discontinuous galerkin approximations of the stokes and navier stokes equations numerical schemes to compute approximate solutions of the evolutionary stokes and navier stokes equations are studied the schemes are discontinuous in time and conforming in space and of arbitrarily high order fully discrete error estimates are derived and dependence of the viscosity constant is carefully tracked it is shown that the errors are bounded by projection errors of the exact solution which exhibit optimal rates when the solutions are smooth

convergence analysis and error estimates for a parallel algorithm for solving the navier stokes equations this paper is concerned with the analysis of the convergence and the derivation of error estimates for a parallel algorithm which is used to solve the incompressible navier stokes equations as usual the main idea is to split the main differential operator this allows to consider independently the two main difficulties namely nonlinearity and incompressibility the results justify the observed accuracy of related numerical results

comparing two proposed measures of cochlear mechanical filter bandwidth based on stimulus frequency otoacoustic emissions it has been hypothesized that the sharpness of the cochlear mechanical filter is related to two measures based on stimulus frequency otoacoustic emissions sfoaes the first is the group delay of the sfoae the second is the bandwidth of the sfoae two tone suppression isoinput tuning characteristic a corollary of this is that natural variability in cochlear mechanical bandwidth within a population would lead to a positive correlation between these two sfoae based measures of tuning within that population to test this prediction sfoae group delay and sfoae two tone suppression isoinput tuning characteristics were measured in a sample of 16 audiometrically normal subjects contrary to the prediction no statistically significant correlation was found between the two bandwidth measures cochlear model simulations were used to aid the interpretation of this result these suggested that a positive correlation between the two measures is expected but that it may well be too weak to detect with the given sample size due to the influence on the sfoae measures of random inhomogeneities in basilar membrane impedance

smoothing transformation and spline collocation for nonlinear fractional initial and boundary value problems abstract we construct and justify a class of high order methods for the numerical solution of initial and boundary value problems for nonlinear fractional differential equations of the form d y t f t y t with caputo type fractional derivatives d y of order 0 using an integral equation reformulation of the underlying problem we first regularize the solution by a suitable smoothing transformation after that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established to illustrate the reliability of the proposed algorithms two numerical examples are given

long term dissipativity of time stepping algorithms for an abstract evolution equation with applications to the incompressible mhd and navier stokes equations abstract this paper examines the long term dissipativity and unconditional non linear stability of time integration algorithms for the incompressible mhd equations the results apply as a particular case to the uncoupled incompressible navier stokes equations the analysis is performed in the setting defined by an abstract evolution equation possessing the same long term dissipative structure as the above physical systems a general class of time stepping schemes is analyzed in this context showing the non linear unconditional stability and long term dissipativity of particular members of this family of algorithms in particular algorithms showing these strong properties which additionally involve the solution of a linear problem at each time step are identified a particular linear algorithm is shown to exhibit optimal long term dissipative properties and to possess an algorithmic global attractor these abstract results apply rigorously to galerkin mixed finite element discretizations of the problem to this end it is shown that such galerkin type projection inherits the dissipative properties of the continuum problem computational aspects involved in the numerical implementation of these methods are addressed and illustrated in representative numerical simulations

suitable initial conditions the aim of this note is to draw attention on a computational problem related to the initial simulation of very large time dependant systems the underlying theoretical problem has been the object of many relevant works in theoretical mathematical and in the applied literature as well but it seems that its computational implications are not familiar to many we hope that this note will help

the postprocessed mixed finite element method for the navier stokes equations refined error bounds a postprocessing technique for mixed finite element methods for the incompressible navier stokes equations is analyzed the postprocess which amounts to solving a linear stokes problem is shown to increase the order of convergence of the method to which it is applied by one unit times the logarithm of the mesh diameter in proving the error bounds some superconvergence results are also obtained contrary to previous analysis of the postprocessing technique in the present paper we take into account the loss of regularity suffered by the solutions of the navier stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data as in h g heywood and r rannacher siam j numer anal 25 1988 pp 489 512 where the same hypothesis is assumed no better than fifth order convergence is achieved

mathematical programming methods in the numerical solution of volterra integral and integro differential equations with weakly singular kernel abstract nonlinear volterra integral and integro differential equations with weakly singular kernel are considered and solved numerically using nonlinear mathematical programming methods based on minimax approximations in both cases polynomial and multiquadric approximation are used

flow rate boundary problems for an incompressible fluid in deformable domains formulations and solution methods in this paper we consider the numerical solution of the interaction of an incompressible fluid and an elastic structure in a truncated computational domain as well known in this case there is the problem of prescribing realistic boundary data on the artificial sections when only partial data are available this problem has been investigated extensively for the rigid case in this work we start considering the compliant case by focusing on the flow rate conditions for the fluid we propose three formulations of this problem different algorithms for its numerical solution and carry out several 2d numerical simulations with the aim of comparing the performances of the different algorithms

a rational basis for second kind abel integral equations abstract a rational basis set is constructed using 1 1 rational interpolation the basis is used in a product integration method for solving second kind abel integral equations criteria are established for determining the amount of bias to be used in the basis set when the integral equation has a nonsmooth solution several test problems are solved to illustrate the performance of the approximation method

a posteriori error estimations for mixed finite element approximations to the navier stokes equations a posteriori estimates for mixed finite element discretizations of the navier stokes equations are derived we show that the task of estimating the error in the evolutionary navier stokes equations can be reduced to the estimation of the error in a steady stokes problem as a consequence any available procedure to estimate the error in a stokes problem can be used to estimate the error in the nonlinear evolutionary problem a practical procedure to estimate the error based on the so called postprocessed approximation is also considered both the semidiscrete in space and the fully discrete cases are analyzed some numerical experiments are provided

