We give results for the approximation of a laminate withvarying volume fractions for multi-well energy minimizationproblems modeling martensitic crystals thatcan undergo either an orthorhombicto monoclinic or a cubic to tetragonal transformation. We construct energy minimizing sequences of deformations which satisfythe corresponding boundary condition, and we establish a series of error bounds in terms of the elastic energyfor the approximation of the limiting macroscopicdeformation and the simply laminated microstructure.Finally, we give results for the corresponding finite element approximation of the laminate with varying volume fractions.

We treat the problem of constructing exact theories of rods and shells for thin incompressible bodies. We employ a systematic method that consists inimposing constraints to reduce the number of degrees of freedom of each crosssection to a finite number. We show that it is very difficult to produce theories that exactly preserve the incompressibility and we show that it is impossible to do so for naive theories. In particular, many exact theories have nonlocal effects.

In this paper, several modifications of the quasi-interpolationoperator of Scott andZhang [30]are discussed. The modified operators are defined for non-smooth functionsand are suited for application on anisotropic meshes. Theanisotropy of the elements is reflected in the local stability andapproximation error estimates.As an application, an example is considered where anisotropic finiteelement meshes are appropriate, namely the Poisson problem in domainswith edges.

For domains which are star-shapedw.r.t. at least one point, we give new bounds on theconstants in Jackson-inequalities in Sobolev spaces. Forconvex domains, these bounds do not depend on theeccentricity. For non-convex domains with a re-entrantcorner, the bounds are uniform w.r.t. the exteriorangle. The main tool is a new projection operator ontothe space of polynomials.

This paper deals with a finite element method to solve fluid-structure interaction problems. More precisely it concerns the numerical computation of harmonic hydroelastic vibrations under gravity. It is based on a displacement formulation for both the fluid and the solid. Gravity effects are included on the free surface of the fluid as wellas on the liquid-solid interface. The pressure of the fluid is used as a variable for the theoretical analysis leading to a well posed mixed linear eigenvalue problem.Lowest order triangular Raviart-Thomas elements are used for the fluid and classical piecewise linear elements for the solid. Transmission conditions at the fluid-solid interface are taken into account in a weak sense yielding a non conforming discretization. The method does not present spurious or circulation modes for nonzero frequencies.Convergence is proved and optimal error estimates are given.Finally, numerical results are shown.

In this paper we study the derivation of homogeneous hydrostatic equationsstarting from 2D Euler equations, following for instance[2,9]. We give a convergence result for convex profiles anda divergence result for a particular inflexion profile.

We consider the evolution of a set $\Lambda\subset \mathbb R^2$ according to theHuygens principle: i.e. the domain at time t>0, Λt, is the set ofthe points whose distance from Λ is lower thant. We give some general results for this evolution,with particular care given to the behavior of the perimeter of theevoluted set as a function of time. We define a class of sets (non-trapping sets) for which the perimeter is a continuous function of t, andwe give an algorithm to approximate the evolution.Finally we restrict our attention to the class of sets for which the turning angle of the boundary is greater than -π (see [2]).For this class of sets we prove that the perimeter is a Lipschitz-continuous function of t. This evolution problem is relevant for the applicationsbecause it is used as a model for solid fuel combustion.

We address the homogenization of an eigenvalue problem for the neutron transport equation in a periodic heterogeneous domain, modeling the criticality study of nuclear reactor cores. We prove that the neutron flux, corresponding to the first and unique positive eigenvector, can be factorized in the product of two terms, up to a remainder which goes strongly to zero with the period. One term is the first eigenvector of the transport equation in the periodicity cell. The other term is the first eigenvector of a diffusion equation in the homogenized domain. Furthermore, the corresponding eigenvalue gives a second order corrector for the eigenvalue of the heterogeneous transport problem. This result justifies and improves the engineering procedure used in practice for nuclear reactor cores computations.

One of the main tools in the proof of residual-based a posteriori error estimates is a quasi-interpolation operator due to Clément. We modify this operator in the setting of a partition of unitywith the effect that the approximation error has a local average zero.This results in a new residual-based a posteriori error estimate with a volume contribution which is smaller than in the standard estimate.For an elliptic model problem, we discuss applications to conforming, nonconforming and mixed finite element methods.

In this paper, a multi-parameter error resolutiontechnique is applied into a mixed finite element method for theStokes problem. By using this technique and establishing a multi-parameter asymptotic error expansion for the mixed finite element method, an approximation of higheraccuracy is obtained by multi-processor computers in parallel.

