Alexandre Ern

Alexandre Ern, Jean-Luc Guermond

This paper introduces a quasi-interpolation operator for scalar- and vector-valued finite element spaces constructed on affine, shape-regular meshes with some continuity across mesh interfaces.This operator gives optimal estimates of the best approximation error in any $L^p$-norm assuming regularity in the fractional Sobolev spaces $W^{r,p}$, where $p\in [1,\infty]$ and the smoothness index $r$ can be arbitrarily close to zero. The operator is stable in $L^1$, leaves the corresponding finite element space point-wise invariant whether homogeneous boundary conditions are imposed or not. The theory is illustrated on $H^1$-, $\mathbf{H}(\text{curl})$- and $\mathbf{H}(\text{div})$-conforming spaces.

Jerome Bonelle, Alexandre Ern

Compatible schemes localize degrees of freedom according to the physical nature of the underlying fields and operate a clear distinction between topological laws and closure relations. For elliptic problems, the cornerstone in the scheme design is the discrete Hodge operator linking gradients to fluxes by means of a dual mesh, while a structure-preserving discretization is employed for the gradient and divergence operators. The discrete Hodge operator is sparse, symmetric positive definite and is assembled cellwise from local operators. We analyze two schemes depending on whether the potential degrees of freedom are attached to the vertices or to the cells of the primal mesh. We derive new functional analysis results on the discrete gradient that are the counterpart of the Sobolev embeddings. Then, we identify the two design properties of the local discrete Hodge operators yielding optimal discrete energy error estimates. Additionally, we show how these operators can be built from local nonconforming gradient reconstructions using a dual barycentric mesh. In this case, we also prove an optimal $L^2$-error estimate for the potential for smooth solutions. Links with existing schemes (finite elements, finite volumes, mimetic finite differences) are discussed. Numerical results are presented on three-dimensional polyhedral meshes.

Daniele A. Di Pietro, Jérôme Droniou, Alexandre Ern

We design and analyze an approximation method for advection-diffusion-reaction equations where the (generalized) degrees of freedom are polynomials of order $k\ge0$ at mesh faces. The method hinges on local discrete reconstruction operators for the diffusive and advective derivatives and a weak enforcement of boundary conditions. Fairly general meshes with polytopal and nonmatching cells are supported. Arbitrary polynomial orders can be considered, including the case $k=0$ which is closely related to Mimetic Finite Difference/Mixed-Hybrid Finite Volume methods. The error analysis covers the full range of Péclet numbers, including the delicate case of local degeneracy where diffusion vanishes on a strict subset of the domain. Computational costs remain moderate since the use of face unknowns leads to a compact stencil with reduced communications. Numerical results are presented.

Erik Burman, Alexandre Ern

We design and analyze a Hybrid High-Order (HHO) method on unfitted meshes to approximate elliptic interface problems. The curved interface can cut through the mesh cells in a very general fashion. As in classical HHO methods, the present unfitted method introduces cell and face unknowns in uncut cells, but doubles the unknowns in the cut cells and on the cut faces. The main difference with classical HHO methods is that a Nitsche-type formulation is used to devise the local reconstruction operator. As in classical HHO methods, cell unknowns can be eliminated locally leading to a global problem coupling only the face unknowns by means of a compact stencil. We prove stability estimates and optimal error estimates in the $H^1$-norm. Robustness with respect to cuts is achieved by a local cell-agglomeration procedure taking full advantage of the fact that HHO methods support polyhedral meshes. Robustness with respect to the contrast in the material properties from both sides of the interface is achieved by using material-dependent weights in Nitsche's formulation.

Alexandre Ern, Jean-Luc Guermond

We construct mollification operators in strongly Lipschitz domains that do not invoke non-trivial extensions, are $L^p$ stable for any real number $p\in[1,\infty]$, and commute with the differential operators $\nabla$, $\nabla{\times}$, and $\nabla{\cdot}$. We also construct mollification operators satisfying boundary conditions and use them to characterize the kernel of traces related to the tangential and normal trace of vector fields. We use the mollification operators to build projection operators onto general $H^1$-, $\mathbf{H}(\text{curl})$- and $\mathbf{H}(\text{div})$-conforming finite element spaces, with and without homogeneous boundary conditions. These operators commute with the differential operators $\nabla$, $\nabla{\times}$, and $\nabla{\cdot}$, are $L^p$-stable, and have optimal approximation properties on smooth functions.