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

a nonpolynomial collocation method for fractional terminal value problems in this paper we propose a nonpolynomial collocation method for solving a class of terminal or boundary value problems for differential equations of fractional order 0 1 the approach used is based on the equivalence between a problem of this type and a fredholm integral equation of a particular form taking into account the asymptotic behaviour of the solution of this problem we propose a nonpolynomial collocation method on a uniform mesh we study the order of convergence of the proposed algorithm and a result on optimal order of convergence is obtained in order to illustrate the theoretical results and the performance of the method we present several numerical examples

the numerical solution of weakly singular volterra integral equations by collocation on graded meshes since the solution of a second kind volterra integral equation with weakly singular kernel has in general unbounded derivatives at the left endpoint of the interval of integration its numerical solution by polynomial spline collocation on uniform meshes will lead to poor convergence rates in this paper we investigate the convergence rates with respect to graded meshes and we discuss the problem of how to select the quadrature formulas to obtain the fully discretized collocation equation

the stabilized extrapolated trapezoidal finite element method for the navier stokes equations abstract we consider a fully discrete stabilized finite element method for the navier stokes equations which is unconditionally stable and has second order temporal accuracy of o k 2 hk spatial error the method involves a simple artificial viscosity stabilization of the linear system for the approximation of the new time level connected to anti diffusion of its effects at the old time level lowering the cell reynolds number of the oseen problem to o 1 the method requires only the solution of one linear system arising from an oseen problem per time step the effectiveness of the method is illustrated in several numerical experiments

calculation of incompressible viscous flows by an unconditionally stable projection fem this paper investigates the numerical performance of a finite element implementation of a new incremental fractional step method to compute steady and unsteady incompressible viscous flows under general boundary conditions and using unstructured meshes a variational framework is adopted which accommodates two different spaces for representing and approximating the velocity fields calculated respectively in the viscous and inviscid phases of the method but which leads to a very simple numerical scheme in terms of only one discrete velocity field an unconditionally stable semi implicit approximation of the nonlinear term is used to eliminate any time step restriction as far as the numerical stability is concerned numerical results for five test problems in two dimensions are reported to illustrate the flexibility of the proposed method

first order decoupled finite element method of the three dimensional primitive equations of the ocean this paper is concerned with a first order fully discrete decoupled method for solving the three dimensional 3d primitive equations of the ocean with the dirichlet boundary conditions on the side where a decoupled semi implicit scheme is used for the time discretization and the p 1 p 1 p 1 p 1 p 1 finite element for velocity pressure and density is used for the spatial discretization of these equations the h 1 l 2 h 1 optimal error estimates for the numerical solution u h n p h n theta h n and the l 2 optimal error estimate for u n h theta h n are established under the restriction of 0 h le beta 1 and 0 tau le beta 2 for some positive constants beta 1 and beta 2 moreover numerical investigations are provided to show that the first order decoupled method is of almost unconditional convergence with accuracy mathcal o h tau in the h 1 norm and mathcal o h 2 tau in the l 2 norm for solving the 3d primitive equations of the ocean numerical results are gi

a sequential regularization method for time dependent incompressible navier stokes equations the objective of the paper is to present a method called the sequential regularization method srm for the nonstationary incompressible navier stokes equations from the viewpoint of regularization of differential algebraic equations daes and to provide a way to apply a dae method to partial differential algebraic equations pdaes the srm is a functional iterative procedure it is proved that its convergence rate is o epsilon m where m is the number of the srm iterations and epsilon is the regularization parameter the discretization and implementation issues of the method are considered in particular a simple explicit difference scheme is analyzed and its stability is proved under the usual step size condition of explicit schemes it appears that the srm formulation is new in the navier stokes context unlike other regularizations or pseudocompressibility methods in the navier stokes context the regularization parameter epsilon in the srm need not be very small and the regularized problem in the sequence may be essentially nonstiff in time direction for any epsilon hence the stability condition is independent of epsilon even for explicit time discretization numerical experiments are given to verify our theoretical results

on the regularity of solutions to integral equations with nonsmooth kernels on a union of open intervals the behaviour of a solution to a fredholm integral equation of the second kind on a union of open intervals is examined the kernel of the corresponding integral operator may have diagonal singularities information about them is given through certain estimates the weighted spaces of smooth functions with boundary singularities containing the solution of the integral equation are described

stability and convergence of the crank nicolson adams bashforth scheme for the time dependent navier stokes equations in this paper we study the stability and convergence of the crank nicolson adams bashforth scheme for the two dimensional nonstationary navier stokes equations a finite element method is applied for the spatial approximation of the velocity and pressure the time discretization is based on the crank nicolson scheme for the linear term and the explicit adams bashforth scheme for the nonlinear term moreover we present optimal error estimates and prove that the scheme is almost unconditionally stable and convergent i e stable and convergent when the time step is less than or equal to a constant

analysis of some low order finite element schemes for the navier stokes equations abstract various low order finite element schemes for the navier stokes equations are analysed including the commonly used bilinear velocity piecewise constant pressure element and the algorithms based on the finite element analogue of the mac stencil discussed in a companion paper by performing a truncation error analysis using a variety of different assumptions about how the nodal parameters represent the data it is shown how a more complete picture of the behaviour of the algorithms can be obtained in particular differences in performance on coarse grids using algorithms which have the same asymptotic convergence rate can be explained

