arXiv subjects

Alexandre Ern

Publications and source records attributed to Alexandre Ern.

At least 19 recordsLinked to original sources

arXiv2026

An energy-decreasing algorithm for the finite element approximation of ferronematic equilibrium states

We develop an energy-decreasing algorithm for the finite element approximation of two-dimensional ferronematic equilibrium states. The problem is formulated as the minimization of the harmonic energy of two two-dimensional vector fields, both with prescribed length, together with an additional nonlinear relation on the orientation of the two vectors. The finite element setting is based on piecewise continuous finite elements on a weakly acute triangulation. The computational realization of the energy-decreasing algorithm employs a decomposition-coordination framework and a Uzawa-like iteration. Numerical experiments are presented to illustrate the computational performances of the algorithm.

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arXiv2026

Bounded, Commuting, Discrete-trace Preserving Projections

We construct bounded, commuting projections for the three-dimensional de Rham complex with the additional property that the projections preserve the trace of functions/fields if the latter is a piecewise polynomial in the appropriate trace space. The projections are locally defined and stable in the graph norm. More precisely, the part of the graph norm involving the exterior derivative only involves the oscillation of this derivative in a narrow strip of elements touching the boundary and weighted by the local mesh size. Moreover, the projections are $L^2$-stable locally when acting on functions/fields whose exterior derivative is a piecewise polynomial in the appropriate space. We present two salient applications of the present bounded, commuting, discrete-trace preserving projections: the construction of stable liftings of piecewise polynomial data and an optimality result on the discrete versus continuous extension of piecewise polynomial data.

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arXiv2026

Proximal Discontinuous Galerkin Methods for Variational Inequalities

We introduce a family of proximal discontinuous Galerkin methods for variational inequalities, focusing on the obstacle problem as a didactic example. Each member of this family is born from applying a different well-known nonconforming finite element discretization to the Bregman proximal point method. We explicitly treat four examples: the symmetric interior penalty discontinuous Galerkin, the enriched Galerkin, the hybridizable interior penalty and the hybrid high-order methods. We formulate a unified analysis framework for this family of methods and prove the existence and uniqueness of solutions, energy dissipation, and error estimates for both the primal and dual variables. Remarkably, the proximal hybrid high-order method with piecewise constant cell unknowns and piecewise affine facet unknowns leads to the first higher-order convergence result for any proximal Galerkin method.

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arXiv2026

$hp$-a posteriori error estimates for hybrid high-order methods applied to biharmonic problems

We derive a residual-based $hp$-a posteriori error estimator for hybrid high-order (HHO) methods on simplicial meshes applied to the biharmonic problem posed on two- and three-dimensional polytopal Lipschitz domains. The a posteriori error estimator hinges on an error decomposition into conforming and nonconforming components. To bound the nonconforming error, we use a $C^1$-partition of unity constructed via Alfeld splittings, combined with local Helmholtz decompositions on vertex stars where the key contribution is to show that the stability constant only depends on the mesh shape-regularity. For the conforming error, we design two residual-based estimators, each associated with a specific interpolation operator. In the first setting, the upper bound on the conforming error involves only the stabilization term and the data oscillation, but hinges on an assumption that we verify numerically. In the second setting, the bound additionally incorporates bulk residuals, normal flux jumps, and tangential jumps. Numerical experiments confirm the theoretical findings on the error upper bound and also illustrate numerically that the proposed estimators lead to moderate effectivity indices.

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arXiv2025

Improved $L^2$-error estimates for the wave equation discretized using hybrid nonconforming methods on simplicial meshes

We present improved $L^2$-error estimates on the time-integrated primal variable for the wave equation in its first-order formulation. The space discretization relies on a hybrid nonconforming method, such as the hybridizable discontinuous Galerkin, the hybrid high-order or the weak Galerkin methods. We consider both equal-order and mixed-order settings on simplices, and include the lowest-order case with piecewise constant unknowns on the faces and in the cells. Our main result is a superclose, resp., optimal bound on the above error in the equal-, resp., mixed-order case. A key result of independent interest to achieve these estimates are novel approximation estimates for an interpolation operator inspired from the hybridizable discontinuous Galerkin literature.