Alexandre Ern, Iain Smears, Martin Vohralík

We consider the a posteriori error analysis of approximations of parabolic problems based on arbitrarily high-order conforming Galerkin spatial discretizations and arbitrarily high-order discontinuous Galerkin temporal discretizations. Using equilibrated flux reconstructions, we present a posteriori error estimates for a norm composed of the $L^2(H^1)\cap H^1(H^{-1})$-norm of the error and the temporal jumps of the numerical solution. The estimators provide guaranteed upper bounds for this norm, without unknown constants. Furthermore, the efficiency of the estimators with respect to this norm is local in both space and time, with constants that are robust with respect to the mesh-size, time-step size, and the spatial and temporal polynomial degrees. We further show that this norm, which is key for local space-time efficiency, is globally equivalent to the $L^2(H^1)\cap H^1(H^{-1})$-norm of the error, with polynomial-degree robust constants. The proposed estimators also have the practical advantage of allowing for very general refinement and coarsening between the timesteps.

Alexandre Ern, Jean-Luc Guermond

We derive $H_{\text{curl}}$-error estimates and improved $L^2$-error estimates for the Maxwell equations approximated using edge finite elements. These estimates only invoke the expected regularity pickup of the exact solution in the scale of the Sobolev spaces, which is typically lower than $\frac12$ and can be arbitrarily close to $0$ when the material properties are heterogeneous. The key tools for the analysis are commuting quasi-interpolation operators in $H_{\text{curl}}$- and $H_{\text{div}}$-conforming finite element spaces and, most crucially, newly-devised quasi-interpolation operators delivering optimal estimates on the decay rate of the best-approximation error for functions with Sobolev smoothness index arbitrarily close to $0$. The proposed analysis entirely bypasses the technique known in the literature as the discrete compactness argument.

Mickaël Abbas, Alexandre Ern, Nicolas Pignet

We devise and evaluate numerically Hybrid High-Order (HHO) methods for hyperelastic materials undergoing finite deformations. The HHO methods use as discrete unknowns piecewise polynomials of order $k\ge1$ on the mesh skeleton, together with cell-based polynomials that can be eliminated locally by static condensation. The discrete problem is written as the minimization of the broken nonlinear elastic energy where a local reconstruction of the displacement gradient is used. Two HHO methods are considered: a stabilized method where the gradient is reconstructed as a tensor-valued polynomial of order $k$ and a stabilization is added to the discrete energy functional, and an unstabilized method which reconstructs a stable higher-order gradient and circumvents the need for stabilization. Both methods satisfy the principle of virtual work locally with equilibrated tractions. We present a numerical study of both HHO methods on test cases with known solution and on more challenging three-dimensional test cases including finite deformations with strong shear layers and cavitating voids. We assess the computational efficiency of both methods, and we compare our results to those obtained with an industrial software using conforming finite elements and to results from the literature. Both methods exhibit robust behavior in the quasi-incompressible regime.

Pierre Sochala, Alexandre Ern, Serge Piperno

Robust and accurate schemes are designed to simulate the coupling between subsurface and overland flows. The coupling conditions at the interface enforce the continuity of both the normal flux and the pressure. Richards' equation governing the subsurface flow is discretized using a Backward Differentiation Formula and a symmetric interior penalty Discontinuous Galerkin method. The kinematic wave equation governing the overland flow is discretized using a Godunov scheme. Both schemes individually are mass conservative and can be used within single-step or multi-step coupling algorithms that ensure overall mass conservation owing to a specific design of the interface fluxes in the multi-step case. Numerical results are presented to illustrate the performances of the proposed algorithms.

Fabien Casenave, Alexandre Ern, Guillaume Sylvand

We consider the convected Helmholtz equation modeling linear acoustic propagation at a fixed frequency in a subsonic flow around a scattering object. The flow is supposed to be uniform in the exterior domain far from the object, and potential in the interior domain close to the object. Our key idea is the reformulation of the original problem using the Prandtl--Glauert transformation on the whole flow domain, yielding (i) the classical Helmholtz equation in the exterior domain and (ii) an anisotropic diffusive PDE with skew-symmetric first-order perturbation in the interior domain such that its transmission condition at the coupling boundary naturally fits the Neumann condition from the classical Helmholtz equation. Then, efficient off-the-shelf tools can be used to perform the BEM-FEM coupling, leading to two novel variational formulations for the convected Helmholtz equation. The first formulation involves one surface unknown and can be affected by resonant frequencies, while the second formulation avoids resonant frequencies and involves two surface unknowns. Numerical simulations are presented to compare the two formulations.

