We develop a functional analytical framework for a linearperidynamic model of a spring network system in any space dimension.Various properties of the peridynamic operators are examined forgeneral micromodulus functions. These properties are utilized toestablish the well-posedness of both the stationary peridynamicmodel and the Cauchy problem of the time dependent peridynamicmodel. The connections to the classical elastic models are alsoprovided.

The penalty method when applied to the Stokes problem provides a very efficient algorithm for solving any discretization of this problem since it gives rise to a system of two equations where the unknowns are uncoupled. For a spectral or spectral element discretization of the Stokes problem, we prove a posteriori estimates that allow us to optimize the penalty parameter as a function of the discretization parameter. Numerical experiments confirm the interest of this technique.

In this paper we construct a new H(div)-conforming projection-basedp-interpolation operator that assumes only H r (K) $\cap$ ${\bf \tilde H}$ -1/2(div, K)-regularity(r > 0) on the reference element (either triangle or square) K.We show that this operator is stable with respect to polynomial degrees andsatisfies the commuting diagram property. We also establish an estimate for theinterpolation error in the norm of the space ${\bf \tilde H}$ -1/2(div, K),which is closely related to the energy spaces for boundary integral formulationsof time-harmonic problems of electromagnetics in three dimensions.

We propose and analyze a finite element method for approximating solutions to the Navier-Stokes-alpha model (NS-α) that utilizes approximate deconvolution and a modified grad-div stabilization and greatlyimproves accuracy in simulations. Standard finite element schemesfor NS-α suffer from two major sources of error if their solutions are considered approximationsto true fluid flow: (1) the consistency error arising from filtering; and (2) the dramatic effect of the large pressure erroron the velocity error that arises from the (necessary) use of the rotational form nonlinearity.The proposed scheme “fixes” these two numericalissues through the combined use of a modified grad-div stabilization that acts in both the momentum and filter equations, and an adapted approximate deconvolution technique designed to work with the altered filter. We provethe scheme is stable, optimally convergent, and the effect of the pressureerror on the velocity error is significantly reduced. Several numerical experiments are given that demonstrate theeffectiveness of the method.

This article is devoted to the presentation of a new contact algorithm for bodies undergoing finite deformations.We only address the kinematic aspect of the contact problem, that is the numerical treatment of the non-intersection constraint. In consequence, mechanical aspects like friction, adhesion or wear are not investigatedand we restrict our analysis to the simplest frictionless case. On the other hand, our method allows us to treat contacts and self-contacts, thin or non-thin structures in a single setting.

In this paper, we continue the study of the Raman amplification in plasmas that we initiated in [Colin and Colin, Diff. Int. Eqs.17 (2004) 297–330; Colin and Colin, J. Comput. Appl. Math.193 (2006) 535–562]. We point out that the Raman instability gives rise to three components. The first one is collinear to the incident laser pulse and counter propagates. In 2-D, the two other ones make a non-zero angle with the initial pulse and propagate forward. Furthermore they are symmetric with respect to the direction of propagation of the incident pulse. We construct a non-linear system taking into account all these components and perform some 2-D numerical simulations.