arXiv subjects

Théophile Chaumont-Frelet

Publications and source records attributed to Théophile Chaumont-Frelet.

At least 19 recordsLinked to original sources

arXiv2026

On $p$-robust convergence and optimality of adaptive FEM driven by equilibrated-flux estimators

Building on existing $hp$-adaptive algorithms driven by equilibrated-flux estimators from [ESAIM Math. Model. Numer. Anal. 57 (2023), 329--366] and the references therein, we propose a novel $h$-adaptive algorithm for a fixed polynomial degree $p$. We consider a conforming finite element discretization of the Poisson equation in two or three space dimensions. Supposing piecewise polynomial right-hand side of degree $p-1$, we show that the algorithm yields error contraction at each step, with a contraction factor that is independent of $p$ provided that a certain {\sl a posteriori} verifiable criterion is satisfied. We further show that this algorithm converges at optimal algebraic rate $s$ if the Dörfler marking parameter is chosen below some specified $p$-independent upper threshold. The constants involved here are $p$-robust, although they may depend on the rate $s$. The theoretical results are supported by numerical experiments, in which the {\sl a posteriori} criterion is always satisfied for one or a few local mesh refinement steps by newest-vertex bisection.

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arXiv2026

A HDG method with transmission variables for time-harmonic wave propagation problems with constant coefficients

Iterative finite element solvers for time-harmonic wave problems are notoriously slow to converge, owing to fundamental properties of these problems. We present a variant of the hybridizable discontinuous Galerkin (HDG) method that is better suited to fast iterative solution. Unlike the standard hybridization strategy, which eliminates physical unknowns by introducing an auxiliary numerical flux on element faces, our approach instead introduces a transmission variable on those faces. For Helmholtz problems, this reformulation, known as CHDG, has been shown to significantly accelerate the convergence of iterative schemes relative to standard HDG. The present work extends CHDG to a general framework covering wave propagation problems with constant coefficients, capable of handling diverse wave types in a unified manner. We prove that the resulting hybridized system is well-posed and amenable to fixed-point iteration. As a practical application, we apply the method to the time-harmonic linearized Euler equations with a uniform subsonic mean flow. The method is illustrated through two-dimensional numerical benchmarks involving both sound and vorticity waves, with a systematic comparison of the convergence behaviour of several iterative schemes across a range of configurations.

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arXiv2026

Spectral coarse spaces based on indefinite operators: the $H_k$-GenEO method

GenEO (`Generalised Eigenvalue problems on the Overlap') is a method for constructing coarse spaces used in the preconditioning of iterative solvers for discrete PDEs. This method combines a (small) number of modes of local PDE eigenproblems to obtain a global coarse space. A coarse solve is then combined with local solves of the global PDE to obtain the preconditioner. A substantial theory for GenEO has been developed for the case when the local elgenproblems are positive semi-definite. This has been applied mostly to positive definite global PDEs, but also recently extended to the case of convection--diffusion--reaction problems, which may be neither self-adjoint, nor positive definite. However, when the global problem is highly indefinite, coarse spaces built from positive semi-definite local eigenproblems fail to be robust in practice. In this paper we consider highly indefinite global PDE problems, characterised by a large parameter $k$ (allowing also highly variable coefficients), and we develop a new spectral coarse space built from solving eigenvalue problems based on \textit{local copies of the global problem}. We put no constraint on the diameters of the local domains, thus allowing the local eigenvalue problems to be indefinite. The new method (which we call $H_k$-GenEO) is seen to be much more robust as $k$ increases than methods based on positive semi-definite eigenproblems. We provide sufficient conditions for robustness of the preconditioned GMRES iterative method, in terms of the tolerance of the local eigenproblems and the size of the subdomains for the local PDE solves. In practice the method is observed to be robust with respect to $k$ under even weaker conditions on the local eigenproblem tolerance. The experiments also suggest the method can be resilient to high variation in PDE coefficients.

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arXiv2026

The pollution effect for FEM approximations of the Ginzburg-Landau equation

In this paper, we investigate the approximation properties of solutions to the Ginzburg-Landau equation (GLE) in finite element spaces. Special attention is given to how the errors are influenced by coupling the mesh size $h$ and the polynomial degree $p$ of the finite element space to the size of the so-called Ginzburg-Landau material parameter $κ$. As observed in previous works, the finite element approximations to the GLE are suffering from a numerical pollution effect, that is, the best-approximation error in the finite element space converges under mild coupling conditions between $h$ and $κ$, whereas the actual finite element solutions possess poor accuracy in a large pre-asymptotic regime which depends on $κ$. In this paper, we provide a new error analysis that allows us to quantify the pre-asymptotic regime and the corresponding pollution effect in terms of explicit resolution conditions. In particular, we are able to prove that higher polynomial degrees reduce the pollution effect, i.e., the accuracy of the best-approximation is achieved under relaxed conditions for the mesh size. We provide both error estimates in the $H^1$- and the $L^2$-norm and we illustrate our findings with numerical examples.