a perspective on the numerical treatment of volterra equations abstract we discuss the properties and numerical treatment of various types of volterra and abel volterra integral and integro differential equations

application of finite element method in aeroelasticity the subject of this paper is the numerical simulation of the interaction of two dimensional incompressible viscous flow and a vibrating airfoil which can rotate around the elastic axis and oscillate in the vertical direction the numerical simulation consists of the finite element approximation of the navier stokes equations coupled with the system of ordinary differential equations describing the airfoil motion the arbitrary lagrangian eulerian ale formulation of the navier stokes equations stabilization the finite element discretization and coupling of both models is discussed moreover the reynolds averaged navier stokes rans system of equations together with the spallart almaras turbulence model is also discussed the computational results of aeroelastic calculations are presented and compared with the nastran code solutions

new pod error expressions error bounds and asymptotic results for reduced order models of parabolic pdes the derivations of existing error bounds for reduced order models of time varying partial differential equations pdes constructed using proper orthogonal decomposition pod have relied on bounding the error between the pod data and various pod projections of that data furthermore the asymptotic behavior of the model reduction error bounds depends on the asymptotic behavior of the pod data approximation error bounds we consider time varying data taking values in two different hilbert spaces h and v with v subset h and prove exact expressions for the pod data approximation errors considering four different pod projections and the two different hilbert space error norms furthermore the exact error expressions can be computed using only the pod eigenvalues and modes and we prove the errors converge to zero as the number of pod modes increases we consider the pod error estimation approaches of kunisch and volkwein siam j numer anal 40 2002 pp 492 515 and chapelle gariah an

identification of otoacoustic emissions components by means of adaptive approximations clicks and a set of tone bursts covering the same frequency band were applied as a stimuli evoking otoacoustic emissions oae recorded otoacoustic emissions were decomposed into the basic waveforms by means of high resolution adaptive time frequency approximation method based on the matching pursuit algorithm the method allows for description of the signal components in terms of frequencies time occurrences time spans and energy the analysis of oae s energy density distributions in time frequency space revealed that click responses can be considered as linear superpositions of responses to tone bursts the frequency latency relationship was studied and compared with earlier works the method made possible the exhaustive description of the resonant modes specific for given subject ear they were characterized not only by the close frequencies appearing for different tones but they usually had similar latencies and time spans short time and long time resonant modes were identified the second ones

spectral methods for weakly singular volterra integral equations with smooth solutions we propose and analyze a spectral jacobi collocation approximation for the linear volterra integral equations vies of the second kind with weakly singular kernels in this work we consider the case when the underlying solutions of the vies are sufficiently smooth in this case we provide a rigorous error analysis for the proposed method which shows that the numerical errors decay exponentially in the infinity norm and weighted sobolev space norms numerical results are presented to confirm the theoretical prediction of the exponential rate of convergence

on error estimates of the projection methods for the navier stokes equations second order schemes we present in this paper a rigorous error analysis of several projection schemes for the approximation of the unsteady incompressible navier stokes equations the error analysis is accomplished by interpreting the respective projection schemes as second order time discretizations of a perturbed system which approximates the navier stokes equations numerical results in agreement with the error analysis are also presented

petrov galerkin methods for linear volterra integro differential equations in this paper we study a class of petrov galerkin solutions that have global optimal convergence rates for linear volterra integro differential equations these solutions also possess certain local and global superconvergence asymptotic expansions of the errors in these solutions are established which can be used to form higher order approximations by richardson extrapolation and defect corrections several postprocessing techniques are introduced to enhance these solutions as by products these higher order numerical approximations can be used to generate a posteriori error estimators representative numerical results are also provided

modified spline collocation for linear fractional differential equations we propose and analyze a class of high order methods for the numerical solution of initial value problems for linear multi term fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the initial value problem we first regularize the solution by a suitable smoothing transformation after that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established theoretical results are verified by some numerical examples

a staggered space time discontinuous galerkin method for the three dimensional incompressible navier stokes equations on unstructured tetrahedral meshes in this paper we propose a novel arbitrary high order accurate semi implicit space time discontinuous galerkin method for the solution of the three dimensional incompressible navier stokes equations on staggered unstructured curved tetrahedral meshes as is typical for space time dg schemes the discrete solution is represented in terms of space time basis functions this allows to achieve very high order of accuracy also in time which is not easy to obtain for the incompressible navier stokes equations similarly to staggered finite difference schemes in our approach the discrete pressure is defined on the primary tetrahedral grid while the discrete velocity is defined on a face based staggered dual grid while staggered meshes are state of the art in classical finite difference schemes for the incompressible navier stokes equations their use in high order dg schemes is still quite rare a very simple and efficient picard iteration is used in order to derive a space time pressure correction algorithm that achieves also high order of accuracy in time and that avoids the direct solution of global nonlinear systems formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse five point block system for the scalar pressure which is conveniently solved with a matrix free gmres algorithm from numerical experiments we find that the linear system seems to be reasonably well conditioned since all simulations shown in this paper could be run without the use of any preconditioner even up to very high polynomial degrees for a piecewise constant polynomial approximation in time and if pressure boundary conditions are specified at least in one point the resulting system is in addition symmetric and positive definite this allows us to use even faster iterative solvers like the conjugate gradient method the flexibility and accuracy of high order space time dg methods on curved unstructured meshes allows to discretize even complex physical domains with very coarse grids in both space and time the proposed method is verified for approximation polynomials of degree up to four in space and time by solving a series of typical 3d test problems and by comparing the obtained numerical results with available exact analytical solutions or with other numerical or experimental reference data to the knowledge of the authors this is the first time that a space time discontinuous galerkin finite element method is presented for the three dimensional incompressible navier stokes equations on staggered unstructured tetrahedral grids high order staggered space time dg scheme for the 3d incompressible navier stokes equations primal grid based on curved tetrahedra dual grid based on non standard five point hexahedra new discretization of the viscous terms on the main grid using lifting operators on the dual grid very simple space time pressure correction algorithm providing also high order in time all linear systems have maximum sparsity and use the smallest possible optimal five point stencil

