition of unity on the convergence of the Symmetrized Optimized Restricted Additive Schwarz (SORAS) preconditioner for the reaction-convection-diffusion equation. We focus on two kinds of partitions of unity, and study the dependence on the overlap and on the number of subdomains. In particular, the second kind of partition of unity, which is non-zero in the interior of the whole overlapping region, gives more favorable convergence properties, especially when increasing the overlap width, in comparison with the first kind of partition of unity, whose gradient is zero on the subdomain interfaces and which would be the natural choice for ORAS solver instead.

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arXiv2022

A new perspective on the fictitious space lemma

In the present contribution we propose a new proof of the so-called fictitious space lemma. For the proof, we exhibit an explicit expression for the inverse of additive Schwarz preconditioners in terms of Moore-Penrose pseudo inverse of the map associated to the decomposition over the subdomain partition.

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arXiv2022

A Directional Equispaced interpolation-based Fast Multipole Method for oscillatory kernels

Fast Multipole Methods (FMMs) based on the oscillatory Helmholtz kernel can reduce the cost of solving N-body problems arising from Boundary Integral Equations (BIEs) in acoustic or electromagnetics. However, their cost strongly increases in the high-frequency regime. This paper introduces a new directional FMM for oscillatory kernels (defmm - directional equispaced interpolation-based fmm), whose precomputation and application are FFT-accelerated due to polynomial interpolations on equispaced grids. We demonstrate the consistency of our FFT approach, and show how symmetries can be exploited in the Fourier domain. We also describe the algorithmic design of defmm, well-suited for the BIE non-uniform particle distributions, and present performance optimizations on one CPU core. Finally, we exhibit important performance gains on all test cases for defmm over a state-of-the-art FMM library for oscillatory kernels.

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arXiv2021

Nonlocal Optimized Schwarz Methods for time-harmonic electromagnetics

We introduce a new domain decomposition strategy for time harmonic Maxwell's equations that is valid in the case of automatically generated subdomain partitions with possible presence of cross-points. The convergence of the algorithm is guaranteed and we present a complete analysis of the matrix form of the method. The method involves transmission matrices responsible for imposing coupling between subdomains. We discuss the choice of such matrices, their construction and the impact of this choice on the convergence of the domain decomposition algorithm. Numerical results and algorithms are provided.

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arXiv2021

Non-self adjoint impedance in Generalized Optimized Schwarz Methods

We present a convergence theory for Optimized Schwarz Methods that rely on a non-local exchange operator and covers the case of coercive possibly non-self-adjoint impedance operators. This analysis also naturally deals with the presence of cross-points in subdomain partitions of arbitrary shape. In the particular case of self-adjoint impedance, we recover the theory proposed in [Claeys & Parolin, 2021].

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arXiv2021

Continuity estimates for Riesz potentials on polygonal boundaries

Riesz potentials are well known objects of study in the theory of singular integrals that have been the subject of recent, increased interest from the numerical analysis community due to their connections with fractional Laplace problems and proposed use in certain domain decomposition methods. While the L$^p$-mapping properties of Riesz potentials on flat geometries are well-established, comparable results on rougher geometries for Sobolev spaces are very scarce. In this article, we study the continuity properties of the surface Riesz potential generated by the $1/\sqrt{x}$ singular kernel on a polygonal domain $\Omega \subset \mathbb{R}^2$. We prove that this surface Riesz potential maps L$^{2}(\partial\Omega)$ into H$^{+1/2}(\partial\Omega)$. Our proof is based on a careful analysis of the Riesz potential in the neighbourhood of corners of the domain $\Omega$. The main tool we use for this corner analysis is the Mellin transform which can be seen as a counterpart of the Fourier transform that is adapted to corner geometries.

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arXiv2020

Analysis of the SORAS domain decomposition preconditioner for non-self-adjoint or indefinite problems

We analyze the convergence of the one-level overlapping domain decomposition preconditioner SORAS (Symmetrized Optimized Restricted Additive Schwarz) applied to a generic linear system whose matrix is not necessarily symmetric/self-adjoint nor positive definite. By generalizing the theory for the Helmholtz equation developed in [I.G. Graham, E.A. Spence, and J. Zou, SIAM J.Numer.Anal., 2020], we identify a list of assumptions and estimates that are sufficient to obtain an upper bound on the norm of the preconditioned matrix, and a lower bound on the distance of its field of values from the origin. We stress that our theory is general in the sense that it is not specific to one particular boundary value problem. Moreover, it does not rely on a coarse mesh whose elements are sufficiently small. As an illustration of this framework, we prove new estimates for overlapping domain decomposition methods with Robin-type transmission conditions for the heterogeneous reaction-convection-diffusion equation (to prove the stability assumption for this equation we consider the case of a coercive bilinear form, which is non-symmetric, though).

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arXiv2020

Local Multiple Traces Formulation for Electromagnetics: Stability and Preconditioning for Smooth Geometries

We consider the time-harmonic electromagnetic transmission problem for the unit sphere. Appealing to a vector spherical harmonics analysis, we prove the first stability result of the local multiple trace formulation (MTF) for electromagnetics, originally introduced by Hiptmair and Jerez-Hanckes [Adv. Comp. Math. 37 (2012), 37-91] for the acoustic case, paving the way towards an extension to general piecewise homogeneous scatterers. Moreover, we investigate preconditioning techniques for the local MTF scheme and study the accumulation points of induced operators. In particular, we propose a novel second-order inverse approximation of the operator. Numerical experiments validate our claims and confirm the relevance of the preconditioning strategies.

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