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arXiv2025

$\boldsymbol{H}(\textbf{curl})$-reconstruction of piecewise polynomial fields with application to $hp$-a posteriori nonconforming error analysis for Maxwell's equations

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 partition of unity induced by hat basis functions in combination with local Helmholtz decompositions. Our main focus is on homogeneous tangential boundary conditions. We prove that the difference between the reconstructed $\boldsymbol{H}_0(\textbf{curl})$-field and the original, piecewise polynomial field, measured in the broken curl norm and in the $\boldsymbol{L}^2$-norm, can be bounded in terms of suitable jump norms of the original field. The bounds are always $h$-optimal, and $p$-suboptimal by $\frac12$-order for the broken curl norm and by $\frac32$-order for the $\boldsymbol{L}^2$-norm. An auxiliary result of independent interest is a novel broken-curl, divergence-preserving Poincar\'{e} inequality on vertex patches. Moreover, the $\boldsymbol{L}^2$-norm estimate can be improved to $\frac12$-order suboptimality under a (reasonable) assumption on the uniform elliptic regularity pickup for a Poisson problem with Neumann conditions over the vertex patches. We also discuss extensions of the $\boldsymbol{H}_0(\textbf{curl})$-reconstruction operator to the prescription of mixed boundary conditions, to agglomerated polytopal meshes, and to convex domains. Finally, we showcase an important application of the $\boldsymbol{H}(\textbf{curl})$-reconstruction operator to the $hp$-a posteriori nonconforming error analysis of Maxwell's equations. We focus on the (symmetric) interior penalty discontinuous Galerkin (dG) approximation of some simplified forms of Maxwell's equations.

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arXiv2025

Elasto-acoustic wave propagation in geophysical media using hybrid high-order methods on general meshes

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-discretization of coupled elasto-acoustic waves in the time domain using a first-order formulation. Explicit and singly diagonal implicit Runge--Kutta (ERK & SDIRK) schemes are used for the time discretization. We show that an efficient implementation of explicit (resp. implicit) time schemes calls for a static condensation of the face (resp. cell) unknowns. Crucially, both static condensation procedures only involve block-diagonal matrices. Then, we provide numerical estimates for the CFL stability limit of ERK schemes and present a comparative study on the efficiency of explicit versus implicit schemes. Our findings indicate that implicit time schemes remain competitive in many situations. Finally, simulations in a 2D realistic geophysical configuration are performed, illustrating the geometrical flexibility of the HHO method: both hybrid (triangular and quadrilateral) and nonconforming (with hanging nodes) meshes are easily handled, delivering results of comparable accuracy to a reference spectral element software based on tensorized elements.

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arXiv2025

Unfitted hybrid high-order methods stabilized by polynomial extension for elliptic interface problems

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 framework, the interface can cut through the mesh cells in a very general fashion, and the polynomial unknowns are doubled in the cut cells and the cut faces. In order to avoid the ill-conditioning issues caused by the presence of small cut cells, the novel approach introduced herein is to use polynomial extensions in the definition of the gradient reconstruction operator. Stability and consistency results are established, leading to optimally decaying error estimates. The theory is illustrated by numerical experiments.

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arXiv2025

Optimal morphings for model-order reduction for poorly reducible problems with geometric variability

We propose a new model-order reduction framework to poorly reducible problems arising from parametric partial differential equations with geometric variability. In such problems, the solution manifold exhibits a slowly decaying Kolmogorov $N$-width, so that standard projection-based model order reduction techniques based on linear subspace approximations become ineffective. To overcome this difficulty, we introduce an optimal morphing strategy: For each solution sample, we compute a bijective morphing from a reference domain to the sample domain such that, when all the solution fields are pulled back to the reference domain, their variability is reduced. We formulate a global optimization problem on the morphings that maximizes the energy captured by the first $r$ modes of the mapped fields obtained from Proper Orthogonal Decomposition, thus maximizing the reducibility of the dataset. Finally, using a non-intrusive Gaussian Process regression on the reduced coordinates, we build a fast surrogate model that can accurately predict new solutions, highlighting the practical benefits of the proposed approach for many-query applications. The framework is general, independent of the underlying partial differential equation, and applies to scenarios with either parameterized or non-parameterized geometries.