Victor Calo, Matteo Cicuttin, Quanling Deng et al.

We study the approximation of the spectrum of a second-order elliptic differential operator by the Hybrid High-Order (HHO) method. The HHO method is formulated using cell and face unknowns which are polynomials of some degree $k\geq0$. The key idea for the discrete eigenvalue problem is to introduce a discrete operator where the face unknowns have been eliminated. Using the abstract theory of spectral approximation of compact operators in Hilbert spaces, we prove that the eigenvalues converge as $h^{2t}$ and the eigenfunctions as $h^{t}$ in the $H^1$-seminorm, where $h$ is the mesh-size, $t\in [s,k+1]$ depends on the smoothness of the eigenfunctions, and $s>\frac12$ results from the elliptic regularity theory. The convergence rates for smooth eigenfunctions are thus $h^{2k+2}$ for the eigenvalues and $h^{k+1}$ for the eigenfunctions. Our theoretical findings, which improve recent error estimates for Hybridizable Discontinuous Galerkin (HDG) methods, are verified on various numerical examples including smooth and non-smooth eigenfunctions. Moreover, we observe numerically in one dimension for smooth eigenfunctions that the eigenvalues superconverge as $h^{2k+4}$ for a specific value of the stabilization parameter.

Patrik Daniel, Alexandre Ern, Iain Smears et al.

We propose a new practical adaptive refinement strategy for $hp$-finite element approximations of elliptic problems. Following recent theoretical developments in polynomial-degree-robust a posteriori error analysis, we solve two types of discrete local problems on vertex-based patches. The first type involves the solution on each patch of a mixed finite element problem with homogeneous Neumann boundary conditions, which leads to an ${\mathbf H}(\mathrm{div},Ω)$-conforming equilibrated flux. This, in turn, yields a guaranteed upper bound on the error and serves to mark mesh vertices for refinement via Dörfler's bulk-chasing criterion. The second type of local problems involves the solution, on patches associated with marked vertices only, of two separate primal finite element problems with homogeneous Dirichlet boundary conditions, which serve to decide between $h$-, $p$-, or $hp$-refinement. Altogether, we show that these ingredients lead to a computable guaranteed bound on the ratio of the errors between successive refinements (error reduction factor). In a series of numerical experiments featuring smooth and singular solutions, we study the performance of the proposed $hp$-adaptive strategy and observe exponential convergence rates. We also investigate the accuracy of our bound on the reduction factor by evaluating the ratio of the predicted reduction factor relative to the true error reduction, and we find that this ratio is in general quite close to the optimal value of one.

Matteo Cicuttin, Alexandre Ern, Simon Lemaire

We devise a Hybrid High-Order (HHO) method for highly oscillatory elliptic problems that is capable of handling general meshes. The method hinges on discrete unknowns that are polynomials attached to the faces and cells of a coarse mesh; those attached to the cells can be eliminated locally using static condensation. The main building ingredient is a reconstruction operator, local to each coarse cell, that maps onto a fine-scale space spanned by oscillatory basis functions. The present HHO method generalizes the ideas of some existing multiscale approaches, while providing the first complete analysis on general meshes. It also improves on those methods, taking advantage of the flexibility granted by the HHO framework. The method handles arbitrary orders of approximation $k\geq 0$. For face unknowns that are polynomials of degree $k$, we devise two versions of the method, depending on the polynomial degree $(k-1)$ or $k$ of the cell unknowns. We prove, in the case of periodic coefficients, an energy-error estimate of the form $\left(\varepsilon^{\frac{1}{2}}+H^{k+1}+(\varepsilon/H)^{\frac{1}{2}}\right)$, and we illustrate our theoretical findings on some test-cases.

Fabien Casenave, Alexandre Ern, Tony Lelièvre

The reduced basis method is a model reduction technique yielding substantial savings of computational time when a solution to a parametrized equation has to be computed for many values of the parameter. Certification of the approximation is possible by means of an a posteriori error bound. Under appropriate assumptions, this error bound is computed with an algorithm of complexity independent of the size of the full problem. In practice, the evaluation of the error bound can become very sensitive to round-off errors. We propose herein an explanation of this fact. A first remedy has been proposed in [F. Casenave, Accurate \textit{a posteriori} error evaluation in the reduced basis method. \textit{C. R. Math. Acad. Sci. Paris} \textbf{350} (2012) 539--542.]. Herein, we improve this remedy by proposing a new approximation of the error bound using the Empirical Interpolation Method (EIM). This method achieves higher levels of accuracy and requires potentially less precomputations than the usual formula. A version of the EIM stabilized with respect to round-off errors is also derived. The method is illustrated on a simple one-dimensional diffusion problem and a three-dimensional acoustic scattering problem solved by a boundary element method.

