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108 results on '"Chaumont-Frelet, Théophile"'

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

Author

Chaumont-Frelet, Théophile, Galkowski, Jeffrey, and Spence, Euan A.

Subjects
Mathematics - Numerical Analysis
Abstract

We prove sharp wavenumber-explicit error bounds for first- or second-type-N\'ed\'elec-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-type N\'ed\'elec elements or higher-than-lowest-order second-type N\'ed\'elec elements. Furthermore, they are the first wavenumber-explicit results, even in the asymptotic regime, for Maxwell scattering problems with a non-empty scatterer.

Published
2024

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

Author

Chaumont-Frelet, Théophile, Gedicke, Joscha, and Mascotto, Lorenzo

Subjects
Mathematics - Numerical Analysis, 65N12, 65N15
Abstract

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. 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.

Published
2024

3. Polynomial-degree-robust $\protect H({\protect \bf curl})$-stability of discrete minimization in a tetrahedron

Author

Chaumont-Frelet, Théophile, Ern, Alexandre, and Vohralík, Martin

Subjects
Mathematics, QA1-939
Abstract

We prove that the minimizer in the Nédélec polynomial space of some degree $p\ge 0$ of a discrete minimization problem performs as well as the continuous minimizer in $H({\bf curl})$, up to a constant that is independent of the polynomial degree $p$. The minimization problems are posed for fields defined on a single non-degenerate tetrahedron in $\mathbb{R}^3$ with polynomial constraints enforced on the curl of the field and its tangential trace on some faces of the tetrahedron. This result builds upon [L. Demkowicz, J. Gopalakrishnan, J. Schöberl, SIAM J. Numer. Anal. 47 (2009), 3293–3324] and [M. Costabel, A. McIntosh, Math. Z. 265 (2010), 297–320] and is a fundamental ingredient to build polynomial-degree-robust a posteriori error estimators when approximating the Maxwell equations in several regimes leading to a curl-curl problem.

Published
2021
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4. The geometric error is less than the pollution error when solving the high-frequency Helmholtz equation with high-order FEM on curved domains

Author

Chaumont-Frelet, Théophile and Spence, Euan A.

Subjects
Mathematics - Numerical Analysis
Abstract

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.

Published
2024

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

Author

Chaumont-Frelet, Théophile, Moiola, Andrea, and Spence, Euan A.

Subjects
Mathematics - Analysis of PDEs
Abstract

We consider the time-harmonic Maxwell equations posed in $\mathbb{R}^3$. We prove a priori bounds on the solution for $L^\infty$ coefficients $\epsilon$ and $\mu$ satisfying certain monotonicity properties, with these bounds valid for arbitrarily-large frequency, and explicit in the frequency and properties of $\epsilon$ and $\mu$. The class of coefficients covered includes (i) certain $\epsilon$ and $\mu$ 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 $\epsilon$ and $\mu$ 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.

Published
2023
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6. A stable local commuting projector and optimal $hp$ approximation estimates in ${\boldsymbol H}(\mathrm{curl})$

Author

Chaumont-Frelet, Théophile and Vohralík, Martin

Subjects
Mathematics - Numerical Analysis
Abstract

We design an operator from the infinite-dimensional Sobolev space ${\boldsymbol H}(\mathrm{curl})$ to its finite-dimensional subspace formed by the N\'ed\'elec 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 N\'ed\'elec 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.

Published
2022

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

Author

Chaumont-Frelet, Théophile and Spence, Euan A.

Subjects
Mathematics - Analysis of PDEs
Abstract

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.

Published
2021

8. $p$-robust equilibrated flux reconstruction in ${\boldsymbol H}(\mathrm{curl})$ based on local minimizations. Application to a posteriori analysis of the curl-curl problem

Author

Chaumont-Frelet, Théophile and Vohralík, Martin

Subjects
Mathematics - Numerical Analysis
Abstract

We present a local construction of H(curl)-conforming piecewise polynomials satisfying a prescribed curl constraint. We start from a piecewise polynomial not contained in the H(curl) space but satisfying a suitable orthogonality property. The procedure employs minimizations in vertex patches and the outcome is, up to a generic constant independent of the underlying polynomial degree, as accurate as the best-approximations over the entire local versions of H(curl). This allows to design guaranteed, fully computable, constant-free, and polynomial-degree-robust a posteriori error estimates of Prager-Synge type for N\'ed\'elec finite element approximations of the curl-curl problem. A divergence-free decomposition of a divergence-free H(div)-conforming piecewise polynomial, relying on over-constrained minimizations in Raviart-Thomas spaces, is the key ingredient. Numerical results illustrate the theoretical developments.