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arXiv2025

Hybrid high-order methods for elasto-acoustic wave propagation in the time domain

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 with polynomial degree k>=0 for the face unknowns, together with O(1)- and O(1/h)-stabilizations. An energy-error estimate is established in the time-continuous case. A numerical spectral analysis is performed, showing that O(1)-stabilization is required to avoid excessive CFL limitations for explicit time discretizations. Moreover, the spectral radius of the stiffness matrix is fairly independent of the geometry of the mesh cells. For analytical solutions on general meshes, optimal convergence rates of order (k+1) are shown in both equal- and mixed-order settings using O(1)-stabilization, whereas order (k+2) is achieved in the mixed-order setting using O(1/h)-stabilization. Test cases with a Ricker wavelet as an initial condition showcase the relevance of the proposed method for the simulation of elasto-acoustic wave propagation across media with contrasted material properties.

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arXiv2025

Local L2-bounded commuting projections using discrete local problems on Alfeld splits

We construct projections onto the classical finite element spaces based on Lagrange, N\'ed\'elec, Raviart-Thomas, and discontinuous elements on shape-regular simplicial meshes. Our projections are defined locally, are bounded in the L2-norm, and commute with the corresponding differential operators. Such projections are a fundamental tool in finite element stability and error analysis. Moreover, to the best of our knowledge, the present construction is the first in the literature where local $L^2$-stability is fully established. The cornerstone of our construction are local weight functions which are piecewise polynomials built using the Alfeld split of local patches from the original simplicial mesh. This way, the L2-stability of the projections is established by invoking discrete Poincar\'e inequalities on these local stars, for which we provide here an original, constructive proof. Another important novelty is that we extend the construction of the projections so as to preserve homogeneous boundary conditions on a subset of the domain boundary. Altogether, the material is presented using the language of vector calculus, and links to the formalism of finite element exterior calculus are provided.

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arXiv2024

Discrete Poincar\'e inequalities: a review on proofs, equivalent formulations, and behavior of constants

We investigate discrete Poincar\'e inequalities on piecewise polynomial subspaces of the Sobolev spaces H(curl) and H(div) in three space dimensions. We characterize the dependence of the constants on the continuous-level constants, the shape regularity and cardinality of the underlying tetrahedral mesh, and the polynomial degree. One important focus is on meshes being local patches (stars) of tetrahedra from a larger tetrahedral mesh. We also review various equivalent results to the discrete Poincar\'e inequalities, namely stability of discrete constrained minimization problems, discrete inf-sup conditions, bounds on operator norms of piecewise polynomial vector potential operators (Poincar\'e maps), and existence of graph-stable commuting projections.

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arXiv2024

$hp$-error analysis of mixed-order hybrid high-order methods for elliptic problems on simplicial meshes

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 $\frac12$-order $p$-suboptimal error estimate. This result is, to our knowledge, the first of this kind for hybrid nonconforming methods and matches the state-of-the-art for other nonconforming methods (as discontinuous Galerkin methods) with general (mixed Dirichlet/Neumann) boundary conditions. Our second main result is a residual-based $hp$-a posteriori upper error bound, comprising residual, normal flux jump, tangential jump, and stabilization estimators (plus data oscillation terms). The first three terms are $p$-optimal and only the latter is $\frac12$-order $p$-suboptimal. This result is, to our knowledge, the first $hp$-a posteriori error estimate for HHO methods. A novel approach based on the partition-of-unity provided by hat basis functions and on local Helmholtz decompositions on vertex stars is devised to estimate the nonconforming error. Finally, we establish local lower error bounds. Remarkably, the normal flux jump estimator is only $\frac12$-order $p$-suboptimal, as it can be bounded by the stabilization owing to the local conservation property of HHO methods. Numerical examples illustrate the theory.