Alexandre Ern, Iain Smears, Martin Vohralik

We consider the a posteriori error analysis of fully discrete approximations of parabolic problems based on conforming $hp$-finite element methods in space and an arbitrary order discontinuous Galerkin method in time. Using an equilibrated flux reconstruction, we present a posteriori error estimates yielding guaranteed upper bounds on the $L^2(H^1)$-norm of the error, without unknown constants and without restrictions on the spatial and temporal meshes. It is known from the literature that the analysis of the efficiency of the estimators represents a significant challenge for $L^2(H^1)$-norm estimates. Here we show that the estimator is bounded by the $L^2(H^1)$-norm of the error plus the temporal jumps under the one-sided parabolic condition $h^2 \lesssim τ$. This result improves on earlier works that required stronger two-sided hypotheses such as $h \simeq τ$ or $h^2\simeq τ$; instead our result now encompasses the practically relevant case for computations and allows for locally refined spatial meshes. The constants in our bounds are robust with respect to the mesh and time-step sizes, the spatial polynomial degrees, and also with respect to refinement and coarsening between time-steps, thereby removing any transition condition.

Alexandre Ern, Iain Smears, Martin Vohralík

We establish the existence of liftings into discrete subspaces of $\mathbf{H}(\mathrm{div})$ of piecewise polynomial data on locally refined simplicial partitions of polygonal/polyhedral domains. Our liftings are robust with respect to the polynomial degree. This result has important applications in the a posteriori error analysis of parabolic problems, where it permits the removal of so-called transition conditions that link two consecutive meshes. It can also be used in a the posteriori error analysis of elliptic problems, where it allows the treatment of meshes with arbitrary numbers of hanging nodes between elements. We present a constructive proof based on the a posteriori error analysis of an auxiliary elliptic problem with $H^{-1}$ source terms, thereby yielding results of independent interest. In particular, for such problems, we obtain guaranteed upper bounds on the error along with polynomial-degree robust local efficiency of the estimators.

Amina Benaceur, Alexandre Ern, Virginie Ehrlacher et al.

We investigate new developments of the combined Reduced-Basis and Empirical Interpolation Methods (RB-EIM) for parametrized nonlinear parabolic problems. In many situations, the cost of the EIM in the offline stage turns out to be prohibitive since a significant number of nonlinear time-dependent problems need to be solved using the high-fidelity (or full-order) model. In the present work, we develop a new methodology, the Progressive RB-EIM (PREIM) method for nonlinear parabolic problems.The purpose is to reduce the offline cost while maintaining the accuracy of the RB approximation in the online stage. The key idea is a progressive enrichment of both the EIM approximation and the RB space, in contrast to the standard approach where the EIM approximation and the RB space are built separately. PREIM uses high-fidelity computations whenever available and RB computationsotherwise. Another key feature of each PREIM iteration is to select twice the parameter in a greedy fashion, the second selection being made after computing the high-fidelity solution for the firstly selected value of the parameter. Numerical examples are presented on nonlinear heat transfer problems.

Alexandre Ern, Jean-Luc Guermond

The converse of Fortin's Lemma in Banach spaces is established in this Note.

Fabien Casenave, Alexandre Ern, Tony Lelièvre

The Empirical Interpolation Method (EIM) is a greedy procedure that constructs approximate representations of two-variable functions in separated form. In its classical presentation, the two variables play a non-symmetric role. In this work, we give an equivalent definition of the EIM approximation, in which the two variables play symmetric roles. Then, we give a proof for the existence of this approximation, and extend it up to the convergence of the EIM, and for any norm chosen to compute the error in the greedy step. Finally, we introduce a way to compute a separated representation in the case where the number of selected values is different for each variable. In the case of a physical field measured by sensors, this is useful to discard a broken sensor while keeping the information provided by the associated selected field.

Zhaonan Dong, Alexandre Ern

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 Poincaré 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.

Alexandre Ern, Ruma R. Maity

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.

Alexandre Ern, Johnny Guzmán, Pratyush Potu

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.

Alexandre Ern, Brendan Keith, Dohyun Kim et al.

