In this paper, we construct and analyze finite element methods for the three dimensionalMonge-Ampère equation. We derive methods using the Lagrange finite element space such thatthe resulting discrete linearizations are symmetric and stable. With this in hand, we thenprove the well-posedness of the method, as well as derive quasi-optimal error estimates.We also present some numerical experiments that back up the theoretical findings.

We introduce a family of mixed discontinuous Galerkin (DG) finite element methods fornearly and perfectly incompressible linear elasticity. These mixed methods allow thechoice of polynomials of any order k ≥ 1 for the approximation of thedisplacement field, and of order k or k − 1 for thepressure space, and are stable for any positive value of the stabilization parameter. Weprove the optimal convergence of the displacement and stress fields in both cases, witherror estimates that are independent of the value of the Poisson’s ratio. These estimatesdemonstrate that these methods are locking-free. To this end, we prove the correspondinginf-sup condition, which for the equal-order case, requires a construction to establishthe surjectivity of the space of discrete divergences on the pressure space. In theparticular case of near incompressibility and equal-order approximation of thedisplacement and pressure fields, the mixed method is equivalent to a displacement methodproposed earlier by Lew et al. [Appel. Math. Res. express3 (2004) 73–106]. The absence of locking of this displacementmethod then follows directly from that of the mixed method, including the uniform errorestimate for the stress with respect to the Poisson’s ratio. We showcase the performanceof these methods through numerical examples, which show that locking may appear ifDirichlet boundary conditions are imposed strongly rather than weakly, as we do here.

In the present work we investigate the numerical simulation of liquid-vapor phase changein compressible flows. Each phase is modeled as a compressible fluid equipped with its ownequation of state (EOS). We suppose that inter-phase equilibrium processes in the mediumoperate at a short time-scale compared to the other physical phenomena such as convectionor thermal diffusion. This assumption provides an implicit definition of an equilibriumEOS for the two-phase medium. Within this framework, mass transfer is the result of localand instantaneous equilibria between both phases. The overall model is strictlyhyperbolic. We examine properties of the equilibrium EOS and we propose a discretizationstrategy based on a finite-volume relaxation method. This method allows to cope with theimplicit definition of the equilibrium EOS, even when the model involves complex EOS’s forthe pure phases. We present two-dimensional numerical simulations that shows that themodel is able to reproduce mechanism such as phase disappearance and nucleation.

The Hartree-Fock equation is widely accepted as the basic model of electronic structure calculation which serves as a canonical starting point for more sophisticated many-particle models. We have studied the s∗-compressibility for Galerkin discretizations of the Hartree-Fock equation in wavelet bases. Our focus is on the compression of Galerkin matrices from nuclear Coulomb potentials and nonlinear terms in the Fock operator which hitherto has not been discussed in the literature. It can be shown that the s∗-compressibility is in accordance with convergence rates obtained from best N-term approximation for solutions of the Hartree-Fock equation. This is a necessary requirement in order to achieve numerical solutions for these equations with optimal complexity using the recently developed adaptive wavelet algorithms of Cohen, Dahmen and DeVore.

The aim of this paper is to derive a general model for reduced viscous and resistive Magnetohydrodynamics (MHD) and to study its mathematical structure. The model is established for arbitrary density profiles in the poloidal section of the toroidal geometry of Tokamaks. The existence of global weak solutions, on the one hand, and the stability of the fundamental mode around initial data, on the other hand, are investigated.

We examine an elliptic optimal control problem with control and state constraints inℝ3. An improved error estimate of 𝒪(hs) with 3/4 ≤ s ≤ 1 − ε is proven for a discretisationinvolving piecewise constant functions for the control and piecewise linear for the state.The derived order of convergence is illustrated by a numerical example.

We examine the composition of the L∞ norm with weaklyconvergent sequences of gradient fields associated with the homogenization of second orderdivergence form partial differential equations with measurable coefficients. Here thesequences of coefficients are chosen to model heterogeneous media and are piecewiseconstant and highly oscillatory. We identify local representation formulas that in thefine phase limit provide upper bounds on the limit superior of theL∞ norms of gradient fields. The local representationformulas are expressed in terms of the weak limit of the gradient fields and localcorrector problems. The upper bounds may diverge according to the presence of roughinterfaces. We also consider the fine phase limits for layered microstructures and forsufficiently smooth periodic microstructures. For these cases we are able to provideexplicit local formulas for the limit of the L∞ norms of theassociated sequence of gradient fields. Local representation formulas for lower bounds areobtained for fields corresponding to continuously graded periodic microstructures as wellas for general sequences of oscillatory coefficients. The representation formulas areapplied to problems of optimal material design.

We consider the symmetric FEM-BEM coupling for the numerical solution of a (nonlinear)interface problem for the 2D Laplacian. We introduce some new a posteriorierror estimators based on the (h − h/2)-errorestimation strategy. In particular, these include the approximation error for the boundarydata, which allows to work with discrete boundary integral operators only. Using theconcept of estimator reduction, we prove that the proposed adaptive algorithm isconvergent in the sense that it drives the underlying error estimator to zero. Numericalexperiments underline the reliability and efficiency of the considered adaptivemesh-refinement.

An abstract framework for constructing stable decompositions of the spaces correspondingto general symmetric positive definite problems into “local” subspaces and a global“coarse” space is developed. Particular applications of this abstract framework includepractically important problems in porous media applications such as: the scalar elliptic(pressure) equation and the stream function formulation of its mixed form, Stokes’ andBrinkman’s equations. The constant in the corresponding abstract energy estimate is shownto be robust with respect to mesh parameters as well as the contrast, which is defined asthe ratio of high and low values of the conductivity (or permeability). The derived stabledecomposition allows to construct additive overlapping Schwarz iterative methods withcondition numbers uniformly bounded with respect to the contrast and mesh parameters. Thecoarse spaces are obtained by patching together the eigenfunctions corresponding to thesmallest eigenvalues of certain local problems. A detailed analysis of the abstractsetting is provided. The proposed decomposition builds on a method of Galvis and Efendiev[Multiscale Model. Simul. 8 (2010) 1461–1483] developedfor second order scalar elliptic problems with high contrast. Applications to the finiteelement discretizations of the second order elliptic problem in Galerkin and mixedformulation, the Stokes equations, and Brinkman’s problem are presented. A number ofnumerical experiments for these problems in two spatial dimensions are provided.

We consider the effect of surface roughness on solid-solid contact in a Stokes flow.Various models for the roughness are considered, and a unified methodology is given toderive the corresponding asymptotics of the drag force in the close-contact limit. In thisway, we recover and clarify the various expressions that can be found in previousstudies.

In this paper we study the temporal convergence of a locally implicit discontinuous Galerkin method for the time-domain Maxwell’s equations modeling electromagnetic waves propagation. Particularly, we wonder whether the method retains its second-order ordinary differential equation (ODE) convergence under stable simultaneous space-time grid refinement towards the true partial differential equation (PDE) solution. This is not a priori clear due to the component splitting which can introduce order reduction

In this paper we propose and analyze stable variational formulations for convection diffusion problems starting from concepts introduced by Sangalli. We derive efficient and reliable a posteriori error estimators that are based on these formulations. The analysis of resulting adaptive solution concepts, when specialized to the setting suggested by Sangalli’s work, reveals partly unexpected phenomena related to the specific nature of the norms induced by the variational formulation. Several remedies, based on other specifications, are explored and illustrated by numerical experiments.