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arXiv2024

Elasticity-based morphing technique and application to reduced-order modeling

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 elasticity equations, onto every target shape. In particular, our approach does not assume any knowledge of a boundary parametrization, and the computation of the boundary deformation is not required beforehand. Furthermore, constraints can be imposed on specific points, lines and surfaces in the reference domain to ensure alignment with their counterparts in the target domain after morphing. Additionally, we show how the proposed methodology can be integrated in an offline and online paradigm, which is useful in reduced-order modeling involving variable shapes. This framework facilitates the efficient computation of the morphings in various geometric configurations, thus improving the versatility and applicability of the approach. The robustness and computational efficiency of the methodology is illustrated on two-dimensional test cases, including the regression problem of the drag and lift coefficients of airfoils of non-parameterized variable shapes.

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arXiv2023

A reduced basis method for frictional contact problems formulated with Nitsche's method

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 the method is to treat the nonlinearity resulting from the contact and friction conditions by means of the Empirical Interpolation Method. The proposed algorithm is applied to the Hertz contact problem between two half-disks with parameter-dependent radius. We also highlight the benefits of the present approach with respect to the mixed (primal-dual) formulation.

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arXiv2022

$C^0$-hybrid high-order methods for biharmonic problems

We devise and analyze $C^0$-conforming hybrid high-order (HHO) methods to approximate biharmonic problems with either clamped or simply supported boundary conditions. $C^0$-conforming HHO methods hinge on cell unknowns which are $C^0$-conforming polynomials of order $(k+2)$ approximating the solution in the mesh cells and on face unknowns which are polynomials of order $k\ge0$ approximating the normal derivative of the solution on the mesh skeleton. Such methods deliver $O(h^{k+1})$ $H^2$-error estimates for smooth solutions. An important novelty in the error analysis is to lower the minimal regularity requirement on the exact solution. The technique to achieve this has broader applicability than just $C^0$-conforming HHO methods, and to illustrate this point, we outline the error analysis for the well-known $C^0$-conforming interior penalty discontinuous Galerkin (IPDG) methods as well. The present technique does not require bubble functions or a $C^1$-smoother to evaluate the right-hand side in case of rough loads. Finally, numerical results including comparisons to various existing methods showcase the efficiency of the proposed $C^0$-conforming HHO methods.

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arXiv2022

Local decay rates of best-approximation errors using vector-valued finite elements for fields with low regularity and integrable curl or divergence

We estimate best-approximation errors using vector-valued finite elements for fields with low regularity in the scale of fractional-order Sobolev spaces. By assuming additionally that the target field has a curl or divergence property, we establish upper bounds on these errors that can be localized to the mesh cells. These bounds are derived using the quasi-interpolation errors with or without boundary prescription derived in [A. Ern and J.-L. Guermond, ESAIM Math. Model. Numer. Anal., 51 (2017), pp.~1367--1385]. By using the face-to-cell lifting operators analyzed in [A. Ern and J.-L. Guermond, Found. Comput. Math., (2021)], and exploiting the additional assumption made on the curl or the divergence of the target field, a localized upper bound on the quasi-interpolation error is derived. As an illustration, we show how to apply these results to the error analysis of the curl-curl problem associated with Maxwell's equations.

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arXiv2021

Hybrid high-order method for singularly perturbed fourth-order problems on curved domains

We propose a novel hybrid high-order method (HHO) to approximate singularly perturbed fourth-order PDEs on domains with a possibly curved boundary. The two key ideas in devising the method are the use of a Nitsche-type boundary penalty technique to weakly enforce the boundary conditions and a scaling of the weighting parameter in the stabilization operator that compares the singular perturbation parameter to the square of the local mesh size. With these ideas in hand, we derive stability and optimal error estimates over the whole range of values for the singular perturbation parameter, including the zero value for which a second-order elliptic problem is recovered. Numerical experiments illustrate the theoretical analysis.

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