Published
2021

9. Stable broken H(curl) polynomial extensions and p-robust a posteriori error estimates by broken patchwise equilibration for the curl-curl problem

Author

Chaumont-Frelet, Théophile, Ern, Alexandre, and Vohralík, Martin

Subjects
Mathematics - Numerical Analysis, 65N30, 78M10, 65N15
Abstract

We study extensions of piecewise polynomial data prescribed in a patch of tetrahedra sharing an edge. We show stability in the sense that the minimizers over piecewise polynomial spaces with prescribed tangential component jumps across faces and prescribed piecewise curl in elements are subordinate in the broken energy norm to the minimizers over the broken H(curl) space with the same prescriptions. Our proofs are constructive and yield constants independent of the polynomial degree. We then detail the application of this result to the a posteriori error analysis of the curl-curl problem discretized with N\'ed\'elec finite elements of arbitrary order. The resulting estimators are reliable, locally efficient, polynomial-degree-robust, and inexpensive. They are constructed by a broken patchwise equilibration which, in particular, does not produce a globally H(curl)-conforming flux. The equilibration is only related to edge patches and can be realized without solutions of patch problems by a sweep through tetrahedra around every mesh edge. The error estimates become guaranteed when the regularity pick-up constant is explicitly known. Numerical experiments illustrate the theoretical findings.

Published
2020

10. Polynomial-degree-robust H(curl)-stability of discrete minimization in a tetrahedron

Author

Chaumont-Frelet, Théophile, Ern, Alexandre, and Vohralík, Martin

Subjects
Mathematics - Numerical Analysis, 65N15, 65N30, 76M10
Abstract

We prove that the minimizer in the N\'ed\'elec polynomial space of some degree p > 0 of a discrete minimization problem performs as well as the continuous minimizer in H(curl), up to a constant that is independent of the polynomial degree p. The minimization problems are posed for fields defined on a single non-degenerate tetrahedron in R^3 with polynomial constraints enforced on the curl of the field and its tangential trace on some faces of the tetrahedron. This result builds upon [L. Demkowicz, J. Gopalakrishnan, J. Sch\"oberl SIAM J. Numer. Anal. 47 (2009), 3293--3324] and [M. Costabel, A. McIntosh, Math. Z. 265 (2010), 297--320] and is a fundamental ingredient to build polynomial-degree-robust a posteriori error estimators when approximating the Maxwell equations in several regimes leading to a curl-curl problem.

Published
2020

11. A generalized finite element method for problems with sign-changing coefficients

Author

Chaumont-Frelet, Théophile and Verfürth, Barbara

Subjects
Mathematics - Numerical Analysis, 65N30, 65N12, 65N15, 78A48, 35J20
Abstract

Problems with sign-changing coefficients occur, for instance, in the study of transmission problems with metamaterials. In this work, we present and analyze a generalized finite element method in the spirit of the Localized Orthogonal Decomposition, that is especially efficient when the negative and positive materials exhibit multiscale features. We derive optimal linear convergence in the energy norm independently of the potentially low regularity of the exact solution. Numerical experiments illustrate the theoretical convergence rates and show the applicability of the method for a large class of sign-changing diffusion problems.

Published
2020

23. An Analysis of High-Frequency Helmholtz Problems in Domains with Conical Points and Their Finite Element Discretisation.

