Abstract: Publication date: May 2011
Source:Applied Numerical Mathematics, Volume 61, Issue 5
Author(s): H.M. Felix , A. Sri Ranga , D.O. Veronese
A positive measure ψ defined on [ a , b ] such that its moments μ n = ∫ a b t n d ψ ( t ) exist for n = 0 , ± 1 , ± 2 , … , is called a strong positive measure on [ a , b ] . If 0 ⩽ a < b ⩽ ∞ then the sequence of (monic) polynomials { Q n } , defined by ∫ a b t − n + s Q n ( t ) d ψ ( t ) = 0 , s = 0 , 1 , … , n − 1 , is known to exist. We refer to these polynomials as the L-orthogonal polynomials with respect to the strong positive measure ψ. The purpose of this manuscript is to consider some properties of the kernel polynomials associated with these L-orthogonal polynomials. As applications, we consider the quadrature rules associated with these kernel polynomials. Associated eigenvalue problems and numerical evaluation of the nodes and weights of such quadrature rules are also considered.

Abstract: Publication date: April 2012
Source:Applied Numerical Mathematics, Volume 62, Issue 4
Author(s): M. Aurada , S. Ferraz-Leite , P. Goldenits , M. Karkulik , M. Mayr , D. Praetorius
For a boundary integral formulation of the 2D Laplace equation with mixed boundary conditions, we consider an adaptive Galerkin BEM based on an ( h − h / 2 ) -type error estimator. We include the resolution of the Dirichlet, Neumann, and volume data into the adaptive algorithm. In particular, an implementation of the developed algorithm has only to deal with discrete integral operators. We prove that the proposed adaptive scheme leads to a sequence of discrete solutions, for which the corresponding error estimators tend to zero. Under a saturation assumption for the non-perturbed problem which is observed empirically, the sequence of discrete solutions thus converges to the exact solution in the energy norm.

Abstract: Publication date: June 2012
Source:Applied Numerical Mathematics, Volume 62, Issue 6
Author(s): Markus Aurada , Samuel Ferraz-Leite , Dirk Praetorius
A posteriori error estimation and related adaptive mesh-refining algorithms have themselves proven to be powerful tools in nowadays scientific computing. Contrary to adaptive finite element methods, convergence of adaptive boundary element schemes is, however, widely open. We propose a relaxed notion of convergence of adaptive boundary element schemes. Instead of asking for convergence of the error to zero, we only aim to prove estimator convergence in the sense that the adaptive algorithm drives the underlying error estimator to zero. We observe that certain error estimators satisfy an estimator reduction property which is sufficient for estimator convergence. The elementary analysis is only based on Dörfler marking and inverse estimates, but not on reliability and efficiency of the error estimator at hand. In particular, our approach gives a first mathematical justification for the proposed steering of anisotropic mesh-refinements, which is mandatory for optimal convergence behavior in 3D boundary element computations.

Abstract: Publication date: May 2014
Source:Applied Numerical Mathematics, Volume 79
Author(s): Oszkár Bíró , Gergely Koczka , Kurt Preis
An efficient finite element method to take account of the nonlinearity of the magnetic materials when analyzing three-dimensional eddy current problems is presented in this paper. The problem is formulated in terms of vector and scalar potentials approximated by edge and node based finite element basis functions. The application of Galerkin techniques leads to a large, nonlinear system of ordinary differential equations in the time domain. The excitations are assumed to be time-periodic and the steady-state periodic solution is of interest only. This is represented either in the frequency domain as a finite Fourier series or in the time domain as a set of discrete time values within one period for each finite element degree of freedom. The former approach is the (continuous) harmonic balance method and, in the latter one, discrete Fourier transformation will be shown to lead to a discrete harmonic balance method. Due to the nonlinearity, all harmonics, both continuous and discrete, are coupled to each other. The harmonics would be decoupled if the problem were linear, therefore, a special nonlinear iteration technique, the fixed-point method is used to linearize the equations by selecting a time-independent permeability distribution, the so-called fixed-point permeability in each nonlinear iteration step. This leads to uncoupled harmonics within these steps. As industrial applications, analyses of large power transformers are presented. The first example is the computation of the electromagnetic field of a single-phase transformer in the time domain with the results compared to those obtained by traditional time-stepping techniques. In the second application, an advanced model of the same transformer is analyzed in the frequency domain by the harmonic balance method with the effect of the presence of higher harmonics on the losses investigated. Finally a third example tackles the case of direct current (DC) bias in the coils of a single-phase transformer.

Abstract: Publication date: Available online 14 May 2015
Source:Applied Numerical Mathematics
Author(s): R. Andreev , O. Scherzer , W. Zulehner
We consider the simultaneous estimation of an optical flow field and an illumination source term in a movie sequence. The particular optical flow equation is obtained by assuming that the image intensity is a conserved quantity up to possible sources and sinks which represent varying illumination. We formulate this problem as an energy minimization problem and propose a space-time simultaneous discretization for the optimality system in saddle-point form. We investigate a preconditioning strategy that renders the discrete system well-conditioned uniformly in the discretization resolution. Numerical experiments complement the theory.

Abstract: Publication date: Available online 29 April 2015
Source:Applied Numerical Mathematics
Author(s): N. Tuncer , A. Madzvamuse , A.J. Meir
In this paper we present a robust, efficient and accurate finite element method for solving reaction-diffusion systems on stationary spheroidal surfaces (these are surfaces which are deformations of the sphere such as ellipsoids, dumbbells, and heart-shaped surfaces) with potential applications in the areas of developmental biology, cancer research, wound healing, tissue regeneration, and cell motility among many others where such models are routinely used. Our method is inspired by the radially projected finite element method introduced in [39]. (Hence the name “projected” finite element method). The advantages of the projected finite element method are that it is easy to implement and that it provides a conforming finite element discretization which is “logically” rectangular. To demonstrate the robustness, applicability and generality of this numerical method, we present solutions of reaction-diffusion systems on various spheroidal surfaces. We show that the method presented in this paper preserves positivity of the solutions of reaction-diffusion equations which is not generally true for Galerkin type methods. We conclude that surface geometry plays a pivotal role in pattern formation. For a fixed set of model parameter values, different surfaces give rise to different pattern generation sequences of either spots or stripes or a combination (observed as circular spot-stripe patterns). These results clearly demonstrate the need for detailed theoretical analytical studies to elucidate how surface geometry and curvature influence pattern formation on complex surfaces.