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arXiv2026

Sharp error bounds for edge-element discretisations of the high-frequency Maxwell equations

We prove sharp wavenumber-explicit error bounds for first- or second-family-Nédélec-element (a.k.a. edge-element) conforming discretisations, of arbitrary (fixed) order, of the variable-coefficient time-harmonic Maxwell equations posed in a bounded domain with perfect electric conductor (PEC) boundary conditions. The PDE coefficients are allowed to be piecewise regular and complex-valued; this set-up therefore includes scattering from a PEC obstacle and/or variable real-valued coefficients, with the radiation condition approximated by a perfectly matched layer (PML). In the analysis of the $h$-version of the finite-element method, with fixed polynomial degree $p$, applied to the time-harmonic Maxwell equations, the $\textit{asymptotic regime}$ is when the meshwidth, $h$, is small enough (in a wavenumber-dependent way) that the Galerkin solution is quasioptimal independently of the wavenumber, while the $\textit{preasymptotic regime}$ is the complement of the asymptotic regime. The results of this paper are the first preasymptotic error bounds for the time-harmonic Maxwell equations using first-family Nédélec elements or higher-than-lowest-order second-family Nédélec elements. Furthermore, they are the first wavenumber-explicit results, even in the asymptotic regime, for Maxwell scattering problems with a non-empty scatterer.

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arXiv2025

A generalized Hessian-based error estimator for an IPDG formulation of the biharmonic problem in two dimensions

We consider a two dimensional biharmonic problem and its discretization by means of a symmetric interior penalty discontinuous Galerkin method. A novel split of an error measure based on a generalized Hessian into two terms measuring the conformity and nonconformity of the scheme is proven. This splitting is the departing point for the design of a new error estimator, which is provably reliable and efficient for polynomial degree larger than~$3$, and does not involve any DG stabilization. Such an error estimator can be bounded from above by the standard DG residual error estimator. Numerical results assess the theoretical predictions, including the efficiency of the proposed estimator, for all polynomial degrees larger than or equal to~$2$.

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arXiv2025

Optimal convergence rates of an adaptive finite element method for unbounded domains

We consider linear reaction-diffusion equations posed on unbounded domains, and discretized by adaptive Lagrange finite elements. To obtain finite-dimensional spaces, it is necessary to introduce a truncation boundary, whereby only a bounded computational subdomain is meshed, leading to an approximation of the solution by zero in the remainder of the domain. We propose a residual-based error estimator that accounts for both the standard discretization error as well as the effect of the truncation boundary. This estimator is shown to be reliable and efficient under appropriate assumptions on the triangulation. Based on this estimator, we devise an adaptive algorithm that automatically refines the mesh and pushes the truncation boundary towards infinity. We prove that this algorithm converges and even achieves optimal rates in terms of the number of degrees of freedom. We finally provide numerical examples illustrating our key theoretical findings.

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arXiv2025

Commuting quasi-interpolators and Maxwell compactness for a polytopal de Rham complex

We establish Maxwell compactness results for the Discrete De Rham (DDR) polytopal complex: sequences in this polytopal complex with bounded discrete $\boldsymbol{H}(\mathbf{curl})$ (resp. discrete $\boldsymbol{H}(\mathrm{div})$) norm and orthogonal to discrete gradients (resp. discrete curls) have $L^2$-relatively compact potential reconstructions. The proof of these results hinges on the design of novel quasi-interpolators, that map the minimal-regularity de Rham spaces onto the discrete DDR spaces and form a commuting diagram. A full set of (primal and adjoint) consistency properties is established for these quasi-interpolators, which paves the way to convergence proofs, under minimal-regularity assumptions, of DDR schemes for partial differential equations based on the de Rham complex. Our analysis is performed with generic mixed boundary conditions, also covering the cases of no boundary conditions or fully homogeneous boundary conditions, and leverages recently introduced liftings from the DDR complex to the continuous de Rham complex.

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arXiv2025

A posteriori error estimates and space-adaptive mesh refinements for time-dependent scattering problems

This work studies a posteriori error estimates and their use for time-dependent acoustic scattering problems, formulated as a time-dependent boundary integral equation based on a single-layer ansatz. The integral equation is discretized by the convolution quadrature method in time and by boundary elements in space. We prove the reliability of an error estimator of residual type and study the resulting space-adaptive mesh refinements. Moreover, we present a simple modification of the convolution quadrature method based on temporal shifts, which recovers, for the boundary densities, the full classical temporal convergence order $2m-1$ of the temporal convolution quadrature method based on the $m$-stage convolution quadrature semi-discretization. We numerically observe that the adaptive scheme yields asymptotically optimal meshes for an acoustic scattering problem in two dimensions.