Author

Chaumont-Frelet, Théophile and Nicaise, Serge

Subjects
POLLUTION
Abstract

We consider Helmholtz problems in three-dimensional domains featuring conical points. We focus on the high-frequency regime and derive novel sharp upper-bounds for the stress intensity factors of the singularities associated with the conical points. We then employ these new estimates to analyse the stability of finite element discretisations. Our key result is that lowest-order Lagrange finite elements are stable under the assumption that " ω 2 ⁢ h is small". This assumption is standard and well known in the case of smooth domains, and we show that it naturally extends to domain with conical points, even when using uniform meshes. [ABSTRACT FROM AUTHOR]

Published
2023
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24. Decay of coefficients and approximation rates in Gabor Gaussian frames

Author

Chaumont-Frelet, Théophile, Ingremeau, Maxime, Modélisation et méthodes numériques pour le calcul d'interactions onde-matière nanostructurée (ATLANTIS), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Laboratoire Jean Alexandre Dieudonné (LJAD), and Inria

Subjects
Phase-space analysis, Approximation theory, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Gabor frames, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

The aim of this note is to present a self-contained proof of the fact that a function can be approximated using a linear combination of Gaussian coherent states, with a number of terms controlled in terms of the smoothness and of the decay at infinity of the function. This result can easily be obtained using advanced results on modulation spaces, but the proof presented here is completely elementary and self-contained.

Published
2023

25. Flux approximation on unfitted meshes and application to multiscale hybrid-mixed methods

Author

Chaumont-Frelet, Théophile, Paredes, Diego, Valentin, Frédéric, Modélisation et méthodes numériques pour le calcul d'interactions onde-matière nanostructurée (ATLANTIS), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Laboratoire Jean Alexandre Dieudonné (LJAD), Universidad de Concepción - University of Concepcion [Chile], and Laboratorio Nacional de Computação Cientifica [Rio de Janeiro] (LNCC / MCT)

Subjects
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

The flux variable determines the approximation quality of hybridizationbased numerical methods. This work proves that approximating flux variables in discontinuous polynomial spaces from the L2 orthogonal projection is super-convergent on meshes that are not aligned with jumping coefficient interfaces. The results assume only the local regularity of exact solutions in physical partitions. Based on the proposed flux approximation, we demonstrate that the mixed hybrid multiscale (MHM) finite element method is superconvergent in unfitted meshes, supporting the numerics presented in MHM seminal works.

Published
2022

26. A stable local commuting projector and optimal hp approximation estimates in H(curl)

Author

Chaumont-Frelet, Théophile, Vohralík, Martin, Modélisation et méthodes numériques pour le calcul d'interactions onde-matière nanostructurée (ATLANTIS), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Laboratoire Jean Alexandre Dieudonné (LJAD), Simulation for the Environment: Reliable and Efficient Numerical Algorithms (SERENA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), and European Project: 647134,H2020 ERC,ERC-2014-CoG,GATIPOR(2015)

Subjects
Sobolev space H(curl), Minimal regularity, Piecewise polynomial, Commuting projector, Error bound, Constrained-unconstrained equivalence, Nédélec space, hp approximation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

We design an operator from the infinite-dimensional Sobolev space H(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 H(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 L2-norm, up to data oscillation; 5) it has optimal (local-best) approximation properties; 6) it satisfies the commuting property with its sibling operator on H(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 H(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 H(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 H(curl)setting, including hp data oscillation terms.

Published
2022

27. Efficient approximation of high-frequency Helmholtz solutions by Gaussian coherent states

Author

Chaumont-Frelet, Théophile, Dolean, Victorita, Ingremeau, Maxime, Modélisation et méthodes numériques pour le calcul d'interactions onde-matière nanostructurée (ATLANTIS), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Laboratoire Jean Alexandre Dieudonné (LJAD), Department of Mathematics and Statistics [Univ Strathclyde], and University of Strathclyde [Glasgow]

Subjects
Mathematics - Analysis of PDEs, FOS: Mathematics, high-frequency problems, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Helmholtz equation, Mathematics - Numerical Analysis, Numerical Analysis (math.NA), Gabor frames, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract

We introduce new finite-dimensional spaces specifically designed to approximate the solutions to high-frequency Helmholtz problems with smooth variable coefficients in dimension $d$. These discretization spaces are spanned by Gaussian coherent states, that have the key property to be localised in phase space. We carefully select the Gaussian coherent states spanning the approximation space by exploiting the (known) micro-localisation properties of the solution. For a large class of source terms (including plane-wave scattering problems), this choice leads to discrete spaces that provide a uniform approximation error for all wavenumber $k$ with a number of degrees of freedom scaling as $k^{d-1/2}$, which we rigorously establish. In comparison, for discretization spaces based on (piecewise) polynomials, the number of degrees of freedom has to scale at least as $k^d$ to achieve the same property. These theoretical results are illustrated by one-dimensional numerical examples, where the proposed discretization spaces are coupled with a least-squares variational formulation.