the piecewise polynomial collocation method for nonlinear weakly singular volterra equations second kind volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration using an adaptation of the analysis originally developed for nonlinear weakly singular fredholm integral equations we present a complete discussion of the optimal global and local order of convergence of piecewise polynomial collocation methods on graded grids for nonlinear volterra integral equations with algebraic or logarithmic singularities in their kernels

a numerical approach to the solution of abel integral equations of the second kind with nonsmooth solution abstract in this paper we study the feasibility of numerical solutions analogous to those methods introduced by bownds 1979 1982 and bownds and wood 1976 1979 for volterra integral equations of the second kind with weakly singular kernels

asymptotic behavior and time discretization analysis for the non stationary navier stokes problem the asymptotic behavior and the euler time discretization analysis are presented for the two dimensional non stationary navier stokes problem if the data and f t satisfy a uniqueness condition corresponding to the stationary navier stokes problem we then obtain the convergence of the non stationary navier stokes problem to the stationary navier stokes problem and the uniform boundedness stability and error estimates of the euler time discretization for the non stationary navier stokes problem

product integration methods for solving a system of nonlinear volterra integral equations in this paper the technique of subtracting out singularities is used to derive explicit and implicit product euler schemes with order one convergence and a product trapezoidal scheme with order two convergence for a system of volterra integral equations with a weakly singular kernel the convergence proofs of the numerical schemes are presented these are nonstandard since the nonlinear function involved in the integral equation system does not satisfy a global lipschitz condition

mixed finite element formulation and error estimates based on proper orthogonal decomposition for the nonstationary navier stokes equations in this paper proper orthogonal decomposition pod is used for model reduction of mixed finite element mfe for the nonstationary navier stokes equations and error estimates between a reference solution and the pod solution of reduced mfe formulation are derived the basic idea of this reduction technique is that ensembles of data are first compiled from transient solutions computed equation system derived with the usual mfe method for the nonstationary navier stokes equations or from physics system trajectories by drawing samples of experiments and interpolation or data assimilation and then the basis functions of the usual mfe method are substituted with the pod basis functions reconstructed by the elements of the ensemble to derive the pod reduced mfe formulation for the nonstationary navier stokes equations it is shown by considering numerical simulation results obtained for the illustrating example of cavity flows that the error between pod solution of reduced mfe formulation and the reference solution is consistent with theoretical results moreover it is also shown that this result validates the feasibility and efficiency of the pod method

stability and convergence of semi implicit time stepping algorithm for stationary incompressible magnetohydrodynamics abstract for the stationary incompressible magnetohydrodynamics mhd equations we provide a new uniqueness assumption a0 and show the exponential stability of the solution then the semi implicit time stepping algorithm is used to solve the stationary mhd equations the algorithm is proved to be unconditionally stable the discrete velocity and magnetic field are bounded in l 0 l for any space and time step sizes the error estimates for the algorithm are deduced under the uniqueness conditions finally numerical experiments are performed to testify our theoretical analysis

a hybrid collocation method for volterra integral equations with weakly singular kernels the commonly used graded piecewise polynomial collocation method for weakly singular volterra integral equations may cause serious round off error problems due to its use of extremely nonuniform partitions and the sensitivity of such time dependent equations to round off errors the singularity preserving nonpolynomial collocation method is known to have only local convergence to overcome the shortcoming of these well known methods we introduce a hybrid collocation method for solving volterra integral equations of the second kind with weakly singular kernels in this hybrid method we combine a singularity preserving nonpolynomial collocation method used near the singular point of the derivative of the solution and a graded piecewise polynomial collocation method used for the rest of the domain we prove the optimal order of global convergence for this method the convergence analysis of this method is based on a singularity expansion of the exact solution of the equations we prove that the solutions of such equations can be decomposed into two parts with one part being a linear combination of some known singular functions which reflect the singularity of the solutions and the other part being a smooth function a numerical example is presented to demonstrate the effectiveness of the proposed method and to compare it to the graded collocation method

high order direct stokes solvers with or without temporal splitting numerical investigations of their comparative properties a recently proposed direct stokes solver which decouples the velocity and pressure operators without calling for a temporal scheme is numerically analyzed in comparison first with the splitting scheme proposed by g karniadakis m israeli and s orszag in j comput phys 97 1991 pp 414 443 and with the unique grid bbb p n bbb p n 2 uzawa approach for the space accuracy and computational costs the chebyshev collocation approximation is used to analyze the spectra of the continuous temporal evolution operators and their discrete time versions from the first to the fourth order an explicit boundary condition is also involved in the proposed stokes solver and it is numerically shown that the trace of bf delta bf u on the boundary must be evaluated through its bf nabla times bf nabla times contribution only otherwise the ellipticity is lost before proceeding to the time discretization the explicit evaluation of the rotational boundary term does not prevent the first and second order in time schemes from being unconditionally stable while the schemes built at the next two higher orders are limited by the usual explicit cal o n 4 stability criterion on the time step the proposed bf u p decoupling gives this limitation a less restrictive coefficient and supplies the expected temporal orders on the whole explored range of time step sizes the effective accuracy obtained for the navier stokes 2d solutions has been measured for the re 1000 lid driven cavity problem and has been found equivalent to what is supplied by the much more expensive uzawa decoupling method