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 cell unknowns leads to the first higher-order convergence result for any proximal Galerkin method.

Alexandre Ern, Pietro Zanotti

Hybrid High-Order methods for elliptic diffusion problems have been originally formulated for loads in the Lebesgue space $L^2(Ω)$. In this paper we devise and analyze a variant thereof, which is defined for any load in the dual Sobolev space $H^{-1}(Ω)$. The main feature of the present variant is that its $H^1$-norm error can be bounded only in terms of the $H^1$-norm best error in a space of broken polynomials. We establish this estimate with the help of recent results on the quasi-optimality of nonconforming methods. We prove also an improved error bound in the $L^2$-norm by duality. Compared to previous works on quasi-optimal nonconforming methods, the main novelties are that Hybrid High-Order methods handle pairs of unknowns, and not a single function, and, more crucially, that these methods employ a reconstruction that is one polynomial degree higher than the discrete unknowns. The proposed modification affects only the formulation of the discrete right-hand side. This is obtained by properly mapping discrete test functions into $H^1_0(Ω)$.

Alexandre Ern, Martin Vohralík, Mohammad Zakerzadeh

We propose a reconstruction-based a posteriori error estimate for linear advection problems in one space dimension. In our framework, a stable variational ultra-weak formulation is adopted, and the equivalence of the $L_2$-norm of the error with the dual graph norm of the residual is established. This dual norm is showed to be localizable over vertex-based patch subdomains of the computational domain under the condition of the orthogonality of the residual to the piecewise affine hat functions. We show that this condition is valid for some well-known numerical methods including continuous/discontinuous Petrov--Galerkin and discontinuous Galerkin methods. Consequently, a well-posed local problem on each patch is identified, which leads to a global conforming reconstruction of the discrete solution. We prove that this reconstruction provides a guaranteed upper bound on the $L_2$ error. Moreover, up to a constant, it also gives local lower bounds on the $L_2$ error, where the generic constant is proven to be independent of mesh-refinement, polynomial degree of the approximation, and the advective velocity. This leads to robustness of our estimates with respect to the advection as well as the polynomial degree. All the above properties are verified in a series of numerical experiments, additionally leading to asymptotic exactness. Motivated by these results, we finally propose a heuristic extension of our methodology to any space dimension, achieved by solving local least-squares problems on vertex-based patches. Though not anymore guaranteed, the resulting error indicator is numerically robust with respect to both advection velocity and polynomial degree, for a collection of two-dimensional test cases including discontinuous solutions.

Thomas Boiveau, Virginie Ehrlacher, Alexandre Ern et al.

We devise a space-time tensor method for the low-rank approximation of linear parabolic evolution equations. The proposed method is a stable Galerkin method, uniformly in the discretization parameters, based on a Minimal Residual formulation of the evolution problem in Hilbert--Bochner spaces. The discrete solution is sought in a trial space composed of tensors of discrete functions in space and in time and is characterized as the unique minimizer of a discrete functional where the dual norm of the residual is evaluated in a space semi-discrete test space. The resulting global space-time linear system is solved iteratively by a greedy algorithm. Numerical results are presented to illustrate the performances of the proposed method on test cases including non-selfadjoint and time-dependent differential operators in space. The results are also compared to those obtained using a fully discrete Petrov--Galerkin setting to evaluate the dual residual norm.

Mickaël Abbas, Alexandre Ern, Nicolas Pignet

We devise and evaluate numerically a Hybrid High-Order (HHO) method for incremental associative plasticity with small deformations. The HHO method uses as discrete unknowns piecewise polynomials of order $k\ge1$ on the mesh skeleton, together with cell-based polynomials that can be eliminated locally by static condensation. The HHO method supports polyhedral meshes with non-matching interfaces, is free of volumetric-locking and the integration of the behavior law is performed only at cell-based quadrature nodes. Moreover, the principle of virtual work is satisfied locally with equilibrated tractions. Various two- and three-dimensional test cases from the literature are presented including comparison against known solutions and against results obtained with an industrial software using conforming and mixed finite elements.

Daniele A. Di Pietro, Alexandre Ern

We show how to recover equilibrated face tractions for the hybrid high-order method for linear elasticity recently introduced in [D. A. Di Pietro and A. Ern, A hybrid high-order locking-free method for linear elasticity on general meshes, Comput. Meth. Appl. Mech. Engrg., 2015, 283:1-21], and prove that these tractions are optimally convergent.