Published
2022

28. Frequency-explicit a posteriori error estimates for discontinuous Galerkin discretizations of Maxwell's equations

Author

Chaumont-Frelet, Théophile, Vega, Patrick, Modélisation et méthodes numériques pour le calcul d'interactions onde-matière nanostructurée (ATLANTIS), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Laboratoire Jean Alexandre Dieudonné (LJAD), and Pontificia Universidad Católica de Valparaíso (PUCV)

Subjects
a posteriori error estimates, discontinuous Galerkin methods, Mathematics - Analysis of PDEs, Maxwell's equations, FOS: Mathematics, high-frequency problems, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], hp-adaptivity, Numerical Analysis (math.NA), Mathematics - Numerical Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract

We propose a new residual-based a posteriori error estimator for discontinuous Galerkin discretizations of time-harmonic Maxwell's equations in first-order form. We establish that the estimator is reliable and efficient, and the dependency of the reliability and efficiency constants on the frequency is analyzed and discussed. The proposed estimates generalize similar results previously obtained for the Helmholtz equation and conforming finite element discretization of Maxwell's equations. In addition, for the discontinuous Galerkin scheme considered here, we also show that the proposed estimator is asymptotically constant-free for smooth solutions. We also present two-dimensional numerical examples that highlight our key theoretical findings and suggest that the proposed estimator is suited to drive $h$- and $hp$-adaptive iterative refinements., arXiv admin note: substantial text overlap with arXiv:2009.09204

Published
2022

30. p-ROBUST EQUILIBRATED FLUX RECONSTRUCTION IN H(curl) BASED ON LOCAL MINIMIZATIONS: APPLICATION TO A POSTERIORI ANALYSIS OF THE CURL-CURL PROBLEM.

Author

CHAUMONT-FRELET, THÉOPHILE and VOHRALÍK, MARTIN

Subjects
*SOBOLEV spaces, *A posteriori error analysis, *POLYNOMIALS
Abstract

We present a local construction of H(curl)-conforming piecewise polynomials satisfying a prescribed curl constraint. We start from a piecewise polynomial not contained in the H(curl) space but satisfying a suitable orthogonality property. The procedure employs minimizations in vertex patches and the outcome is, up to a generic constant independent of the underlying polynomial degree, as accurate as the best-approximations over the entire local versions of H(curl). This allows to design guaranteed, fully computable, constant-free, and polynomial-degree-robust a posteriori error estimates of Prager-Synge type for Nédélec finite element approximations of the curl-curl problem. A divergence-free decomposition of a divergence-free H(div)-conforming piecewise polynomial, relying on over-constrained minimizations in Raviart-Thomas spaces, is the key ingredient. Numerical results illustrate the theoretical developments. [ABSTRACT FROM AUTHOR]

Published
2023
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31. A multiscale hybrid-mixed method for Helmholtz problems in periodic structures

Author

Chaumont-Frelet, Théophile, kassali, zakaria, Lanteri, Stéphane, Modélisation et méthodes numériques pour le calcul d'interactions onde-matière nanostructurée (ATLANTIS), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), and Laboratoire Jean Alexandre Dieudonné (LJAD)

Subjects
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience

Published
2022

32. Efficient a posteriori estimates for the wave equation

Author

Chaumont-Frelet, Théophile, Modélisation et méthodes numériques pour le calcul d'interactions onde-matière nanostructurée (ATLANTIS), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), and Laboratoire Jean Alexandre Dieudonné (LJAD)

Subjects
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience

Published
2022

34. Scattering by finely-layered obstacles: frequency-explicit stability and homogenization

Author

Chaumont-Frelet, Théophile, Spence, Euan, Modélisation et méthodes numériques pour le calcul d'interactions onde-matière nanostructurée (ATLANTIS), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Laboratoire Jean Alexandre Dieudonné (LJAD), Department of Mathematical Sciences [Bath], and University of Bath [Bath]

