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 L2-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.

The reduced basis method is a model reduction technique yielding substantial savings ofcomputational time when a solution to a parametrized equation has to be computed for manyvalues of the parameter. Certification of the approximation is possible by means of ana posteriori error bound. Under appropriate assumptions, this errorbound is computed with an algorithm of complexity independent of the size of the fullproblem. In practice, the evaluation of the error bound can become very sensitive toround-off errors. We propose herein an explanation of this fact. A first remedy has beenproposed in [F. Casenave, Accurate a posteriori error evaluation in thereduced basis method. C. R. Math. Acad. Sci. Paris 350(2012) 539–542.]. Herein, we improve this remedy by proposing a new approximationof the error bound using the empirical interpolation method (EIM). This method achieveshigher levels of accuracy and requires potentially less precomputations than the usualformula. 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 athree-dimensional acoustic scattering problem solved by a boundary element method.

We analyze a two-stage implicit-explicit Runge–Kutta scheme for time discretization of advection-diffusion equations. Space discretization uses continuous, piecewise affine finite elements with interelement gradient jump penalty; discontinuous Galerkin methods can be considered as well. The advective and stabilization operators are treated explicitly, whereas the diffusion operator is treated implicitly. Our analysis hinges on L2-energy estimates on discrete functions in physical space. Our main results are stability and quasi-optimal error estimates for smooth solutions under a standard hyperbolic CFL restriction on the time step, both in the advection-dominated and in the diffusion-dominated regimes. The theory is illustrated by numerical examples.

We investigate unilateral contact problems with cohesive forces, leading tothe constrained minimization of a possibly nonconvex functional. Weanalyze the mathematical structure of the minimization problem. The problem is reformulated in terms of a three-field augmentedLagrangian, and sufficient conditions for the existence of a localsaddle-point are derived. Then, we derive and analyze mixed finiteelement approximations to the stationarity conditions of the three-fieldaugmented Lagrangian. The finite element spaces for the bulk displacement andthe Lagrange multiplier must satisfy a discrete inf-sup condition, whilediscontinuous finite element spaces spanned by nodal basis functions areconsidered for the unilateral contact variable so as to use collocationmethods. Two iterative algorithms are presented and analyzed, namely anUzawa-type method within a decomposition-coordination approach and anonsmooth Newton's method. Finally, numerical results illustrating thetheoretical analysis are presented.

We analyze Euler-Galerkin approximations (conforming finite elements inspace and implicit Euler in time) to coupled PDE systems in which one dependentvariable, say u, is governed by an elliptic equation and the other,say p, by a parabolic-like equation. The underlying application is theporoelasticity system within the quasi-static assumption. Differentpolynomial orders are used for the u- and p-components toobtain optimally convergent a priori bounds for allthe terms in the error energy norm.Then, a residual-type a posteriori error analysis is performed. Upon extending theideas of Verfürth for the heat equation [Calcolo40 (2003)195–212],an optimally convergent bound is derived for the dominant term in theerror energy norm and an equivalence result between residual anderror is proven. The error bound can be classically split intotime error, space error and data oscillation terms. Moreover, upon extending the elliptic reconstruction techniqueintroduced by Makridakis and Nochetto [SIAM J. Numer. Anal.41 (2003) 1585–1594],an optimally convergent bound is derived for the remaining terms in theerror energy norm. Numerical results are presented toillustrate the theoretical analysis.

A continuous finite element method to approximate Friedrichs' systems isproposed and analyzed. Stability is achieved by penalizing the jumpsacross meshinterfaces of the normal derivative of some components of the discrete solution. The convergence analysis leads to optimal convergence ratesin the graph norm and suboptimal of order ½ convergence rates inthe L 2-norm. A variant of the method specialized toFriedrichs' systems associated with elliptic PDE's in mixed form andreducing the number of nonzero entries in the stiffness matrix is alsoproposed and analyzed. Finally, numerical results are presented to illustrate thetheoretical analysis.

This paper derives upper and lower bounds for the $\ell^p$ -conditionnumber of the stiffness matrix resulting from the finite elementapproximation of a linear, abstract model problem. Sharp estimates interms of the meshsize h are obtained. The theoretical results areapplied to finite element approximations of elliptic PDE's invariational and in mixed form, and to first-order PDE's approximatedusing the Galerkin–Least Squares technique or bymeans of a non-standard Galerkin technique in L 1(Ω). Numerical simulations are presented to illustrate thetheoretical results.

We analyze residual and hierarchicala posteriori error estimates for nonconforming finite elementapproximations of elliptic problems with variable coefficients.We consider a finite volume box scheme equivalent toa nonconforming mixed finite element method in a Petrov–Galerkinsetting. We prove thatall the estimators yield global upper and local lower bounds for the discretizationerror. Finally, we present results illustrating the efficiency of theestimators, for instance, in the simulation of Darcy flows throughheterogeneous porous media.