volterra integral equations and a new gronwall inequality part ii the nonlinear case we consider nonlinear singular volterra integral equations of the second kind we generalise the transformation method introduced in part i of this paper 6 to cope with both the nonlinearity and slightly more general singular kernels we also consider a particular class of nonlinear equation for which the solution behaviour is known using this a priori knowledge we propose a modification of the transformation technique which results in a numerical method with good asymptotic stability properties applying the general theory of part i of this paper we prove convergence of this scheme

polynomial spline collocation methods for the nonlinear basset equation abstract we discuss the application of a class of spline collocation methods to a first order volterra integro differential equation arising in fluid dynamics since this equation possesses a weakly singular kernel its solutions even for analytic data are not smooth on the entire interval of integration it is shown that while the use of uniform grids leads to poor convergence rates high order convergence is restored on suitably graded grids

error estimates for the spectral approximation of the nonstationary stokes problem this paper analyzes a discretization of the nonstationary stokes problem with periodic nonperiodic boundary conditions fourier legendre spectral approximations are considered with respect to the space variables while the time discretization is based on the second order crank nicolson scheme optimal convergence estimates are proved for the velocity field

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

superconvergence of numerical solutions to volterra integral equations with singularities in this paper we discuss the beta polynomial discrete collocation method based on practical meshes for volterra integral equations with weakly singular kernels it will be shown that superconvergence properties may be obtained by using appropriate collocation parameters and by a simple cheap postprocessing of the original discrete collocation solution

error analysis of fully discrete velocity correction methods for incompressible flows a fully discrete version of the velocity correction method proposed by guermond and shen 2003 for the time dependent navier stokes equations is introduced and analyzed it is shown that when accounting for space discretization additional consistency terms which vanish when space is not discretized have to be added to establish stability and optimal convergence error estimates are derived for both the standard version and the rotational version of the method these error estimates are consistent with those by guermond and shen 2003 as far as time discretiztion is concerned and are optimal in space for finite elements satisfying the inf sup condition

a reliable technique for solving the weakly singular second kind volterra type integral equations abstract in this paper the adomian decomposition method and the phenomenon of the self canceling noise terms are used for solving the weakly singular volterra type linear and nonlinear integral equations the solution obtained is in the form of a convergent power series with elegantly computable terms comparing this scheme with many collocation type methods that use the nonpolynomial basis shows that the present approach is effective and powerful many test modeling problems from mathematical physics linear and nonlinear are discussed to illustrate the effectiveness and the performance of the decomposition method

analysis of numerical errors in large eddy simulation we consider the question of numerical errors in large eddy simulation it is often claimed that straightforward discretization and solution using centered methods of models for large eddy motion can simulate the motion of turbulent flows with complexity independent of the reynolds number and dependence only on the resolution delta of the eddies sought this report considers this question analytically is it possible to prove error estimates for discretizations of actually used large eddy models whose error constants depend only on delta but not re we consider the most common simplest and most mathematically tractable model and the most mathematically clear discretization in two cases we prove such an error estimate and try to explain why our technique of proof fails in the most general case our analysis aims to assume as little time regularity on the true solution as possible

semi discrete stabilized finite element methods for navier stokes equations with nonlinear slip boundary conditions based on regularization procedure based on the pressure projection stabilized methods the semi discrete finite element approximation to the time dependent navier stokes equations with nonlinear slip boundary conditions is considered in this paper because this class of boundary condition includes the subdifferential property then the variational formulation is the navier stokes type variational inequality problem using the regularization procedure we obtain a regularized problem and give the error estimate between the solutions of the variational inequality problem and the regularized problem with respect to the regularized parameter varepsilon which means that the solution of the regularized problem converges to the solution of the navier stokes type variational inequality problem as the parameter varepsilon longrightarrow 0 moreover some regularized estimates about the solution of the regularized problem are also derived under some assumptions about the physical data the pressure projection stabilized finite element methods are used to the regularized problem and some optimal error estimates of the finite element approximation solutions are derived

on convergent schemes for diffuse interface models for two phase flow of incompressible fluids with general mass densities we are concerned with convergence results for fully discrete finite element schemes suggested in grun and klingbeil j comput phys in press doi 10 1016 j jcp 2013 10 028 they were developed for the diffuse interface model in h abels h garcke and g grun math models methods appl sci 22 2012 1150013 which describes two phase flow of immiscible incompressible viscous fluids we formulate general conditions on discretization spaces and projection operators which allow us to prove compactness of discrete solutions with respect to both time and space and which hence permit us to establish convergence of the scheme to a generalized solution we identify a simple quantitative and physical criterion to decide whether this generalized solution is in fact a weak solution in this case our analysis provides another pathway to establish existence of weak solutions to the aforementioned model in two and in three space dimensions our argument is particularly based on higher regularity results

stability and convergence of second order time discrete projection method for the linearized oldroyd model abstract in this paper we consider the second order time discrete projection method for the linearized oldroyd model based on the time iterative discrete scheme by the projection method the original problem is decoupled into two linear subproblems and each subproblem can be solved easily unconditional stability and the corresponding convergence results of the numerical solutions are derived our main results are that the convergence orders in time for the velocity in l 2 norm is second order and for the pressure in h 1 norm is first order finally some numerical examples are provided to verify the performances of the developed numerical method

