The aim of this paper is to find estimates of the Green's function of stationary discrete shock profiles and discrete boundary layers of the modified Lax–Friedrichs numerical scheme, by using techniques developed by Zumbrun and Howard [CITE] in the continuous viscous setting.

Semi–smooth Newton methods are analyzed for a class of variational inequalities in infinite dimensions. It is shown that they are equivalent to certain active set strategies. Global and local super-linear convergence are proved. To overcome the phenomenon of finite speed of propagation of discretized problems a penalty version is used as the basis for a continuation procedure to speed up convergence. The choice of the penalty parameter can be made on the basis of an L∞ estimate for the penalized solutions. Unilateral as well as bilateral problems are considered.

We propose a discontinuous Galerkin method for linear elasticity, based on discontinuous piecewise linear approximation of the displacements. We show optimal order a priori error estimates, uniform in the incompressible limit, and thus locking is avoided. The discontinuous Galerkin method is closely related to the non-conforming Crouzeix–Raviart (CR) element, which in fact is obtained when one of the stabilizing parameters tends to infinity. In the case of the elasticity operator, for which the CR element is not stable in that it does not fulfill a discrete Korn's inequality, the discontinuous framework naturally suggests the appearance of (weakly consistent) stabilization terms. Thus, a stabilized version of the CR element, which does not lock, can be used for both compressible and (nearly) incompressible elasticity. Numerical results supporting these assertions are included. The analysis directly extends to higher order elements and three spatial dimensions.

In this paper we introduce numerical schemes for a one-dimensional kinetic model of the Boltzmann equation with dissipative collisions and variable coefficient of restitution. In particular, we study the numerical passage of the Boltzmann equation with singular kernel to nonlinear friction equations in the so-called quasi elastic limit. To this aim we introduce a Fourier spectral method for the Boltzmann equation [CITE] and show that the kernel modes that define the spectral method have the correct quasi elastic limit providing a consistent spectral method for the limiting nonlinear friction equation.

We propose and analyze a domain decomposition method on non-matching grids for partial differential equations with non-negative characteristic form. No weak or strong continuity of the finite element functions, their normal derivatives, or linear combinations of the two is imposed across the boundaries of the subdomains. Instead, we employ suitable bilinear forms defined on the common interfaces, typical of discontinuous Galerkin approximations. We prove an error bound which is optimal with respect to the mesh–size and suboptimal with respect to the polynomial degree. Our analysis is valid for arbitrary shape–regular meshes and arbitrary partitions into subdomains. Our method can be applied to advective, diffusive, and mixed–type equations, as well, and is well-suited for problems coupling hyperbolic and elliptic equations. We present some two-dimensional numerical results that support our analysis for the case of linear finite elements.

Let f be an odd function of a class C2 such that ƒ(1) = 0,ƒ'(0) < 0,ƒ'(1) > 0 and $x\mapsto f(x)/x$ increases on [0,1]. We approximate the positive solution of Δu + ƒ(u) = 0, on $ \xR_{+}^{2}$ with homogeneous Dirichlet boundary conditions by the solution of $-\Delta u_{L}+f(u_{L})=0,$ on ]0,L[2 with adequate non-homogeneous Dirichlet conditions. We show that the error uL - u tends to zero exponentially fast, in the uniform norm.

A sparse algebraic multigrid method is studied as a cheap and accurate way to compute approximations of Schur complements of matrices arising from the discretization of some symmetric and positive definite partial differential operators. The construction of such a multigrid is discussed and numerical experiments are used to verify the properties of the method.

We consider a family of shell finite elements with quadratic displacements across the thickness. These elements are very attractive, but compared to standard general shell elements they face another source of numerical locking in addition to shear and membrane locking. This additional locking phenomenon – that we call “pinching locking” – is the subject of this paper and we analyse a numerical strategy designed to overcome this difficulty. Using a model problem in which only this specific source of locking is present, we are able to obtain error estimates independent of the thickness parameter, which shows that pinching locking is effectively treated. This is also confirmed by some numerical experiments of which we give an account.

We establish an asymptotic representation formula for the steady state voltage perturbations caused by low volume fraction internal conductivity inhomogeneities. This formula generalizes and unifies earlier formulas derived for special geometries and distributions of inhomogeneities.

The main goal of this article is to establish a priori and a posteriori error estimates for the numerical approximation of some non linear elliptic problems arising in glaciology. The stationary motion of a glacier is given by a non-Newtonian fluid flow model which becomes, in a first two-dimensional approximation, the so-called infinite parallel sided slab model. The approximation of this model is made by a finite element method with piecewise polynomial functions of degree 1. Numerical results show that the theoretical results we have obtained are almost optimal.

We consider in this paper the problem of finding appropriate models for Large Eddy Simulations of turbulent incompressible flows from a mathematical point of view. The Smagorinsky model is analyzed and the vorticity formulation of the Navier–Stokes equations is used to explore more efficient subgrid-scale models as minimal regularizations of these equations. Two classes of variants of the Smagorinsky model emerge from this approach: a model based on anisotropic turbulent viscosity and a selective model based on vorticity angles. The efficiency of these models is demonstrated by comparisons with reference results on decaying turbulence experiments.