Subjects
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]
Abstract

International audience

Published
2022

36. Bridging the Multiscale Hybrid-Mixed and Multiscale Hybrid High-Order methods

Author

Chaumont-Frelet, Théophile, Ern, Alexandre, Lemaire, Simon, Valentin, Frédéric, Modélisation et méthodes numériques pour le calcul d'interactions onde-matière nanostructurée (ATLANTIS), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), Simulation for the Environment: Reliable and Efficient Numerical Algorithms (SERENA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Reliable numerical approximations of dissipative systems (RAPSODI ), Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe, Laboratorio Nacional de Computação Cientifica [Rio de Janeiro] (LNCC / MCT), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Alexandre Dieudonné (JAD), Laboratoire Paul Painlevé - UMR 8524 (LPP), Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Inria Lille - Nord Europe, Centre National de la Recherche Scientifique (CNRS)-Université de Lille, Laboratoire Jean Alexandre Dieudonné (LJAD), and Université de Lille-Centre National de la Recherche Scientifique (CNRS)

Subjects
Numerical Analysis (math.NA), 010103 numerical & computational mathematics, 01 natural sciences, Highly heterogeneous diffusion, 010101 applied mathematics, High-order methods, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], General polytopal meshes, Mathematics - Numerical Analysis, 0101 mathematics, Multiscale methods, ComputingMilieux_MISCELLANEOUS, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; We establish the equivalence between the Multiscale Hybrid-Mixed (MHM) and the Multiscale Hybrid High-Order (MsHHO) methods for a variable diffusion problem with piecewise polynomial source term. Under the idealized assumption that the local problems defining the multiscale basis functions are exactly solved, we prove that the equivalence holds for general polytopal (coarse) meshes and arbitrary approximation orders. We also leverage the interchange of properties to perform a unified convergence analysis, as well as to improve on both methods.

Published
2022

37. A hybridizable discontinuous Galerkin method with characteristic variables for Helmholtz problems

Author

Modave, Axel, Chaumont-Frelet, Théophile, Propagation des Ondes : Étude Mathématique et Simulation (POEMS), Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA), École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS), Modélisation et méthodes numériques pour le calcul d'interactions onde-matière nanostructurée (ATLANTIS), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), and ANR-21-CE46-0010,WavesDG,Méthodes d'éléments finis discontinus de type Galerkin spécifiques pour la propagation des ondes en régime harmonique(2021)

Subjects
FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], FOS: Physical sciences, Mathematics - Numerical Analysis, Numerical Analysis (math.NA), Computational Physics (physics.comp-ph), Physics - Computational Physics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], [PHYS.MECA.ACOU]Physics [physics]/Mechanics [physics]/Acoustics [physics.class-ph]
Abstract

A new hybridizable discontinuous Galerkin method, named the CHDG method, is proposed for solving time-harmonic scalar wave propagation problems. This method relies on a standard discontinuous Galerkin scheme with upwind numerical fluxes and high-order polynomial bases. Auxiliary unknowns corresponding to characteristic variables are defined at the interface between the elements, and the physical fields are eliminated to obtain a reduced system. The reduced system can be written as a fixed-point problem that can be solved with stationary iterative schemes. Numerical results with 2D benchmarks are presented to study the performance of the approach. Compared to the standard HDG approach, the properties of the reduced system are improved with CHDG, which is more suited for iterative solution procedures. The condition number of the reduced system is smaller with CHDG than with the standard HDG method. Iterative solution procedures with CGN or GMRES required smaller numbers of iterations with CHDG.

Published
2022
Full Text
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40. Guaranteed error estimates for finite element discretizations of Helmholtz problems

Author

Chaumont-Frelet, Théophile, Ern, Alexandre, Vohralík, Martin, Modélisation et méthodes numériques pour le calcul d'interactions onde-matière nanostructurée (ATLANTIS), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Alexandre Dieudonné (JAD), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), Simulation for the Environment: Reliable and Efficient Numerical Algorithms (SERENA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), and Laboratoire Jean Alexandre Dieudonné (LJAD)

Subjects
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], ComputingMilieux_MISCELLANEOUS, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience

Published
2021

41. Polynomial-degree-robust a posteriori error estimation for Maxwell's equations

Author

Chaumont-Frelet, Théophile, Ern, Alexandre, Vohralík, Martin, Modélisation et méthodes numériques pour le calcul d'interactions onde-matière nanostructurée (ATLANTIS), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Alexandre Dieudonné (JAD), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), Simulation for the Environment: Reliable and Efficient Numerical Algorithms (SERENA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), and Laboratoire Jean Alexandre Dieudonné (LJAD)

