We present a domain decomposition theory on an interface problemfor the linear transport equation between a diffusive and a non-diffusive region.To leading order, i.e. up to an error of the order of the mean free path in thediffusive region, the solution in the non-diffusive region is independent of thedensity in the diffusive region. However, the diffusive and the non-diffusive regionsare coupled at the interface at the next order of approximation. In particular, ouralgorithm avoids iterating the diffusion and transport solutions as is done in mostother methods — see for example Bal–Maday (2002). Our analysis is based instead on an accurate description of the boundarylayer at the interface matching the phase-space density of particles leaving thenon-diffusive region to the bulk density that solves the diffusion equation.

This paper presents a model based on spectral hyperviscosity for the simulation of 3D turbulent incompressible flows. One particularity of this model is that the hyperviscosity is active only at the short velocity scales, a feature which is reminiscent of Large Eddy Simulation models. We propose a Fourier–Galerkin approximation of the perturbedNavier–Stokes equations and we show that, as the cutoff wavenumbergoes to infinity, the solution of the modelconverges (up to subsequences) to a weak solution which is dissipativein the sense defined by Duchon and Robert (2000).

We show that it is possible to construct a class of entropicschemes for the multicomponent Euler system describing a gas or fluidhomogeneous mixture at thermodynamic equilibrium by applying a relaxation technique. Afirst order Chapman–Enskog expansion shows that the relaxed systemformally converges when the relaxation frequencies go to the infinitytoward a multicomponent Navier–Stokes system with the classical Fick andNewton laws, with a thermal diffusion which can be assimilated to a Soret effect in the case of a fluid mixture,and with also a pressure diffusion or a density diffusion respectively for a gas or fluid mixture. We also discuss on the link between the convexity of the entropies of each species and the existence of the Chapman–Enskog expansion.

Models of two phase flows in porous media, used in petroleumengineering, lead to a system of two coupled equations with ellipticand parabolic degenerate terms, and two unknowns,the saturation and the pressure.For the purpose of their approximation, a coupled scheme, consisting ina finite volume method together witha phase-by-phase upstream weighting scheme, is used in the industrial setting.This paper presents a mathematical analysis of this coupled scheme, first showingthat it satisfies some a priori estimates:the saturation is shown to remain in a fixed interval, anda discrete L2(0,T;H1 (Ω)) estimate is proved for both the pressureand a function of the saturation. Thanks to these properties,a subsequence of the sequence of approximate solutions is shown toconverge to a weak solutionof the continuous equationsas the size of the discretization tends to zero.

Interest in meshfree methods in solving boundary-value problems has grownrapidly in recent years. A meshless method that has attracted considerableinterest in the community of computational mechanics is built around theidea of modified local Shepard's partition of unity. For these kinds ofapplications it is fundamental to analyze the order of the approximation inthe context of Sobolev spaces. In this paper, we study two differenttechniques for building modified local Shepard's formulas, and we provide atheoretical analysis for error estimates of the approximation in Sobolevnorms. We derive Jackson-type inequalities for h-p cloud functionsusing the first construction. These estimates are important in the analysisof Galerkin approximations based on local Shepard's formulas or h-pcloud functions.

We consider a non-conforming stabilized domaindecomposition technique forthe discretization of the three-dimensional Laplace equation.The aim is to extend the numerical analysis of residual error indicators tothis model problem. Two formulations of the problem are consideredand the error estimators are studied for both. In thefirst one, the error estimator provides upper and lower bounds forthe energy norm of the mortar finite element solution whereas inthe second case, it also estimates the error for the Lagrangemultiplier.

We consider a posteriori error estimators that can be applied to anisotropic tetrahedral finite element meshes, i.e. meshes where the aspect ratio of the elements can be arbitrarily large.Two kinds of Zienkiewicz–Zhu (ZZ) type error estimators are derivedwhich originate from different backgrounds. In the course of the analysis, the first estimator turns out to be a special case of the second one, and both estimators can be expressed using some recovered gradient.The advantage of keeping two different analyses of the estimators is that they allow different and partially novel investigations and results. Both rigorous analytical approaches yield the equivalence of each ZZ error estimator to a known residual error estimator. Thus reliability and efficiency of the ZZ error estimation is obtained.The anisotropic discretizations require analytical tools beyond the standard isotropic methods. Particular attention is paid to the requirements on the anisotropic mesh.The analysis is complemented and confirmed by extensive numerical examples. They show that good results can be obtained for a large class of problems, demonstrated exemplary for the Poisson problem and a singularly perturbed reaction diffusion problem.