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arXiv2025

Adaptive boundary element methods for regularized combined field integral equations

While the exterior Helmholtz problem with Dirichlet boundary conditions is always well-posed, the associated standard boundary integral equations are not if the squared wavenumber agrees with an eigenvalue of the interior Dirichlet problem. Combined field integral equations are not affected by this spurious resonances but are essentially restricted to sufficiently smooth boundaries. For general Lipschitz domains, the latter integral equations are applicable through suitable regularization. Under fairly general assumptions on the regularizing operator, we propose {\sl a posteriori} computable error estimators for corresponding Galerkin boundary element methods of arbitrary polynomial degree. We show that adaptive mesh-refining algorithms steered by these local estimators converge at optimal algebraic rate with respect to the number of underlying boundary mesh elements. In particular, we consider mixed formulations involving the inverse Laplace--Beltrami as regularizing operator. Numerical examples highlight that in the vicinity of spurious resonances the proposed adaptive algorithm is significantly more performant when applied to the regularized combined field equation rather than the standard one.

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arXiv2025

Computable Poincaré--Friedrichs constants for the $L^{p}$ de~Rham complex over convex domains and domains with shellable triangulations

We construct potentials for the exterior derivative, in particular, for the gradient, the curl, and the divergence operators, over domains with shellable triangulations. Notably, the class of shellable triangulations includes local patches (stars) in two or three dimensions. The operator norms of our potentials satisfy explicitly computable bounds that depend only on the geometry. We thus compute upper bounds for constants in Poincaré--Friedrichs inequalities and lower bounds for the eigenvalues of vector Laplacians. As an additional result with independent standing, we establish Poincaré--Friedrichs inequalities with computable constants for the $L^{p}$ de~Rham complex over bounded convex domains, derived as explicit operator norms of regularized Poincaré and Bogovski\uı potential operators. We express all our main results in the calculus of differential forms and treat the gradient, curl, and divergence operators as instances of the exterior derivative. Computational examples illustrate the theoretical findings.

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arXiv2025

A hybridizable discontinuous Galerkin method with transmission variables for time-harmonic electromagnetic problems

The CHDG method is a hybridizable discontinuous Galerkin (HDG) finite element method suitable for the iterative solution of time-harmonic wave propagation problems. Hybrid unknowns corresponding to transmission variables are introduced at the element interfaces and the physical unknowns inside the elements are eliminated, resulting in a hybridized system with favorable properties for fast iterative solution. In this paper, we extend the CHDG method, initially studied for the Helmholtz equation, to the time-harmonic Maxwell equations. We prove that the local problems stemming from hybridization are well-posed and that the fixed-point iteration naturally associated to the hybridized system is contractive. We propose a 3D implementation with a discrete scheme based on nodal basis functions. The resulting solver and different iterative strategies are studied with several numerical examples using a high-performance parallel C++ code.

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arXiv2025

A hybridizable discontinuous Galerkin method with transmission variables for time-harmonic acoustic problems in heterogeneous media

We consider the finite element solution of time-harmonic wave propagation problems in heterogeneous media with hybridizable discontinuous Galerkin (HDG) methods. In the case of homogeneous media, it has been observed that the iterative solution of the linear system can be accelerated by hybridizing with transmission variables instead of numerical traces, as performed in standard approaches. In this work, we extend the HDG method with transmission variables, which is called the CHDG method, to the heterogeneous case with piecewise constant physical coefficients. In particular, we consider formulations with standard upwind and general symmetric fluxes. The CHDG hybridized system can be written as a fixed-point problem, which can be solved with stationary iterative schemes for a class of symmetric fluxes. The standard HDG and CHDG methods are systematically studied with the different numerical fluxes by considering a series of 2D numerical benchmarks. The convergence of standard iterative schemes is always faster with the extended CHDG method than with the standard HDG methods, with upwind and scalar symmetric fluxes.

