A modified Newton method for solving non-symmetric algebraic Riccati equations arising in transport theory

Liang Bao ^{The non-symmetric algebraic Riccati equation arising in transporttheory can be rewritten as a vector equation and the minimalpositive solution of the non-symmetric algebraic Riccati equationcan be obtained by solving the vector equation. In this paper,we apply the modified Newton method to solve the vector equation.Some convergence results are presented. Numerical tests showthat the modified Newton method is feasible and effective, andoutperforms the Newton method.}     

Asymptotic behaviour of a finite-volume scheme for the transient drift-diffusion model

Francis Filbet Université de Lyon, Université Lyon 1, CNRS, UMR 5208, Institut Camille Jordan, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne cedex, France Received on 28 June 2006. Revised on 6 December 2006. In this paper, we propose a finite-volume discretization formultidimensional nonlinear drift-diffusion system. Such a systemarises in semiconductors modelling and is composed of two parabolicequations and an elliptic one. We prove that the numerical solutionconverges to a steady state when time goes to infinity. Severalnumerical tests show the efficiency of the method.     

A refined mixed finite-element method for the stationary Navier Stokes equations with mixed boundary conditions

Serge Nicaise ^{This paper is concerned with the mixed formulation of the Navier–Stokesequations with mixed boundary conditions in 2D polygonal domainsand its numerical approximation. We first describe the regularityof any solution. The problem is then approximated by a mixedfinite-element method where the strain tensor and the antisymmetricgradient tensor, quantities of practical importance, are introducedas new unknowns. An existence result for the finite-elementsolution and convergence results are proved near a nonsingularsolution. Quasi-optimal error estimates are finally presented.}     

Additive Schwarz preconditioning for p-version triangular and tetrahedral finite elements

Jens M. Melenk ^{This paper analyses two-level Schwarz methods for matrices arisingfrom the p-version finite-element method on triangular and tetrahedralmeshes. The coarse level consists of the lowest-order finite-elementspace. On the fine level, we investigate several decompositionswith large or small overlap leading to optimal or close to optimalcondition numbers. The analysis is confirmed by numerical experimentsfor a simple model problem and an elasticity problem on a complexgeometry.}     

A safeguarded dual weighted residual method

Andreas Veeser ^{The dual weighted residual (DWR) method yields reliable a posteriorierror bounds for linear output functionals provided that theerror incurred by the numerical approximation of the dual solutionis negligible. In that case, its performance is generally superiorthan that of global ‘energy norm’ error estimatorswhich are ‘unconditionally’ reliable. We presenta simple numerical example for which neglecting the approximationerror leads to severe underestimation of the functional error,thus showing that the DWR method may be unreliable. We proposea remedy that preserves the original performance, namely a DWRmethod safeguarded by additional asymptotically higher ordera posteriori terms. In particular, the enhanced estimator isunconditionally reliable and asymptotically coincides with theoriginal DWR method. These properties are illustrated via theaforementioned example.}     

An a posteriori error estimator for the Lame equation based on equilibrated fluxes

Katharina Witowski ^{We derive a new a posteriori error estimator for the Lamésystem based on H(div)-conforming elements and equilibratedfluxes. It is shown that the estimator gives rise to an upperbound where the constant is one up to higher-order terms. Thelower bound is also established using Argyris elements. Thereliability and efficiency of the proposed estimator are confirmedby some numerical tests.}     

Statistical solution to the capacity problem in direct-sequence code-division multiple-access communication systems

Tohru Kohda Department of Computer Science and Communication Engineering, Kyushu University, Fukuoka 812-8581, Japan Minoru Tabata Department of Mathematical Sciences, Osaka Prefecture University, Osaka 599-8531, Japan Email: eshima{at}med.oita-u.ac.jp Received on September 21, 2005; Accepted on May 2, 2006 Capacity in the direct-sequence code-division multiple-access(DS/CDMA) communication system was considered according to thecode acquisition performance with the conventional serial-searchmethod, because code acquisition needs a difficult operation.Since capacity in DS/CDMA systems is defined by the maximumnumber of users that can simultaneously transmit their signalswith the same carrier frequencies, assuring a larger capacityis very important in DS/CDMA systems with respect to the economyof frequency source. This paper reconsiders the capacity problemthrough a statistical approach to code acquisition. First, aDS/CDMA system model is reviewed. Second, properties of thecounting method for code acquisition are discussed. It is provedthat the method can acquire the target user's code under anynumber of interference users that simultaneously transmit theirsignals, and that the method can guarantee a considerable precisionin code acquisition. Third, an observation time necessary forcode acquisition is given from a statistical discussion, andthe original counting method is modified into a simpler one.It is concluded that the capacity of the DS/CDMA communicationsystem can be set by the ‘bit error-rate-based capacityor signal-to-noise ratio-based capacity’, i.e. ‘post-acquisition-basedcriterion’, rather than the ‘acquisition-based capacity’.     

Preconditioning by inverting the Laplacian: an analysis of the eigenvalues

Wolfgang Hackbusch ^{We study the eigenvalues of the operator generated by usingthe inverse of the Laplacian as a preconditioner for self-adjointsecond-order elliptic partial differential equations with smoothcoefficients. It is well-known that the spectral condition numberof the preconditioned operator can be bounded by , where k is the uniformly positive coefficientof the second-order elliptic equation. The purpose of this paperis to study the spectrum of the preconditioned operator. Wewill show that there is a strong relation between the spectrumof this operator and the range of the coefficient function.In the continuous case, we prove, both for mappings definedon Sobolev spaces and in terms of generalized functions, thatthe spectrum of the preconditioned operator contains the rangeof the coefficient function k. In the discrete case, we indicateby numerical examples that the entire discrete spectrum is approximatelygiven by values of k.}     

Adaptive frame methods for elliptic operator equations: the steepest descent approach

Massimo Fornasier^¶ Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università "La Sapienza" in Roma, Via Antonio Scarpa, 16/B, I-00161 Roma, Italy Rob Stevenson^|| Department of Mathematics, Utrecht University, PO Box 80.010, NL-3508 TA Utrecht, The Netherlands ^{This paper is concerned with the development of adaptive numericalmethods for elliptic operator equations. We are particularlyinterested in discretization schemes based on wavelet frames.We show that by using three basic subroutines an implementable,convergent scheme can be derived, which, moreover, has optimalcomputational complexity. The scheme is based on adaptive steepestdescent iterations. We illustrate our findings by numericalresults for the computation of solutions of the Poisson equationwith limited Sobolev smoothness on intervals in 1D and L-shapeddomains in 2D.}