This paper deals with the solution of problems involving partial differential equations in  $\mathbb{R}^3$ . For three dimensional case, methods are useful if they require neither domain boundary regularity nor regularity for the exact solution of the problem. A new domain decomposition method is therefore presented whichuses low degree finite elements. The numerical approximation of the solution is easy, and optimal error bounds are obtained according to suitablenorms.

We consider the numerical approximation of a first orderstationary hyperbolic equation by the method of characteristics with pseudo time step k using discontinuous finite elements on a mesh ${\cal T}_h$ . For this method, we exhibit a “natural” norm || ||h,k for which we show that the discrete variational problem $P_h^k$ is well posed and weobtain an error estimate. We show that when k goes to zero problem $(P_h^k)$ (resp. the || ||h,k norm)has as a limit problem (P h ) (resp. the || ||h norm) associated to the Galerkin discontinuousmethod. This extends to two and three space dimension our previous results obtained in one space dimension.

The minimization of nonconvex functionals naturally arises inmaterials sciences where deformation gradients in certain alloys exhibitmicrostructures. For example, minimizing sequences of the nonconvexEricksen-James energy can be associated with deformations in martensitic materials thatare observed in experiments[2,3]. — From the numericalpoint of view, classical conforming and nonconforming finite elementdiscretizations have been observed to give minimizerswith their quality being highlydependent on the underlying triangulation, see [8,24,26,27] for a survey. Recently, a newapproach has been proposed and analyzed in [15,16]that is based on discontinuous finite elements to reduce the pollution effectof a general triangulation on the computed minimizer.The goal of the present paper isto propose and analyzean adaptive method,giving a more accurate resolution of laminated microstructure on arbitrary grids.

The equilibrium configurations of a one-dimensional variational model thatcombines terms expressing the bulk energy of a deformable crystal and itssurface energy are studied. After elimination of the displacement, theproblem reduces to the minimization of a nonconvex and nonlocal functional ofa single function, the thickness. Depending on a parameter which strengthensone of the terms comprising the energy at the expense of the other, it isshown that this functional may have a stable absolute minimum or only aminimizing sequence in which the term corresponding to the bulk energy isforced to zero by the production of a crack in the material.

The fluctuation splitting schemes were introduced by Roe in the beginning of the80's and have been then developed since then, essentially thanks to Deconinck.In this paper, the fluctuation splittingschemes formalism is recalled. Then, the hyperbolic/elliptic decomposition of thethree dimensional Euler equations is presented. This decomposition leads to an acousticsubsystem and two scalar advection equations, one of them being the entropy advection.Thanks to this decomposition, the two scalar equations are treated with the well knownPSI scalar fluctuation splitting scheme, and the acoustic subsystem is treatedwith the Lax Wendroff matrix fluctuation splitting scheme. Anadditional viscous term is introduced in order to reduce the oscillatory behavior of theLax Wendroff scheme. An implicit form leadsto a robust scheme which enables computations over a large rangeof Mach number. This fluctuation splitting scheme, called the Lax Wendroff - PSI scheme, produceslittlespurious entropy, thus leading to accurate drag predictions.Numerical results obtained with this Lax Wendroff PSI scheme are presented and compared to areference Euler code, based on a Lax Wendroff scheme.

We prove existence (uniqueness is easy) of a weak solution to a boundary value problem for an equation like $(v-1)^+_t = v_{xx} + F(v)_x$ where the function $F: \Bbb R\rightarrow\Bbb R$ is onlysupposed to be locally lipschitz continuous. In order to replace the lack of compactness in t on v<1, we use nonlinear semigroup theory.

Linear Force-free (or Beltrami) fields are three-components divergence-free fields solutions of the equation curlB = α B, where α is a real number.Such fields appear in many branches of physics like astrophysics, fluid mechanics, electromagnetics and plasma physics. In this paper,we deal with some related boundary value problems in multiply-connected bounded domains, in half-cylindrical domains and in exterior domains.

Based on QR-like decomposition with column pivoting, a new and efficient numerical method for solving symmetric matrix inverse eigenvalue problems is proposed, which is suitable for both the distinct and multiple eigenvalue cases. A locally quadratic convergence analysis is given. Some numerical experiments are presented to illustrate our results.

A justification of the two-dimensional nonlinear “membrane”equations for a plate made of a Saint Venant-Kirchhoff material hasbeen given by Fox et al. [9] by means of the method of formal asymptotic expansions applied to the three-dimensional equations of nonlinear elasticity. This model, which retains the material-frame indifference of the originalthree dimensional problem in the sense that its energy density isinvariant under the rotations of ${\mathbb{R}}^3$ , is equivalent to finding thecritical points of a functional whose nonlinear part depends on the firstfundamental form of the unknown deformed surface. We establish here an existence result for these equations in the case of themembrane submitted to a boundary condition of “tension”, and we show that thesolution found in our analysis is injective and is the unique minimizer of thenonlinear membrane functional, which is not sequentially weakly lowersemi-continuous.We also analyze the behaviour of the membrane when the “tension” goes toinfinityand we conclude that a “well-extended” membrane may undergo largeloadings.