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arXiv2025

The geometric error is less than the pollution error when solving the high-frequency Helmholtz equation with high-order FEM on curved domains

We consider the $h$-version of the finite-element method, where accuracy is increased by decreasing the meshwidth $h$ while keeping the polynomial degree $p$ constant, applied to the Helmholtz equation. Although the question "how quickly must $h$ decrease as the wavenumber $k$ increases to maintain accuracy?" has been studied intensively since the 1990s, none of the existing rigorous wavenumber-explicit analyses take into account the approximation of the geometry. In this paper we prove that for nontrapping problems solved using straight elements the geometric error is order $kh$, which is then less than the pollution error $k(kh)^{2p}$ when $k$ is large; this fact is then illustrated in numerical experiments. More generally, we prove that, even for problems with strong trapping, using degree four (in 2-d) or degree five (in 3-d) polynomials and isoparametric elements ensures that the geometric error is smaller than the pollution error for most large wavenumbers.

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arXiv2025

Generalised gradients for virtual elements and applications to a posteriori error analysis

We rewrite the standard nodal virtual element method as a generalised gradient method. This re-formulation allows for computing a reliable and efficient error estimator by locally reconstructing broken fluxes and potentials on elemental subtriangulations. We prove the usual upper and lower bounds with constants independent of the stabilisation of the method and, under technical assumptions on the mesh, the degree of accuracy.

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arXiv2023

A stable local commuting projector and optimal $hp$ approximation estimates in ${\boldsymbol H}(\mathrm{curl})$

We design an operator from the infinite-dimensional Sobolev space ${\boldsymbol H}(\mathrm{curl})$ to its finite-dimensional subspace formed by the Nédélec piecewise polynomials on a tetrahedral mesh that has the following properties: 1) it is defined over the entire ${\boldsymbol H}(\mathrm{curl})$, including boundary conditions imposed on a part of the boundary; 2) it is defined locally in a neighborhood of each mesh element; 3) it is based on simple piecewise polynomial projections; 4) it is stable in the ${\boldsymbol L}^2$-norm, up to data oscillation; 5) it has optimal (local-best) approximation properties; 6) it satisfies the commuting property with its sibling operator on ${\boldsymbol H}(\mathrm{div})$; 7) it is a projector, i.e., it leaves intact objects that are already in the Nédélec piecewise polynomial space. This operator can be used in various parts of numerical analysis related to the ${\boldsymbol H}(\mathrm{curl})$ space. We in particular employ it here to establish the two following results: i) equivalence of global-best, tangential-trace-and curl-constrained, and local-best, unconstrained approximations in ${\boldsymbol H}(\mathrm{curl})$ including data oscillation terms; and ii) fully $h$- and $p$- (mesh-size- and polynomial-degree-) optimal approximation bounds valid under the minimal Sobolev regularity only requested elementwise. As a result of independent interest, we also prove a $p$-robust equivalence of curl-constrained and unconstrained best-approximations on a single tetrahedron in the ${\boldsymbol H}(\mathrm{curl})$-setting, including $hp$ data oscillation terms.

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arXiv2023

Explicit bounds for the high-frequency time-harmonic Maxwell equations in heterogeneous media

We consider the time-harmonic Maxwell equations posed in $\mathbb{R}^3$. We prove a priori bounds on the solution for $L^\infty$ coefficients $ε$ and $μ$ satisfying certain monotonicity properties, with these bounds valid for arbitrarily-large frequency, and explicit in the frequency and properties of $ε$ and $μ$. The class of coefficients covered includes (i) certain $ε$ and $μ$ for which well-posedness of the time-harmonic Maxwell equations had not previously been proved, and (ii) scattering by a penetrable $C^0$ star-shaped obstacle where $ε$ and $μ$ are smaller inside the obstacle than outside. In this latter setting, the bounds are uniform across all such obstacles, and the first sharp frequency-explicit bounds for this problem at high-frequency.

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arXiv2022

Scattering by finely-layered obstacles: frequency-explicit bounds and homogenization

We consider the scalar Helmholtz equation with variable, discontinuous coefficients, modelling transmission of acoustic waves through an anisotropic penetrable obstacle. We first prove a well-posedness result and a frequency-explicit bound on the solution operator, with both valid for sufficiently-large frequency and for a class of coefficients that satisfy certain monotonicity conditions in one spatial direction, and are only assumed to be bounded (i.e., $L^\infty$) in the other spatial directions. This class of coefficients therefore includes coefficients modelling transmission by penetrable obstacles with a (potentially large) number of layers (in 2-d) or fibres (in 3-d). Importantly, the frequency-explicit bound holds uniformly for all coefficients in this class; this uniformity allows us to consider highly-oscillatory coefficients and study the limiting behaviour when the period of oscillations goes to zero. In particular, we bound the $H^1$ error committed by the first-order bulk correction to the homogenized transmission problem, with this bound explicit in both the period of oscillations of the coefficients and the frequency of the Helmholtz equation; to our knowledge, this is the first homogenization result for the Helmholtz equation that is explicit in these two quantities and valid without the assumption that the frequency is small.

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