Alexandre Ern
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View article: Model-order reduction with optimal morphings for poorly reducible problems with geometric variability
Model-order reduction with optimal morphings for poorly reducible problems with geometric variability Open
We propose a new model-order reduction framework for poorly reducible problems arising from parametric partial differential equations with geometric variability. In such problems, standard projection-based model-order reduction techniques …
View article: Asymptotic optimality of the edge finite element approximation of the time-harmonic Maxwell’s equations
Asymptotic optimality of the edge finite element approximation of the time-harmonic Maxwell’s equations Open
We analyze the conforming approximation of the time-harmonic Maxwell’s equations using Nédélec (edge) finite elements. We prove that the approximation is asymptotically optimal, i.e., the approximation error in the energy norm is bounded b…
View article: Explicit Runge--Kutta schemes with hybrid high-order and weakGalerkin methods for the wave equation in first-order form
Explicit Runge--Kutta schemes with hybrid high-order and weakGalerkin methods for the wave equation in first-order form Open
We derive energy error estimates in a fully-discrete setting for the first-order wave equation in its Friedrichs formulation, using third- and fourth-order explicit Runge--Kutta (ERK3 and ERK4) schemes in time combined with hybrid high-ord…
View article: A stabilized hybridized Nitsche method for sign-changing elliptic PDEs
A stabilized hybridized Nitsche method for sign-changing elliptic PDEs Open
In this paper, we present and analyze a stabilized hybridized Nitsche method for elliptic problems with sign-changing coefficients without imposing symmetry assumptions on the mesh around the material interfaces. The use of a stabilized pr…
View article: $\boldsymbol{H}(\textbf{curl})$-reconstruction of piecewise polynomial fields with application to $hp$-a posteriori nonconforming error analysis for Maxwell's equations
$\boldsymbol{H}(\textbf{curl})$-reconstruction of piecewise polynomial fields with application to $hp$-a posteriori nonconforming error analysis for Maxwell's equations Open
We devise and analyse a novel $\boldsymbol{H}(\textbf{curl})$-reconstruction operator for piecewise polynomial fields on shape-regular simplicial meshes. The (non-polynomial) reconstruction is devised over the mesh vertex patches using the…
View article: Discrete Poincaré inequalities: a review on proofs, equivalent formulations, and behavior of constants
Discrete Poincaré inequalities: a review on proofs, equivalent formulations, and behavior of constants Open
We investigate discrete Poincaré inequalities on piecewise polynomial subspaces of the Sobolev spaces $\boldsymbol{H}({\mathbf{curl}},\omega )$ and $\boldsymbol{H}({\mathop{\text{div}}},\omega )$ in three space dimensions. We characterize …
View article: Convergence of explicit Runge-Kutta discontinuous Galerkin approximations of the first-order form of Maxwell's equations with low regularity solutions
Convergence of explicit Runge-Kutta discontinuous Galerkin approximations of the first-order form of Maxwell's equations with low regularity solutions Open
We establish a convergence result for the approximation of low-regularity solutions to time-dependent PDE systems that have an involution structure similar to Maxwell's equations and the linear wave equations. The approximation is based on…
View article: Hybrid high-order methods for elasto-acoustic wave propagation in the time domain
Hybrid high-order methods for elasto-acoustic wave propagation in the time domain Open
We devise a Hybrid High-Order (HHO) method for the coupling between the acoustic and elastic wave equations in the time domain. A first-order formulation in time is considered. The HHO method can use equal-order and mixed-order settings wi…
View article: Convergence in operator norm of a stabilized continuous finite element stabilized approximation of Maxwell's equations in first-order form
Convergence in operator norm of a stabilized continuous finite element stabilized approximation of Maxwell's equations in first-order form Open
Maxwell's equations in first-order form with smooth permeability and smooth permittivity are approximated using continuous finite elements. The method is stabilized using the continuous interior penalty technique. Using an existence result…
View article: Elasto-acoustic wave propagation in geophysical media using hybrid high-order methods on general meshes
Elasto-acoustic wave propagation in geophysical media using hybrid high-order methods on general meshes Open
Hybrid high-order (HHO) methods are numerical methods characterized by several interesting properties such as local conservativity, geometric flexibility and high-order accuracy. Here, HHO schemes are studied for the space semi-discretizat…
View article: Damped Energy-norm <i>a posteriori</i> error estimates using <i>C</i><sup>2</sup>-reconstructions for the fully discrete wave equation with the leapfrog scheme
Damped Energy-norm <i>a posteriori</i> error estimates using <i>C</i><sup>2</sup>-reconstructions for the fully discrete wave equation with the leapfrog scheme Open
We derive a posteriori error estimates for the the scalar wave equation discretized in space by continuous finite elements and in time by the explicit leapfrog scheme. Our analysis combines the idea of invoking extra time-regularity for th…
View article: A priori and a posteriori analysis of the discontinuous Galerkin approximation of the time-harmonic Maxwell’s equations under minimal regularity assumptions
A priori and a posteriori analysis of the discontinuous Galerkin approximation of the time-harmonic Maxwell’s equations under minimal regularity assumptions Open
We derive a priori and a posteriori error estimates for the discontinuous Galerkin (dG) approximation of the time-harmonic Maxwell’s equations. Specifically, we consider an interior penalty dG method, and establish error estimates that are…