product integration methods for second kind abel integral equations the construction and convergence of high order product integration methods for the second kind abel equation are discussed and the results of de hoog and weiss are generalised backward difference methods are introduced and numerical results are presented which verify the theoretical rates of convergence

human cochlear tuning estimates from stimulus frequency otoacoustic emissions two objective measures of human cochlear tuning using stimulus frequency otoacoustic emissions sfoae have been proposed one measure used sfoae phase gradient delay and the other two tone suppression 2ts tuning curves here it is hypothesized that the two measures lead to different frequency functions in the same listener two experiments were conducted in ten young adult normal hearing listeners in three frequency bands 1 2 khz 3 4 khz and 5 6 khz experiment 1 recorded sfoae latency as a function of stimulus frequency and experiment 2 recorded 2ts iso input tuning curves in both cases the output was converted into a sharpness of tuning factor based on the equivalent rectangular bandwidth in both experiments sharpness of tuning curves were shown to be frequency dependent yielding sharper relative tuning with increasing frequency only a weak frequency dependence of the sharpness of tuning curves was observed for experiment 2 consistent with objective and behavioural estimates from the literature most importantly the absolute difference between the two tuning estimates was very large and statistically significant it is argued that the 2ts estimates of cochlear tuning likely represents the underlying properties of the suppression mechanism and not necessarily cochlear tuning thus the phase gradient delay estimate is the most likely one to reflect cochlear tuning

error estimates of finite element methods for nonstationary thermal convection problems with temperature dependent coefficients general error estimates are proved for a class of finite element schemes for nonstationary thermal convection problems with temperature dependent coefficients these variable coefficients turn the diffusion and the buoyancy terms to be nonlinear which increases the nonlinearity of the problems an argument based on the energy method leads to optimal error estimates for the velocity and the temperature without any stability conditions error estimates are also provided for schemes modified by approximate coefficients which are used conveniently in practical computations

collocation and iterated collocation methods for a class of weakly singular volterra integral equations we discuss the convergence properties of spline collocation and iterated collocation methods for a weakly singular volterra integral equation associated with certain heat conduction problems this work completes the previous studies of numerical methods for this type of equations with noncompact kernel in particular a global convergence result is obtained and it is shown that discrete superconvergence can be achieved with the iterated collocation if the exact solution belongs to some appropriate spaces some numerical examples illustrate the theoretical results

finite element discretization of the navier stokes equations with a free capillary surface the instationary navier stokes equations with a free capillary boundary are considered in 2 and 3 space dimensions a stable finite element discretization is presented the key idea is the treatment of the curvature terms by a variational formulation in the context of a discontinuous in time space time element discretization stability in weak energy norms can be proved numerical examples in 2 and 3 space dimensions are given

on an efficient second order backward difference newton scheme for mhd system abstract for the 2d 3d time dependent magnetohydrodynamics mhd system we propose a new second order backward difference formula newton scheme sbdfn which is a combination of second order backward difference approximation of the time derivative terms and newton treatment of the nonlinear terms meanwhile the finite element method is applied as spatial discretization firstly the optimal convergence of spatially semi discrete form is deduced secondly the stability and well posedness of sbdfn scheme are provided under c where c is independent of h based on these results optimal error estimates of the scheme in time are proved by negative norm technique finally numerical experiments are carried out to validate the theoretical analysis

smoothing transformation and spline collocation for linear fractional boundary value problems we construct and justify a high order method for the numerical solution of multi point boundary value problems for linear multi term fractional differential equations involving caputo type fractional derivatives using an integral equation reformulation of the boundary value problem we first regularize the solution by a suitable smoothing transformation after that we solve the transformed equation by a piecewise polynomial collocation method on a mildly graded or uniform grid optimal global convergence estimates are derived and a superconvergence result for a special choice of collocation parameters is established to illustrate the reliability of the proposed method some numerical results are given

piecewise polynomial collocation methods for linear volterra integro differential equations with weakly singular kernels in the first part of this paper we study the regularity properties of solutions of linear volterra integro differential equations with weakly singular or other nonsmooth kernels we then use these results in the analysis of two piecewise polynomial collocation methods for solving such equations numerically the main purpose of the paper is the derivation of optimal global convergence estimates and the analysis of the attainable order of local superconvergence at the collocation points

approximation methods for second kind weakly singular volterra integral equations abstract projection methods are applied to obtain the convergence rates for volterra integral equations with weakly singular kernels we consider galerkin and multi galerkin methods and their iterated versions to solve volterra integral equations with weakly singular kernels in the space of piecewise polynomials subspaces based on graded mesh we will show that the iterated multi galerkin method improves over iterated galerkin method in fact we show that iterated multi galerkin solution converges with the convergence rates o n 3 m and o n 3 m log n 2 for algebraic and logarithmic type kernels respectively we prove that iterated galerkin method for algebraic kernel converges with the convergence rate o n 2 m and for logarithmic type kernel converges with the convergence rate o n 2 m log n where n denotes the number of partition points and m is the highest order of the polynomials employed in the approximations theoretical results are justified by the numerical results

