The convergence of a two-scale FEM for elliptic problems in divergence form with coefficients and geometries oscillating at length scale ε << 1 is analyzed. Full elliptic regularity independent of ε is shownwhen the solution is viewed as mapping from the slow into the fast scale.Two-scale FE spaces which are able to resolve the ε scale of thesolution with work independent of ε and withoutanalytical homogenization are introduced. Robustin ε error estimates for the two-scale FE spaces are proved. Numerical experiments confirm thetheoretical analysis.

The stability of flat interfaces with respect to a spatial semidiscretization of a solidification model is analyzed. The considered model is the quasi-static approximation of the Stefan problem with dynamical Gibbs–Thomson law. The stability analysis bases on an argument developed by Mullins and Sekerka for the undiscretized case. The obtained stability properties differ from those with respect to the quasi-static model for certain parameter values and relatively coarse meshes. Moreover, consequences on discretization issues are discussed.

In this work, we analyze hierarchic hp-finite element discretizations of the full, three-dimensional plate problem. Based on two-scale asymptotic expansion of the three-dimensional solution, we give specific mesh design principles for the hp-FEM which allow to resolve the three-dimensional boundary layer profiles at robust, exponential rate.We prove that, as the plate half-thickness ε tends to zero, the hp-discretization is consistent with the three-dimensional solution to any power of ε in the energy norm for the degree $p={\cal O}(\left|{\log \varepsilon}\right|)$ and with ${\cal O}({p^4})$ degrees of freedom.

This paper deals with the diffusion limit of a kinetic equation where thecollisions are modeled by a Lorentz type operator. The main aim is to construct adiscrete scheme to approximate this equation which gives for any value of theKnudsen number, and in particular at the diffusive limit, the right discretediffusion equation with the same value of the diffusion coefficient as in thecontinuous case. We are also naturally interested with a discretization whichcan be used with few velocity discretization points, in order to reduce the cost ofcomputation.

We consider a dynamical one-dimensional nonlinear von Kármán model for beamsdepending on a parameter ε > 0 and studyits asymptotic behavior for t large, as ε → 0. Introducing appropriate dampingmechanisms we show that the energy of solutionsof the corresponding damped models decayexponentially uniformly with respect to theparameter ε. In order for this to be true thedamping mechanism has to have the appropriatescale with respect to ε. In the limit as ε → 0 we obtain damped Berger–Timoshenko beam modelsfor which the energy tends to zero exponentiallyas well. This is done both in the case ofinternal and boundary damping. We address the sameproblem for plates with internal damping.

In the present work the symmetrized sequential-parallel decomposition methodof the third degree precision for the solution of Cauchy abstract problemwith an operator under a split form, is presented. The third degreeprecision is reached by introducing a complex coefficient with the positivereal part. For the considered schema the explicit a priori estimation isobtained.

We prove the convergence of a finite volumemethod for a noncoercive linear elliptic problem, with right-handside in the dual space of the natural energy space of the problem.

A numerically inexpensive globalization strategy of sequential quadratic programming methods (SQP-methods) for control of the instationary Navier Stokes equations is investigated. Based on the proper functional analytic setting a convergence analysis for the globalized methodis given. It is argued that the a priori formidable SQP-step can be decomposed into linear primal and linear adjoint systems, which is amenable for existing CFL-software. A report on a numerical test demonstrates the feasibility of the approach.