In this paper we present a novel exponentially fitted finite elementmethod with triangular elements for the decoupled continuity equations in the drift-diffusion model of semiconductor devices.The continuous problem is first formulated as a variational problem using a weighted inner product. A Bubnov-Galerkinfinite element method with a set of piecewise exponential basis functions is then proposed. The method is shown to be stable and can be regarded asan extension to two dimensions of the well-known Scharfetter-Gummel method.Error estimates for the approximate solution and its associated flux are given. These h-order error bounds depend on some first-order seminorms of the exact solution, the exact fluxand the coefficient function of the convection terms.A method is also proposed for the evaluation of terminal currentsand it is shown that the computed terminal currents are convergent andconservative.

Error estimates in L∞(0,T;L 2(Ω)), L∞(0,T;L 2(Ω)2),L∞(0,T;L ∞(Ω)), L∞(0,T;L∞ (Ω)2), Ω in ${\mathbb R}^2$ , are derived for a mixed finite element method for the initial-boundary value problem for integro-differential equation $$u_t={\rm div}\{a\bigtriangledown u+\int^t_0b_1\bigtriangledown u{\rm d}\tau+\int^t_0{\bf c}u{\rm d}\tau\}+f$$ based on the Raviart-Thomas space V h x Wh ⊂ H(div;Ω) x L2(Ω). Optimal order estimates are obtained for theapproximation of u,ut in L∞(0,T;L 2(Ω)) and the associated velocity p in L∞(0,T;L 2(Ω)2), divp in L∞(0,T;L 2(Ω)). Quasi-optimal order estimates are obtained for the approximation of u in L∞ (0,T;L ∞(Ω)) and p in L∞(0,T;L ∞(Ω)2.

Using the approach in [5] for analysingtime discretization error and assuming more regularity on the initial data, we improve onthe error bound derived in [2] for a fully practical piecewise linear finite element approximation with a backward Euler time discretization of a model for phase separation of a multi-component alloy withnon-smooth free energy.

The evolution of n–dimensional graphs under a weighted curvature flow is approximated by linear finite elements. We obtain optimal error bounds for the normals and the normal velocities of the surfacesin natural norms. Furthermore we prove a global existence result for thecontinuous problem and present some examples of computed surfaces.

We present the semi-conductor Boltzmann equation,which is time-reversible, and indicate that it can be formally derivedby considering the large time and small perturbing potential limitin the Von-Neumann equation (time-reversible). We then rigorously computethe corresponding asymptotics in the case of the Von-Neumann equation onthe Torus. We show that the limiting equation we obtain does not coincidewith the physically realistic model. The former is indeedan equation of Boltzmann type, yet with memory in time, so that it appearsto be time-reversible. We comment on this point, and further describethe properties of the limiting equation.

Low-order nonconforming Galerkin methods will be analyzed for second-order elliptic equations subjected to Robin, Dirichlet, or Neumann boundary conditions. Both simplicial and rectangular elements will be considered in two and three dimensions. The simplicial elements will be based on P 1, as for conforming elements; however, it is necessary to introduce new elements in the rectangular case. Optimal order error estimates are demonstrated in all cases with respect to a broken norm in H 1(Ω) and in the Neumann and Robin cases in L 2(Ω).

In this paper, we consider second order neutrons diffusion problem withcoefficients in L ∞(Ω). Nodal method of the lowest order is applied to approximate the problem's solution. The approximation uses special basis functions [1] in which the coefficientsappear. The rate of convergence obtained is O(h2) in L 2(Ω), with a free rectangular triangulation.

We present a family of high-order, essentially non-oscillatory, central schemes for approximating solutions of hyperbolic systems of conservation laws. These schemes are based on a new centered version of the Weighed Essentially Non-Oscillatory (WENO) reconstruction of point-valuesfrom cell-averages, which is then followed by an accurate approximationof the fluxes via a natural continuous extension of Runge-Kutta solvers.We explicitly construct the third and fourth-order scheme and demonstrate theirhigh-resolution properties in several numerical tests.

The initial-boundary value problem of two-dimensionalincompressible fluid flow in stream function form is considered.A prediction-correction Legendre spectral scheme is proposed, which iseasy to be performed.The numerical solution possesses the accuracy of second-order in time and higher order in space. Thenumerical experiments show the high accuracy of this approach.

Stability and asymptotic stability of the solutions of impulsivenonlinear pa ra bo lic equationsare studied via the method of differential inequalities.