Subjects
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], ComputingMilieux_MISCELLANEOUS, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience

Published
2021

42. p-robust equilibrated flux reconstruction in H(curl) based on local minimizations. Application to a posteriori analysis of the curl-curl problem

Author

Chaumont-Frelet, Théophile, Vohralík, Martin, Modélisation et méthodes numériques pour le calcul d'interactions onde-matière nanostructurée (ATLANTIS), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Alexandre Dieudonné (JAD), Simulation for the Environment: Reliable and Efficient Numerical Algorithms (SERENA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), European Project: 647134,H2020 ERC,ERC-2014-CoG,GATIPOR(2015), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), and Laboratoire Jean Alexandre Dieudonné (LJAD)

Subjects
p-robustness, Sobolev space H(curl), Divergence-free decomposition, Broken polynomial extension, Sobolev space H(div), Equilibrated flux reconstruction, A posteriori error estimate, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience We present a local construction of H(curl)-conforming piecewise polynomials satisfying a prescribed curl constraint. We start from a piecewise polynomial not contained in the H(curl) space but satisfying a suitable orthogonality property. The procedure employs minimizations in vertex patches and the outcome is, up to a generic constant independent of the underlying polynomial degree, as accurate as the best-approximations over the entire local versions of H(curl). This allows to design guaranteed, fully computable, constant-free, and polynomial-degree-robust a posteriori error estimates of Prager-Synge type for Nédélec finite element approximations of the curl-curl problem. A divergence-free decomposition of a divergence-free H(div)-conforming piecewise polynomial, relying on over-constrained minimizations in Raviart-Thomas spaces, is the key ingredient. Numerical results illustrate the theoretical developments.

Published
2021

44. A postprocessing technique for a discontinuous Galerkin discretization of time-dependent Maxwell's equations

Author

Nehmetallah, Georges, Chaumont-Frelet, Théophile, Descombes, Stéphane, Lanteri, Stéphane, Modélisation et méthodes numériques pour le calcul d'interactions onde-matière nanostructurée (ATLANTIS), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Laboratoire Jean Alexandre Dieudonné (LJAD), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), and Laboratoire Jean Alexandre Dieudonné (JAD)

Subjects
time-domain electromagnetics, Maxwell's equations, high-order method, FOS: Mathematics, discontinuous Galerkin method, Numerical Analysis (math.NA), Mathematics - Numerical Analysis, postprocessing, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics::Numerical Analysis
Abstract

We present a novel postprocessing technique for a discontinuous Galerkin (DG) discretization of time-dependent Maxwell's equations that we couple with an explicit Runge-Kutta time-marching scheme. The postprocessed electromagnetic field converges one order faster than the unprocessed solution in the H(curl)-norm. The proposed approach is local, in the sense that the enhanced solution is computed independently in each cell of the computational mesh, and at each time step of interest. As a result, it is inexpensive to compute, especially if the region of interest is localized, either in time or space. The key ideas behind this postprocessing technique stem from hybridizable discontinuous Galerkin (HDG) methods , which are equivalent to the analyzed DG scheme for specific choices of penalization parameters. We present several numerical experiments that highlight the superconvergence properties of the postprocessed electromagnetic field approximation.

Published
2020

45. Stable broken H(curl) polynomial extensions and p-robust quasi-equilibrated a posteriori estimators for Maxwell's equations

Author

Chaumont-Frelet, Théophile, Ern, Alexandre, Vohralík, Martin, Modélisation et méthodes numériques pour le calcul d'interactions onde-matière nanostructurée (ATLANTIS), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015 - 2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015 - 2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015 - 2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015 - 2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Laboratoire Jean Alexandre Dieudonné (JAD), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), Simulation for the Environment: Reliable and Efficient Numerical Algorithms (SERENA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and European Project: 647134,H2020 ERC,ERC-2014-CoG,GATIPOR(2015)

Subjects
Finite element methods, A posteriori error estimates, High order methods, Maxwell's equations, AMS subject classification.65N30, 78M10, 65N15, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

We study extensions of piecewise polynomial data prescribed in a patch of tetrahedra sharing an edge. We show stability in the sense that the minimizers over piecewise polynomial spaces with prescribed tangential component jumps across faces and prescribed piecewise curl in elements are subordinate in the broken energy norm to the minimizers over the broken H(curl) space with the same prescriptions. Our proofs are constructive and yield constants independent of the polynomial degree. We then detail the application of this result to a posteriori error analysis of Maxwell's equations discretized with Nédélec's finite elements of arbitrary order. The resulting estimators are locally efficient, polynomial-degree-robust, and inexpensive since the quasi-equilibration only goes over edge patches and can be realized without solutions of patch problems by a sweep through tetrahedra around every mesh edge. The estimators become guaranteed when the regularity pickup constant is explicitly known. Numerical experiments illustrate the theoretical findings.

