classroom note geometry and convergence of euler s and halley s methods we investigate the global convergence of euler s and halley s methods by using their geometric interpretation a generalization of these methods is also briefly discussed

an iterative penalty method for the finite element solution of the stationary navier stokes equations abstract the objective of this paper is to analyse an iterative procedure for the finite element solution of the stokes and navier stokes stationary problems for the latter case the usual condition on the viscosity and the data that ensures uniqueness is assumed the method is based on the iterative imposition of the incompressibility condition via penalization theoretical and numerical results show that this constraint can be approximated iteratively within the same iterative loop used to deal with the nonlinear term of the equations two particular iterative schemes are analysed namely those based on the picard and newton raphson algorithms

high resolution finite element analysis of shallow water equations in two dimensions abstract we present in this study the taylor galerkin finite element model to stimulate shallow water equations for bore wave propagation in a domain of two dimensions to provide the necessary precision for the prediction of a sharply varying solution profile the generalized taylor galerkin finite element model is constructed through introduction of four parameters this paper also presents the fundamental theory behind the choice of free parameters one set of parameters is theoretically determined to obtain the high order accurate taylor galerkin finite element model the other set of free parameters is determined using the underlying discrete maximum principle to obtain the low order monotonic taylor galerkin finite element model theoretical study reveals that the higher order scheme exhibits dispersive errors near the discontinuity while lower order scheme dissipates the discontinuity a scheme which has a high resolution shock capturing ability as a built in feature is thus needed in the present study notice that lumping of the mass matrix equations is invoked in the low order scheme to allow simulation of the hydraulic problem with discontinuities we check the prediction accuracy against suitable test problems preferably ones for which exact solutions are available based on numerical results it is concluded that the taylor galerkin flux corrected transport tg fct finite element method can render the technique suitable for solving shallow water equations with sharply varying solution profiles

computation of the gamma digamma and trigamma functions this paper gives an approximation for the gamma function that while different has the same form as one by lanczos siam j numer anal b1 1964 pp 86 96 both approximations correct stirlin

sixth order variants of chebyshev halley methods for solving non linear equations abstract in this paper we present a family of new variants of chebyshev halley methods with sixth order convergence compared with chebyshev halley methods the new methods require one additional evaluation of the function the numerical results presented show that the new methods compete with chebyshev halley methods

some recent trends and developments in finite element analysis for incompressible thermal flows a numerical procedure for solving the time dependent incompressible navier stokes and energy equations is presented the present method is based on a set of finite element equations of the primitive variable formulation and a time splitting procedure which has unique features in its formulation as well as in its evaluation from the viewpoint of efficiency and accuracy the applicability of the proposed formulation was verified by computations and comparison with known results

a universal formulation of two equation models for adaptive computation of turbulent flows abstract this paper describes the application of a recently developed universal adaptive finite element algorithm to the simulation of several turbulent flows the objective of the present work is to show how the controlled accuracy of adaptive methods provides the means to perform careful quantitative comparisons of two equation models the formulation uses the logarithmic form of turbulence variables which naturally leads to a simple algorithm applicable to all two equation turbulence models the new methodology is free of ad hoc stability enhancement measures such as clipping and limiters which may often differ from one model to the other such techniques limit the predictive capability of a turbulence model and cloud the issues of a comparison study the present procedure results in one adaptive solver applicable to all two equation models the approach is demonstrated by comparing three popular turbulence models on a few non trivial compressible and incompressible flows we have chosen the following models the standard k model the k model of speziale and the k model of wilcox results show that accurate solutions can be obtained for all models and that systematic comparison of turbulence models can be made

clustering on the unit hypersphere using von mises fisher distributions several large scale data mining applications such as text categorization and gene expression analysis involve high dimensional data that is also inherently directional in nature often such data is l2 normalized so that it lies on the surface of a unit hypersphere popular models such as mixtures of multi variate gaussians are inadequate for characterizing such data this paper proposes a generative mixture model approach to clustering directional data based on the von mises fisher vmf distribution which arises naturally for data distributed on the unit hypersphere in particular we derive and analyze two variants of the expectation maximization em framework for estimating the mean and concentration parameters of this mixture numerical estimation of the concentration parameters is non trivial in high dimensions since it involves functional inversion of ratios of bessel functions we also formulate two clustering algorithms corresponding to the variants of em that we derive our approach provides a theoretical basis for the use of cosine similarity that has been widely employed by the information retrieval community and obtains the spherical kmeans algorithm kmeans with cosine similarity as a special case of both variants empirical results on clustering of high dimensional text and gene expression data based on a mixture of vmf distributions show that the ability to estimate the concentration parameter for each vmf component which is not present in existing approaches yields superior results especially for difficult clustering tasks in high dimensional spaces

a new finite element formulation for computational fluid dynamics viii the galerkin least squares method for advective diffusive equations abstract galerkin least squares finite element methods are presented for advective diffusive equations galerkin least squares represents a conceptual simplification of supg and is in fact applicable to a wide variety of other problem types a convergence analysis and error estimates are presented

