J.M.L. Maubach

A posteriori error estimators for a two-level finite element method for the Navier-Stokes equations

Two- and multilevel truncated Newton finite element discretizations are presently a very promising approach for approximating the (nonlinear) Navier-Stokes equations describing the equilibrium flow of a viscous, incompressible fluid. Their combination with mesh adaptivity is considered in this article. Specifically, locally calculable a posteriori error estimators are derived, with full mathematical support, for the basic two-level discretization of the Navier-Stokes equations.

Defocus and twofold astigmatism correction in HAADF-STEM

A new simultaneous autofocus and twofold astigmatism correction method is proposed for High Angle Annular Dark Field Scanning Transmission Electron Microscopy (HAADF-STEM). The method makes use of a modification of image variance, which has already been used before as an image quality measure for different types of microscopy, but its use is often justified on heuristic grounds. In this paper we show numerically that the variance reaches its maximum at Scherzer defocus and zero astigmatism. In order to find this maximum a simultaneous optimization of three parameters (focus, x- and y-stigmators) is necessary. This is implemented and tested on a FEI Tecnai F20. It successfully finds the optimal defocus and astigmatism with time and accuracy, compared to a human operator.

Iterative autofocus algorithms for scanning electron microscopy

Introduction. A robust and reliable autofocus algorithm is important concern for the automation of a Scanning Electron Microscope (SEM). Comparison of existing autofocus techniques has been done for specific specimen for fluorescence [1] and non-fluorescence microscopy [2-3]. For Scanning Transmission Electron Microscopy some of available algorithms were compared [4]. To the authors’ knowledge broad evolution has not been published yet for SEM.

A Derivative-Based Fast Autofocus Method in Electron Microscopy

Most automatic focusing methods are based on a sharpness function, which delivers a real-valued estimate of an image quality. In this paper, we study an L2-norm derivative-based sharpness function, which has been used before based on heuristic consideration. We give a more solid mathematical foundation for this function and get a better insight into its analytical properties. Moreover an efficient autofocus method is presented, in which an artificial blur variable plays an important role.We show that for a specific choice of the artificial blur control variable, the function is approximately a quadratic polynomial, which implies that after the recording of at least three images one can find the approximate position of the optimal defocus. This provides the speed improvement in comparison with existing approaches, which usually require recording of more than ten images for autofocus. The new autofocus method is employed for the scanning transmission electron microscopy. To be more specific, it has been implemented in the FEI scanning transmission electron microscope and its performance has been tested as a part of a particle analysis application.

An accelerated Poincaré-map method for autonomous oscillators

A novel time-domain method for finding the periodic steady state of a free-running electrical oscillator is introduced. The method is based on the extrapolation technique MPE. The new method is applied to a family of artificial benchmark problems and to a real circuit (Colpitts oscillator). It turns out to have super-linearly convergence properties in both cases. Full implementational details are provided.

Efficient solution of Maxwell's equations for geometries with repeating patterns by an exchange of discretization directions in the aperiodic Fourier modal method

The aperiodic Fourier modal method in contrast-field formulation is a numerical discretization and solution technique for solving scattering problems in electromagnetics. Typically, spectral discretization is used in the finite periodic direction and spatial discretization in the orthogonal direction. In the light of the fact that the structures of interest often have a large width-to-height ratio and that the two discretization approaches have different computational complexities, we propose exchanging the directions for spatial and spectral discretization. Moreover, if the scatterer has repeating patterns, swapping the discretization directions facilitates the reuse of previous computations. Therefore, the new method is suited for scattering from objects with a finite number of periods, such as gratings, memory arrays, metamaterials, etc. Numerical experiments demonstrate a considerable reduction of the computational costs in terms of time and memory. For a specific test case considered in this paper, the new method (based on alternative discretization) is 40 times faster and requires 100 times less memory than the method based on classical discretization.

Optimal algebraic multilevel preconditioning for local refinement along a line

The application of some recently proposed algebraic multilevel methods for the solution of two-dimensional finite element problems on nonuniform meshes is studied. The locally refined meshes are created by the newest vertex mesh refinement method. After the introduction of this refinement technique it is shown that, by combining levels of refinement, a preconditioner of optimal order can be constructed for the case of local refinement along a line. Its relative condition number is accurately estimated. Numerical tests demonstrating the performance of the proposed preconditioners will be reported in a forthcoming paper.

A Parallel Finite Element Method for Convection-Diffusion Problems

The robust Parallel Finite Element Method examined in [5] and [4]. It is an element-wise parallel iterative solution method based on a Red-Black domain decomposition. Convection-diffusion problems are solved in an optimal order for a method which makes use of not more than local communication. For the parallellism, the recent paper [8] shows that a near perfect load-balance can be obtained for two-dimensional problems. This paper proves that one of the conditions which is sufficient in the two-dimensional case, unexpectedly is not so for the three-dimensional case.