Published
2020

48. Wavenumber explicit convergence analysis for finite element discretizations of time-harmonic wave propagation problems with perfectly matched layers

Author

Chaumont-Frelet, Théophile, Gallistl, Dietmar, Nicaise, Serge, Tomezyk, Jérôme, Chaumont-Frelet, Théophile, Laboratoire de Mathématiques de l'INSA de Rouen Normandie (LMI), Institut national des sciences appliquées Rouen Normandie (INSA Rouen Normandie), Institut National des Sciences Appliquées (INSA)-Normandie Université (NU)-Institut National des Sciences Appliquées (INSA)-Normandie Université (NU), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), Department of Applied Mathematics [Twente], University of Twente [Netherlands], Laboratoire de Mathématiques et leurs Applications de Valenciennes - EA 4015 (LAMAV), Centre National de la Recherche Scientifique (CNRS)-Université de Valenciennes et du Hainaut-Cambrésis (UVHC)-INSA Institut National des Sciences Appliquées Hauts-de-France (INSA Hauts-De-France), and Université de Valenciennes et du Hainaut-Cambrésis (UVHC)-Centre National de la Recherche Scientifique (CNRS)-INSA Institut National des Sciences Appliquées Hauts-de-France (INSA Hauts-De-France)

Subjects
[MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

The first part of this paper is devoted to a wavenumber-explicit stability analysis of a planar Helmholtz problem with a perfectly matched layer. We prove that, for a model scattering problem, the H1 norm of the solution is bounded by the right-hand side, uniformly in the wavenumber k in the highly oscillatory regime. The second part proposes two numerical discretizations: an hp finite element method and a multiscale method based on local subspace correction. The stability result is used to relate the choice of parameters in the numerical methods to the wavenumber. A priori error estimates are shown and their sharpness is assessed in numerical experiments.

Published
2018

49. Equivalence of local-best and global-best approximations in <bold>H</bold>(curl)

Author

Chaumont-Frelet, Théophile and Vohralík, Martin

Abstract

We derive results on equivalence of piecewise polynomial approximations of a given function in the Sobolev space H (curl) . We namely show that the global-best approximation of a given H (curl) function in a H (curl) -conforming piecewise polynomial space imposing the continuity of the tangential trace can be bounded above and below by the Hilbertian sum of the respective local approximations from the elementwise spaces without any inter-element continuity requirement. In other words, the approximation of a H (curl) function by tangential-trace-continuous and discontinuous piecewise polynomials has comparable precision. We consider approximations of the curl of the target function in the L 2 -norm, as well as approximations of the target function in the L 2 -norm with a constraint on the curl; in the latter case, the constraint is removed in the local approximations. These best-approximation localizations hold under the minimal H (curl) regularity, on arbitrary shape-regular tetrahedral meshes, and include imposition of conditions on a part of the boundary. They extend to the H (curl) context some recent results from the H 1 and H (div) spaces and have direct applications to a priori and a posteriori error analysis of numerical discretizations related to the H (curl) space, namely Maxwell’s equations. [ABSTRACT FROM AUTHOR]

Published
2021
Full Text
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50. A multiscale hybrid mixed method for time-harmonic maxwell's equations in two dimensions

Author

Chaumont-Frelet, Théophile, Descombes, S., Lanteri, Stephane, Valentin, F., Basque Center for Applied Mathematics (BCAM), Basque Center for Applied Mathematics, Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Numerical modeling and high performance computing for evolution problems in complex domains and heterogeneous media (NACHOS), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), Laboratorio Nacional de Computação Cientifica [Rio de Janeiro] (LNCC / MCT), Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), and COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS)

Subjects
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Published
2018

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