A new finite element, which is continuously differentiable, but only piecewise quadratic polynomials on a type of uniform triangulations, is introduced. We construct a local basis which does not involve nodal values nor derivatives. Different from the traditional finite elements, we have to construct a special, averaging operator which is stable and preserves quadratic polynomials. We show the optimal order of approximation of the finite element in interpolation, and in solving the biharmonic equation. Numerical results are provided confirming the analysis.


We are concerned with a 2D time harmonic wave propagationproblem in a medium including a thin slot whose thickness εis small with respect to the wavelength. In a previous article, we derivedformally an asymptotic expansion of the solution with respect to εusing the method of matched asymptotic expansions. We also proved theexistence and uniqueness of the terms of the asymptotics. In this paper,we complete the mathematical justification of our work by deriving optimal error estimates between the exact solutions and truncated expansions at any order.

In this paper, we consider C 1,1 Hamiltonian systems. We prove the existence of a first derivative of the flow with respect to initial values and show that it satisfies the symplecticity condition almost everywhere in the phase-space. In a second step, we present a geometric integrator for such systems (called the SDH method) based on B-splines interpolation and a splitting method introduced by McLachlan and Quispel [Appl. Numer. Math. 45 (2003) 411–418], and we prove it is convergent, and that it preserves the energy and the volume.

In this paper, the convergence of a Neumann-Dirichlet algorithm to approximateCoulomb's contact problem between two elastic bodies is proved in a continuous setting. In this algorithm, the natural interface between the two bodies is retained as a decomposition zone.

We consider a body immersed in a perfect gas and moving under the action of a constant force.Body and gas are in thermal equilibrium. We assume a stochastic interaction body/medium: when a particle of the medium hits the body, it is absorbed and immediately re-emitted with a Maxwellian distribution. This system gives rise to a microscopic model of friction. We study the approach of the body velocity V(t) to the limiting velocity $V_\infty$ and prove that, under suitable smallness assumptions, the approach to equilibrium is $$|V(t)-V_\infty|\approx \frac{C}{t^{d+1}},$$ where d is the dimension of the space, and C is a positive constant. This approach is not exponential, as typical in friction problems, and even slower than for the same problem with elastic collisions.

The model order reduction methodology of reduced basis (RB)techniques offers efficient treatment of parametrized partial differential equations (P2DEs) by providing both approximate solution procedures and efficient error estimates. RB-methods have so far mainly been applied to finite element schemes for elliptic and parabolic problems. In the current study we extend the methodology to general linear evolution schemes such as finite volume schemes for parabolic and hyperbolic evolution equations. The new theoretic contributions are the formulation of a reduced basis approximation scheme for these general evolution problems and the derivation of rigorous a-posteriori error estimates in various norms. Algorithmically, an offline/online decomposition of the scheme and the error estimators is realized in case of affine parameter-dependence of the problem. This is the basis for a rapid online computation in case of multiple simulation requests.We introduce a new offline basis-generation algorithm based on our a-posteriori error estimator which combines ideas from existing approaches. Numerical experiments for an instationary convection-diffusion problem demonstrate the efficient applicability of the approach.

We present in this paper a pressure correction scheme for the barotropic compressible Navier-Stokes equations, which enjoys an unconditional stability property, in the sense that the energy and maximum-principle-based a priori estimates of the continuous problem also hold for the discrete solution.The stability proof is based on two independent results for general finite volume discretizations, both interesting for their own sake: the L 2-stability of the discrete advection operator provided it is consistent, in some sense, with the mass balance and the estimate of the pressure work by means of the time derivative of the elastic potential.The proposed scheme is built in order to match these theoretical results, and combines a fractional-step time discretization of pressure-correction type with a space discretization associating low order non-conforming mixed finite elements and finite volumes.Numerical tests with an exact smooth solution show the convergence of the scheme.