decoupled schemes for unsteady mhd equations ii finite element spatial discretization and numerical implementation in this article a decoupled fully discrete scheme for solving 2d magnetohydrodynamics mhd equations is proposed the decoupled scheme is used for time discretization and the finite element method is used for spatial discretization firstly the almost unconditional stability t c of this scheme is established then optimal l2 and h1 error estimates of numerical solution are provided finally a numerical example is presented to confirm our theoretical results and show the high efficiency of the decoupled scheme

a stabilized nonconfirming finite element method based on multiscale enrichment for the stationary navier stokes equations abstract in the paper we proposed a stabilized nonconforming finite element method for the stationary incompressible navier stokes equations which is a petrov galerkin approach based on the enrichment of the standard polynomial space for the velocity component with multiscale functions and we studied the stability of the method for p 1 p 0 triangular element q 1 p 0 quadrilateral element and obtained the optimal error estimates of the stabilized nonconforming finite element method for the stationary navier stokes equations

stokes coupling method for the exterior flow part iv stabilized finite element approximation based on the stokes coupling method for solving the two dimensional exterior unsteady navier stokes equations the stabilized finite element approximation combining the boundary element with the stabilized finite element for solving the reduced stokes coupling equations is presented finally the optimal error estimates for the finite element solution are derived

a finite element filtered eddy viscosity method for the navier stokes equations with large reynolds number fluid turbulence is commonly modeled by the navier stokes equations with a large reynolds number the simulation of turbulence model is known to be very difficult we study artificial spectral viscosity models that render the simulation of turbulence tractable the models introduce several parameters we show that the models have solutions that converge in certain parameter limits to solutions of the navier stokes equations we also show using the mathematical analysis how effective choice for the parameter can be made the direct computational simulation of turbulence flow is a formidable task due to the disparate scales that have to be resolved turbulence modeling attempts to mitigate this situation by accounting for the effects of small scale behavior on that at large scales without explicitly esolving the small scales one such approach is to add viscosity to the problem the smagorin sky and ladyzhenskaya models and other eddy viscosity models are examples of this approach unfortunately this approach usually results in over dampening at the large scales i e large scale structures are unphysically smeared out to mitigate this fault of eddy viscosity modeling filtered eddy viscosity methods that add the artificial viscosity only to the high frequency modes were developed in the context of spectral methods we apply the filtered eddy viscosity idea to finite element methods with hierarchical basis functions we prove the existence and uniqueness of the finite element approximation and its convergence to a weak solution of the navier stokes system

convergence analysis of the jacobi spectral collocation methods for volterra integral equations with a weakly singular kernel in this paper a jacobi collocation spectral method is developed for volterra integral equations of second kind with a weakly singular kernel we use some function transformation and variable transformations to change the equation into a new volterra integral equation deflned on the standard interval i1 1 so that the solution of the new equation possesses better regularity and the jacobi orthogonal polynomial theory can be applied conveniently in order to obtain high order accuracy for the approx imation the integral term in the resulting equation is approximated by using jacobi spectral quadrature rules the convergence analysis of this novel method is based on the lebesgue constants corresponding to the lagrange interpolation polynomials poly nomials approximation theory for orthogonal polynomials and the operator theory the spectral rate of convergence for the proposed method is established in the l 1 norm and weighted l 2 norm numerical results are presented to demonstrate the efiectiveness of the proposed method

numerical solution of fractional integro differential equations by a hybrid collocation method fractional integro differential equations have been recently solved by many methods such as adomian decomposition method differential transform method collocation method and taylor expansion approach in this paper a hybrid collocation method is used which combines a non polynomial collocation used on the first subinterval and graded piecewise polynomial collocation used on the rest of the interval a theoretical analysis for the convergence order of the method is presented some numerical examples are given which confirm the theoretical results

nonpolynomial spline collocation for volterra equations with weakly singular kernels volterra integral equations of the second kind with weakly singular kernels possess in general solutions which are not smooth near the left endpoint of the interval of integration since ordinary polynomial spline collocation cannot lead to high order convergence we introduce special nonpolynomial spline spaces which are modelled after the structure of these solutions near the point of nonsmooth behavior collocation in these spaces will once more lead to high order methods analogous results are derived for volterra integro differential equations with weakly singular kernels

two level method based on finite element and crank nicolson extrapolation for the time dependent navier stokes equations a fully discrete two level finite element method the two level method is presented for solving the two dimensional time dependent navier stokes problem the method requires a crank nicolson extrapolation solution u h tau 0 p h tau 0 on a spatial time coarse grid j h tau 0 and a backward euler solution u h tau p h tau on a space time fine grid j h tau the error estimates of optimal order of the discrete solution for the two level method are derived compared with the standard crank nicolson extrapolation method the one level method based on a space time fine grid j h tau the two level method is of the error estimates of the same order as the one level method in the h1 norm for velocity and the l2 norm for pressure however the two level method involves much less work than the one level method

a semi implicit scheme for stationary statistical properties of the infinite prandtl number model we propose a semidiscrete in time semi implicit numerical scheme for the infinite prandtl model for convection besides the usual finite time convergence this scheme enjoys the additional highly desirable feature that the stationary statistical properties of the scheme converge to those of the infinite prandtl number model at vanishing time step one of the key characteristics of the scheme is that it preserves the dissipativity of the infinite prandtl number model uniformly in terms of the time step so far as we know this is the first rigorous result on convergence of stationary statistical properties of numerical schemes for infinite dimensional dissipative complex systems