View article: Unfitted hybrid high-order methods stabilized by polynomial extension for elliptic interface problems
Unfitted hybrid high-order methods stabilized by polynomial extension for elliptic interface problems Open
In this work, we study the design and analysis of a novel hybrid high-order (HHO) method on unfitted meshes. HHO methods rely on a pair of unknowns, combining polynomials attached to the mesh faces and the mesh cells. In the unfitted frame…
View article: Local L2-bounded commuting projections using discrete local problems on Alfeld splits
Local L2-bounded commuting projections using discrete local problems on Alfeld splits Open
We construct projections onto the classical finite element spaces based on Lagrange, Nédélec, Raviart-Thomas, and discontinuous elements on shape-regular simplicial meshes. Our projections are defined locally, are bounded in the L2-norm, a…
View article: A priori and a posteriori analysis of the discontinuous Galerkin approximation of the time-harmonic Maxwell's equations under minimal regularity assumptions
A priori and a posteriori analysis of the discontinuous Galerkin approximation of the time-harmonic Maxwell's equations under minimal regularity assumptions Open
We derive a priori and a posteriori error estimates for the discontinuous Galerkin (dG) approximation of the time-harmonic Maxwell's equations. Specifically, we consider an interior penalty dG method, and establish error estimates that are…
View article: Explicit Runge-Kutta schemes with hybrid high-order methods for the wave equation in first-order form
Explicit Runge-Kutta schemes with hybrid high-order methods for the wave equation in first-order form Open
We analyze the approximation of the acoustic wave equation in its first-order Friedrichs formulation by explicit Runge-Kutta (ERK) schemes in time combined with hybrid high-order (HHO) methods in space. We propose two general assumptions (…
View article: $hp$-error analysis of mixed-order hybrid high-order methods for elliptic problems on simplicial meshes
$hp$-error analysis of mixed-order hybrid high-order methods for elliptic problems on simplicial meshes Open
We present both $hp$-a priori and $hp$-a posteriori error analysis of a mixed-order hybrid high-order (HHO) method to approximate second-order elliptic problems on simplicial meshes. Our main result on the $hp$-a priori error analysis is a…
View article: Bridging the Hybrid High-Order and Hybridizable Discontinuous Galerkin Methods: Summary
Bridging the Hybrid High-Order and Hybridizable Discontinuous Galerkin Methods: Summary Open
International audience
View article: Elasticity-based morphing technique and application to reduced-order modeling
Elasticity-based morphing technique and application to reduced-order modeling Open
The aim of this article is to introduce a new methodology for constructing morphings between shapes that have identical topology. The morphings are obtained by deforming a reference shape, through the resolution of a sequence of linear ela…
View article: A reduced basis method for frictional contact problems formulated with Nitsche’s method
A reduced basis method for frictional contact problems formulated with Nitsche’s method Open
We develop an efficient reduced basis method for the frictional contact problem formulated using Nitsche’s method. We focus on the regime of small deformations and on Tresca friction. The key idea ensuring the computational efficiency of t…
View article: Morphing techniques for model order reduction with non parametric geometrical variabilities
Morphing techniques for model order reduction with non parametric geometrical variabilities Open
National audience
View article: Third-order A-stable alternating implicit Runge-Kutta schemes
Third-order A-stable alternating implicit Runge-Kutta schemes Open
We design pairs of six-stage, third-order, alternating implicit Runge--Kutta (RK) schemes that can be used to integrate in time two stiff operators by an operator-split technique. We also design for each pair a companion explicit RK scheme…
View article: Damped energy-norm a posteriori error estimates for fully discrete approximations of the wave equation using C2-reconstructions with the leapfrog scheme
Damped energy-norm a posteriori error estimates for fully discrete approximations of the wave equation using C2-reconstructions with the leapfrog scheme Open
We derive a posteriori error estimates for the the scalar wave equation discretized in space by continuous finite elements and in time by the explicit leapfrog scheme. Our analysis combines the idea of invoking extra time-regularity for th…
View article: Spectral correctness of the continuous finite element stabilized approximation of the first-order form of Maxwell's equations
Spectral correctness of the continuous finite element stabilized approximation of the first-order form of Maxwell's equations Open
Maxwell's equations in first-order form with smooth permeability and smooth permittivity are approximated using continuous finite elements. The method is stabilized using the continuous interior penalty technique. Using an existence result…
View article: Surface tension effects between two immiscible Stokes fluids: a computational study using unfitted hybrid high-order methods and a level-set scheme
Surface tension effects between two immiscible Stokes fluids: a computational study using unfitted hybrid high-order methods and a level-set scheme Open
We develop an unfitted Hybrid High-Order (HHO) method coupled with a level-set scheme to solve numerically the flow of two immiscible Stokes fluids separated by an unknown interface where surface tension effects are present. The interface …
View article: The Discontinuous Galerkin Approximation of the Grad-Div and Curl-Curl Operators in First-Order Form Is Involution-Preserving and Spectrally Correct
The Discontinuous Galerkin Approximation of the Grad-Div and Curl-Curl Operators in First-Order Form Is Involution-Preserving and Spectrally Correct Open
International audience