Band 15, Heft 3

Enantiomers are chiral objects which differ by their orientation and are thus referred to as left-handed and right-handed enantiomers. In applications they mostly occur as so-called racemic compounds consisting of approximately the same amount of left- and right-handed species which may have completely different properties. Hence, the separation of left- from right-handed enantiomers is an important issue. Conventional technologies are based on gas or high pressure liquid chromatography, capillary electrophoresis, or nuclear magnetic resonance, but typically they are slow and require costly chiral media. A new idea for separation of chiral objects is based on introducing them in certain vorticity patterns, which has been shown to work in theory for an extremely simplified setting by Kostur et al. [Phys. Rev. Lett. 96 (2006), 014502-1–014502-4]. In this paper, we investigate whether these ideas can be successfully adapted to a more realistic setup which can be implemented experimentally. For this purpose, we simulate transport of rigid chiral particles in a fluidic environment by an application of the fictitious domain Lagrange multiplier method due to Glowinski et al. [J. Comput. Phys. 169 (2001), 363–427] which has been designed to study the motion of rigid particles in carrier fluids. Numerical results are presented which illustrate the feasibility of enantiomer separation in flow fields consisting of pairwise counter-rotating vortices. Moreover, a first experimental setup based on surface acoustic wave generated vorticity patterns on the surface of a carrier fluid is devised which reflects the idealized numerical model and gives promising results with respect to properties of particle propagation. These findings may lead to a new technology for enantiomer separation which is both fast and cost-effective.

This paper provides a refined a posteriori error control for the obstacle problem with an affine obstacle which allows for a proof of optimal complexity of an adaptive algorithm. This is the first adaptive mesh-refining finite element method known to be of optimal complexity for some variational inequality. The result holds for first-order conforming finite element methods in any spacial dimension based on shape-regular triangulation into simplices for an affine obstacle. The key contribution is the discrete reliability of the a posteriori error estimator from [Numer. Math. 107 (2007), 455–471] in an edge-oriented modification which circumvents the difficulties caused by the non-existence of a positive second-order approximation [Math. Comp. 71 (2002), 1405–1419].

Based on the powerful tool of variational inequalities, in recent papers convergence rates results on ℓ 1 -regularization for ill-posed inverse problems have been formulated in infinite dimensional spaces under the condition that the sparsity assumption slightly fails, but the solution is still in ℓ 1 . In the present paper, we improve those convergence rates results and apply them to the Cesáro operator equation in ℓ 2 and to specific denoising problems. Moreover, we formulate in this context relationships between Nashed's types of ill-posedness and mapping properties like compactness and strict singularity.

We consider PDE constrained shape optimization in the framework of finite element discretization of the underlying boundary value problem. We present an algorithm tailored to preserve and exploit the approximation properties of the finite element method, and that allows for arbitrarily high resolution of shapes. It employs (i) B-spline based representations of the deformation diffeomorphism, and (ii) superconvergent domain integral expressions for the shape gradient. We provide numerical evidence of the performance of this method both on prototypical well-posed and ill-posed shape optimization problems.

In this work, we study Crank–Nicolson leap-frog (CNLF) methods with fast-slow wave splittings for Navier–Stokes equations (NSE) with a rotation/Coriolis force term, which is a simplification of geophysical flows. We propose a new stabilized CNLF method where the added stabilization completely removes the method's CFL time step condition. A comprehensive stability and error analysis is given. We also prove that for Oseen equations with the rotation term, the unstable mode (for which un+1+un-1≡0$u^{n+1}+u^{n-1}\equiv 0 $) of CNLF is asymptotically stable. Numerical results are provided to verify the stability and the convergence of the methods.

This paper provides a survey on several numerical approximation schemes for stochastic control problems that arise from actuarial science and finance. The problems to be considered include dividend optimization, reinsurance game, and quantile hedging for guaranteed minimum death benefits. To better describe the complicated financial markets and their inherent uncertainty and randomness, the so-called regime-switching models are adopted. Such models are more realistic and versatile, however, far more complicated to handle. Due to the complexity of the construction, the regime-switching diffusion systems can only be solved in very special cases. In general, it is virtually impossible to obtain closed-form solutions. We use Markov chain approximation techniques to construct discrete-time controlled Markov chains to approximate the value function and optimal controls. Examples are presented to illustrate the applicability of the numerical methods.

The paper is concerned with parabolic time-periodic boundary value problems which are of theoretical interest and arise in different practical applications. The multiharmonic finite element method is well adapted to this class of parabolic problems. We study properties of multiharmonic approximations and derive guaranteed and fully computable bounds of approximation errors. For this purpose, we use the functional a posteriori error estimation techniques earlier introduced by S. Repin. Numerical tests confirm the efficiency of the a posteriori error bounds derived.

The paper deals with iterative methods for solving linear operator equations x=Bx+f${x = Bx + f}$ and Ax=f${Ax = f}$ with self-adjoint operators in Hilbert space X in the critical case when ρ(B)=1${\rho (B) = 1}$ and 0∈SpA${0 \in \operatorname{Sp} A}$. The results obtained are based on a theorem by M. A. Krasnosel'skii on the convergence of the successive approximations, their modifications and refinements.

In this paper goal-oriented error control based on the dual weighted residual error method (DWR) is applied to frictional contact problems. The derivation of DWR error controls is done for arbitrary discretization schemes via the introduction of some discrete Lagrange multipliers describing the residual of the discretization. The discrete Lagrange multipliers may be provided by a reconstruction in a post-processing step or by a discretization of a mixed formulation in which they are directly available. The error controls are defined for user-defined functionals (the quantities of interest) which measure the error of the displacement field as well as the normal and tangential contact forces. Numerical experiments confirm the applicability of the estimates within adaptive schemes.