maximum l 2 convergence rates of the crank nicolson scheme to the stokes initial value problem let a denote the stokes operator and d a alpha the domain of its fractional powers a alpha we consider the homogeneous stokes initial value problem with initial data u 0 u 0 in d a 1 varepsilon varepsilon in 0 1 for stokes like equations the range varepsilon in 0 frac 1 4 is of special interest as any solution derived from varepsilon ge frac 1 4 would necessarily have to satisfy an additional in practice unverifiable compatibility condition at time t 0 approximating any strong solution u in c 0 0 infty d a 1 varepsilon in time direction on a finite time interval 0 t with a crank nicolson scheme we show convergence of order o frac tau 2 t 1 varepsilon which is maximal for the assumed regularity of the data and reflects the loss of regularity as t rightarrow 0 the error estimates are derived by energy and semigroup methods combined with a parabolic duality argument

on energy inequality smoothness and large time behavior in l 2 for weak solutions of the navier stokes equations in exterior domains soit un domaine exterieur de r n n 3 4 a frontiere lisse sur le cylindre d espace temps 0 on considere le probleme de navier stokes non stationnaire u t u u u p f dans 0 u 0 dans 0 u 0 sur 0 u x 0 a x avec des donnees prescrites f et a

analysis of a sequential regularization method for the unsteady navier stokes equations the incompressibility constraint makes navier stokes equations difficult a reformulation to a better posed problem is needed before solving it numerically the sequential regularization method srm is a reformulation which combines the penalty method with a stabilization method in the context of constrained dynamical systems and has the benefit of both methods in the paper we study the existence and uniqueness for the solution of the srm and provide a simple proof of the convergence of the solution of the srm to the solution of the navier stokes equations we also give error estimates for the time discretized srm formulation

finite element approximation for the viscoelastic fluid motion problem in this article the variational formulation of the two dimensional viscoelastic fluid motion problem and its finite element approximation are considered an local error estimate for the velocity with h1 norm and the pressure with l2 norm is obtained and a uniform error estimate for the velocity and pressure with the above norms is provided if the given data satisfies the uniqueness condition

euler implicit explicit iterative scheme for the stationary navier stokes equations in this paper a new uniqueness assumption a2 of the solution for the stationary navier stokes equations is presented under assumption a2 the exponential stability of the solution bar u bar p for the stationary navier stokes equations is proven moreover the euler implicit explicit scheme based on the mixed finite element is applied to solve the stationary navier stokes equations finally the almost unconditionally stability is proven and the optimal error estimates uniform in time are provided for the scheme

a defect correction method for unsteady conduction convection problems ii time discretization in this paper a fully discrete defect correction mixed finite element method mfem for solving the non stationary conduction convection problems in two dimension which is leaded by combining the back euler time discretization with the two step defect correction in space is presented in this method we solve the nonlinear equations with an added artificial viscosity term on a finite element grid and correct these solutions on the same grid using a linearized defect correction technique the stability and the error analysis are derived the theory analysis shows that our method is stable and has a good convergence property some numerical results are also given which show that this method is highly efficient for the unsteady conduction convection problems

energy conserving lagrangian approximation for the equations of navier stokes on considere les equations de navier stokes stationnaires dans un domaine cylindrique borne 0 t ou t 0 et r 3 a une frontiere lisse on construit un quotient aux differences lagrangien conservant l energie qui approche le terme convectif non lineaire

a reduced order extrapolation algorithm based on sfve method and pod technique for non stationary stokes equations in this paper a reduced order extrapolation algorithm roea based on proper orthogonal decomposition pod technique and classical stabilized finite volume element sfve method for non stationary stokes equations is established the error estimates between the roea solutions and the classical sfve solutions and the implementation for solving roea are provided some numerical examples are used to verify that the results of numerical computation are consistent with theoretical conclusions moreover it is shown that roea based on sfve formulation and pod method is feasible and efficient for solving non stationary stokes equations

optimal error estimate of the penalty finite element method for the time dependent navier stokes equations a fully discrete penalty finite element method is presented for the two dimensional time dependent navier stokes equations the time discretization of the penalty navier stokes equations is based on the backward euler scheme the spatial discretization of the time discretized penalty navier stokes equations is based on a finite element space pair x h m h which satisfies some approximate assumption an optimal error estimate of the numerical velocity and pressure is provided for the fully discrete penalty finite element method when the parameters e at and h are sufficiently small

error estimates for semi discrete gauge methods for the navier stokes equations the gauge formulation of the navier stokes equations for incompressible fluids is a new projection method it splits the velocity u a in terms of auxiliary nonphysical variables a and and replaces the momentum equation by a heat like equation for a and the incompressibility constraint by a diffusion equation for this paper studies two time discrete algorithms based on this splitting and the backward euler method for a with explicit boundary conditions and shows their stability and rates of convergence for both velocity and pressure the analyses are variational and hinge on realistic regularity requirements on the exact solution and data both neumann and dirichlet boundary conditions are in principle admissible for but a compatibility restriction for the latter is uncovered which limits its applicability

analysis of the pressure stabilized petrov galerkin method for the evolutionary stokes equations avoiding time step restrictions optimal error estimates for the pressure stabilized petrov galerkin method for the evolutionary stokes equations are proved in the case of regular solutions without restriction on the length of the time step these results clarify the instability of the discrete pressure for small time steps reported in the literature first the limit situation of the continuous in time discretization is considered and second the numerical analysis for the backward euler scheme is carried out the main results are applicable to higher order finite elements the analytical results are strongly based on an appropriate approach for computing the initial velocity which is suggested in this paper numerical studies confirm the theoretical results showing in particular that this instability does not occur for the proposed initial condition