In this paper, we study some finite volume schemes for the nonlinear hyperbolic equation ${u_t}(x,t)+\mbox{div}F(x,t,u(x,t))=0$ with the initial condition $u_{0}\in{L^\infty}(\mathbb{R}^N)$ . Passing to the limit in these schemes, we prove the existence of an entropy solution $u\in{L^infty}(\mathbb{R}^N\times\mathbb{R}_+)$ . Proving also uniqueness, we obtain the convergence of the finite volume approximation to the entropy solution in $L^p_{loc}(\mathbb{R}^N\times\mathbb{R}_+)$ , 1 ≤ p ≤ +∞. Furthermore, if ${u_0}\in {L^\infty}\cap\mbox{BV}_{loc}(\mathbb{R}^N)$ , we show that $u\in\mbox{BV}_{loc}(\mathbb{R}^N\times\mathbb{R}_+)$ , which leads to an “ $h^{\frac{1}{4}}$ ” error estimate between the approximate and the entropy solutions (where h defines the size of the mesh).

The computation of glacier movements leads to a system of nonlinear partial differentialequations. The existence and uniqueness of a weak solution is established by using the calculus ofvariations. A discretization by the finite element method is done. Thesolution of the discrete problem is proved to be convergent to the exactsolution. A first simple numerical algorithm is proposed and its convergence numericallystudied.

In this paper we are interested in the numerical modelingof absorbing ferromagnetic materialsobeying the non-linear Landau-Lifchitz-Gilbert law with respect to the propagation and scattering of electromagnetic waves.In this workwe consider the 1D problem. We first show that the corresponding Cauchy problemhas a unique global solution. We then derive a numerical scheme based on an appropriate modificationof Yee's scheme, that we show to preserve some importantproperties of the continuous model such as the conservation of the normof the magnetization and the decay of the electromagnetic energy.Stability is proved under a suitable CFL condition.Some numerical results for the 1D model are presented.

Modern physics theories claim that the dynamics of interfaces between the two-phase is described by the evolution equations involving the curvature and various kinematic energies. We consider the motion of spiral-shaped polygonal curves by its crystalline curvature, which deserves a mathematical model of real crystals. Exploiting the comparison principle, we show the local existence and uniqueness of the solution.

In this paper, a phase field system of Penrose–Fife type withnon–conserved order parameter is considered. A class of time–discrete schemes for an initial–boundary value problem for this phase–field system is presented.In three space dimensions, convergence is proved and an error estimate linear with respect to the time–step sizeis derived.

Error estimates for the mixed finite element solution of4th order elliptic problems with variable coefficients, which, in the particular case of aniso-/ortho-/isotropic plate bending problems, gives a direct, simultaneous approximation to bending moment tensorfield $\Psi= (\psi_{ij})_{1 \le i,j \le 2}$ and displacement field 'u', have been developed considering the combined effect of boundary approximation and numerical integration.

We investigate time harmonic Maxwell equations in heterogeneous media, where thepermeability μ and the permittivity ε are piecewise constant. Theassociated boundary value problem can be interpreted as a transmission problem. Ina very natural way the interfaces can have edges and corners. We give a detaileddescription of the edge and corner singularities of the electromagnetic fields.

This paper presents a stabilization technique for approximating transport equations. The key idea consists in introducing an artificial diffusion based on a two-level decomposition of the approximation space.The technique is proved to have stability and convergenceproperties that are similar to that of the streamline diffusion method.

This work deals with a system of nonlinear parabolic equations arisingin turbulence modelling. The unknowns are the N components of the velocityfield u coupled with two scalar quantities θ and φ. The systempresents nonlinear turbulent viscosity $A(\theta,\varphi)$ and nonlinearsource terms of the form $\theta^2|\nabla u|^2$ and $\theta\varphi|\nabla u|^2$ lying in L 1. Some existence results are shown in this paper, including $L^\infty$ -estimates and positivity for both θ and φ.

The standard discretization of the Stokes andNavier–Stokes equations in vorticity and stream function formulation by affine finiteelements is known for its bad convergence. We present here a modified discretization, weprove that the convergence is improved and we establish a priori error estimates.

We consider space semi-discretizations of the 1-d wave equation in a boundedinterval with homogeneous Dirichlet boundary conditions. We analyze the problemof boundary observability, i.e., the problem of whether the total energy ofsolutions can be estimated uniformly in terms of the energy concentrated on theboundary as the net-spacing h → 0. We prove that, due to the spurious modesthat the numerical scheme introduces at high frequencies, there is no such auniform bound. We prove however a uniform bound in a subspace of solutionsgenerated by the low frequencies of the discrete system. When h → 0 thisfinite-dimensional spaces increase and eventually cover the whole space. Wethus recover the well-known observability property of the continuous systemas the limit of discrete observability estimates as the mesh size tends tozero.We consider both finite-difference and finite-element semi-discretizations.