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

local spectral component decomposition for multi channel image denoising we propose a method for local spectral component decomposition based on the line feature of local distribution our aim is to reduce noise on multi channel images by exploiting the linear correlation in the spectral domain of a local region we first calculate a linear feature over the spectral components of an m channel image which we call the spectral line and then using the line we decompose the image into three components a single m channel image and two gray scale images by virtue of the decomposition the noise is concentrated on the two images and thus our algorithm needs to denoise only the two gray scale images regardless of the number of the channels as a result image deterioration due to the imbalance of the spectral component correlation can be avoided the experiment shows that our method improves image quality with less deterioration while preserving vivid contrast our method is especially effective for hyperspectral images the experimental results demonstrate that our proposed method can compete with the other state of the art denoising methods

a petrov galerkin formulation for the incompressible navier stokes equations using equal order interpolation for velocity and pressure a new petrov galerkin method for the incompressible navier stokes equations is presented the use of the so called optimal upwind parameter in multidimensions is justified by a time scale analysis of the relevant physical processes the resulting procedure circumvents the babuska brezzi condition and allows equal order interpolation for velocity and pressure to be used

high order parameter approximation for von mises fisher distributions abstract this paper concerns the issue of the maximum likelihood estimation mle for the concentration parameters of the von mises fisher vmf distributions which are crucial to directional data analysis in particular we study the numerical approximation approach for solving the implicit nonlinear equation arising from building the mle of the concentration parameter of vmf distributions in addition we address the implementation of i s x the modified bessel function of the first kind which is the most time consuming and fundamental ingredient in the proposed approximation scheme of the mle for the main contribution of this paper is two fold the first is to present a two steps halley based method for exploring a high order approximation of the mle for which can significantly contribute to the improvement of estimation accuracy the second is to develop a novel approach for the implementation of i s x which can make the substantial improvement of computation efficiency for computing the mle approximation for the numerical experiments were conducted to compare the proposed schemes with those in the existing works by tanabe et al 1 and sra 2 the experimental results show that given the same amount of computation as their methods the proposed high order scheme can achieve much more accurate approximations while our implementation of i s x is preferable yet desirable for high dimensional applications

a new family of eighth order iterative methods for solving nonlinear equations a family of eighth order iterative methods with four evaluations for the solution of nonlinear equations is presented kung and traub conjectured that an iteration method without memory based on n evaluations could achieve optimal convergence order 2 n 1 the new family of eighth order methods agrees with the conjecture of kung traub for the case n 4 therefore this family of methods has efficiency index equal to 1 682 numerical comparisons are made with several other existing methods to show the performance of the presented methods

finite element solution of the unsteady navier stokes equations by a fractional step method abstract a finite element method is presented for the numerical simulation of time dependent incompressible viscous flows the method is based on a fractional step approach to the time integration of the navier stokes equations in which only the incompressibility condition is treated implicitly this leads to a computational scheme of extremely simple algorithmic structure that is particularly attractive for cost effective solutions of large scale problems numerical results indicate the versatility and effectiveness of the proposed method

finite element solution of the navier stokes equations for incompressible flow using a segregated algorithm abstract a segregated finite element algorithm for the solution of the supg formulation of the incompressible steady state navier stokes equations is investigated in this paper the method features equal order interpolation for all the flow variables the simplest algorithm is employed which results in symmetric coefficient matrices for the momentum equations the same iterative linear equation solver can therefore be employed for the solution of the momentum and pressure equations the effect of applying the supg weighting functions only to the convective terms is compared with the effect of applying the functions to all the terms in the momentum equations the effect of using an iterative linear equation solver as opposed to a direct linear equation solver is also discussed

incompressible navier stokes computations with supg and gls formulations a comparison study abstract a comparison between different stabilized formulations and the standard galerkin formulation with and without penalty function approach is presented for the lid driven cavity problem computations are done for re 400 1000 5000 and 10 000 employing different grid resolutions results obtained are presented as velocity profiles pressure profiles velocity vectors and pressure contours and are compared with the benchmark solution the galerkin least squares formulation using biquadratic serendipity elements exhibits better stability properties compared to the bilinear interpolation case the supg formulation with constant pressure elements seems less accurate than the gls formulations

computation of unsteady viscous flows around moving bodies using the k e turbulence model on unstructured dynamic grids abstract we consider the numerical solution on unstructured dynamic meshes of the averaged navier stokes equations equipped with the k e turbulence model and a wall function we discuss discretization issues pertaining to conservation laws moving grids and numerical dissipation we also present a robust spring analogy method for constructing dynamic meshes we validate our implementation of this two equation turbulence model and justify its usage for a class of vortex shedding problems by correlating our computational results with experimental data obtained for a flow past a square cylinder we also apply our solution methodology to the two dimensional aerodynamic stability analysis of the tacoma narrows bridge and report numerical results that are in good agreement with observed data

incompressible turbulent flow simulation using a galerkin least squares formulation and a low reynolds number k ϵ model a galerkin least squares formulation using equal order serendipity velocity pressure elements is used in conjunction with a low reynolds number k model to solve a turbulent poiseuille type flow at re 5000 predictions are compared to available experimental results the pressure solution obtained using the gls formulation is in close agreement with the measurements this work illustrates that the gls formulation can be extended successfully to predict the pressure solution for the case of incompressible turbulent flows

eighth order methods with high efficiency index for solving nonlinear equations in this paper we derive a new family of eighth order methods for solving simple roots of nonlinear equations by using weight function methods per iteration these methods require three evaluations of the function and one evaluation of its first derivative which implies that the efficiency indexes are 1 682 numerical comparisons are made to show the performance of the derived methods as shown in the illustration examples

supg stabilized finite element resolution of the navier stokes equations applications to water treatment engineering in this paper an analysis of the viscous incompressible flow has been carried out from the very definition of the governing equations up to the resolution of some practical problems passing through the comprehensive study of the stabilized finite element techniques used in their resolution as a consequence of this analysis a code based upon a realistic interpretation of the forces has been written which allows for the modelling of the open channel flow with optimum results in the resolution of some benchmark and real flow problems related with the wastewater industry

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100