finite element preconditioning for spectral multigrid methods abstract the good performance of finite element preconditioning for the algebraic spectral systems is demonstrated different types of multigrid strategies are compared the most efficient strategy consists of a pure richardson relaxation preconditioned by one v cycle of a finite element multigrid method the good smoothing properties of line relaxation become obvious the computational work for the various methods is estimated

data compatibility and conditional stability for an inverse source problem in the heat equation an inverse problem of determining a nonlinear source term in a heat equation via final observations is investigated by applying integral identity method data compatibilities are obtained with which the inverse source problem here is proved to be solvable furthermore with aid of an integral identity that connects unknown source terms with the known data a conditional stability is constructed theoretical examples are presented showing that theorems and conclusions given in this paper are reasonable and practicable

the method of central difference for the inverse time dependent heat source problem abstract this paper investigates the inverse problem of determining a heat source in a parabolic equation where data are given at some fixed location this problem is ill posed i e the solution if it exists does not depend continuously on the data a central difference regularization method is given and an order optimal stability estimate is obtained numerical results for several benchmark test problems indicate that the central difference regularization method is an accurate and flexible method to determine the unknown time dependent heat source

preconditioned minimal residual methods for chebyshev spectral calculations abstract the problem of preconditioning the pseudospectral chebyshev approximation of an elliptic operator is considered the numerical sensitivity to variations of the coefficients of the operator are investigated for two classes of preconditioning matrices one arising from finite differences the other from finite elements the preconditioned system is solved by a conjugate gradient type method and by a dufort frankel method with dynamical parameters the methods are compared on some test problems with the richardson method 13 and with the minimal residual richardson method 21

a variational multiscale method with bubble stabilization for the oseen problem based on two local gauss integrations abstract we present a variational multiscale method for the oseen equation with higher reynolds number based on the mini element discretization the stabilization term only depends on bubble functions and the best algorithmic feature of our method is using two local gauss integrations to replace projection operator without adding any extra storage a priori error estimate is proved and the effectiveness of the method is illustrated through several numerical experiments

multigrid methods for combined finite difference and fourier problems abstract multigrid methods for combined finite difference and fourier pseudospectral problems are considered applications to two and three dimensional elliptic systems are given suitable relaxation schemes for isotropic and anisotropic problems are presented numerical results demonstrate the efficiency of these methods

a mollification regularization method for the inverse spatial dependent heat source problem in this paper we consider the inverse problem of determining a heat source in a parabolic equation where data are given at a fixed time this problem is ill posed i e the solution if it exists does not depend continuously on the data the mollification method with gauss kernel is proposed to solve this problem an a priori error estimate between the exact solution and its regularized approximation is obtained moreover we also propose a new a posteriori parameter choice rule and get a good error estimate numerical results for several benchmark test problems indicate that the method is an accurate and flexible method to determine the unknown spatial dependent heat source

determination of a spacewise dependent heat source this paper investigates the inverse problem of determining a spacewise dependent heat source in the parabolic heat equation using the usual conditions of the direct problem and information from a supplementary temperature measurement at a given single instant of time the spacewise dependent temperature measurement ensures that the inverse problem has a unique solution but this solution is unstable hence the problem is ill posed for this inverse problem we propose an iterative algorithm based on a sequence of well posed direct problems which are solved at each iteration step using the boundary element method bem the instability is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied numerical results are presented for various typical benchmark test examples which have the input measured data perturbed by increasing amounts of random noise

brownian motion on the hyperbolic plane and selberg trace formula abstract we will show that the relation of the heat kernels for the schrodinger operators with uniform magnetic fields on the hyperbolic plane h 2 the maass laplacians and for the schrodinger operators with morse potentials on r is given by means of a one dimensional fourier transform in the framework of stochastic analysis where the brownian motion on h 2 plays an important role by using this relation we will give the explicit forms of the green functions as a typical related problem we will discuss the selberg trace formula the close relation of the trace formula with the corresponding classical mechanics will also be discussed

analysis of a defect correction method for viscoelastic fluid flow we study a defect correction method for a viscoelastic fluid flow obeying an oldroyd btype constitutive law in the defect step the constitutive equation is computed with the artificially reduced weissenberg parameter for stability and the residual is taken care of in the correction step we show the convergence of the defect correction method and derive an error estimate

subgrid stabilized defect correction method for a steady state natural convection problem in this paper we consider a subgrid stabilized defect correction method for the steady state natural convection problem we state the stabilities and error results for the defect step and the first correction step of the defect correction method the derived theoretical results are supported by several numerical examples

a posteriori steered p robust multigrid with optimal step sizes and adaptive number of smoothing steps we develop a multigrid solver steered by an a posteriori estimator of the algebraic error we adopt the context of a second order elliptic diffusion problem discretized by conforming finite elements of arbitrary polynomial degree p 1 our solver employs zero pre and one post smoothing by the overlapping schwarz block jacobi method and features an optimal choice of the step sizes in the smoothing correction on each level by line search this leads to a simple pythagorean formula of the algebraic error in the next step in terms of the current error and level wise and patch wise error reductions we show the two following results and their equivalence the solver contracts the algebraic error independently of the polynomial degree p and the estimator represents a two sided p robust bound on the algebraic error the p robustness results are obtained by carefully applying the results of schoberl et al ima j numer anal 28 2008 pp 1 24 for one mesh combined with a multilevel stable decomposition for piecewise affine polynomials of xu et al multiscale nonlinear and adaptive approximation springer berlin 2009 pp 599 659 we consider quasi uniform or graded bisection simplicial meshes and prove mild dependence on the number of mesh levels for minimal h 1 regularity and complete independence for h 2 regularity we also present a simple and effective way for the solver to adaptively choose the number of post smoothing steps necessary at each individual level yielding a yet improved error reduction numerical tests confirm p robustness and show the benefits of the adaptive number of smoothing steps

improved condition number for spectral methods introduction d une methode spectrale avec un nombre de conditionnement o n 2 n etant le degre maximum des polynomes

a high accuracy defect correction multigrid method for the steady incompressible navier stokes equations the solution of large sets of equations is required when discrete methods are used to solve fluid flow and heat transfer problems although the cost of the solution is often a drawback when the number of equations in the set becomes large higher order numerical methods can be employed in the discretization of differential equations to decrease the number of equations without losing accuracy for example using a fourth order difference scheme instead of a second order one would reduce the number of equations by approximately half while preserving the same accuracy in a recent paper gupta has developed a fourth order compact method for the numerical solution of navier stokes equations in this paper we propose a defect correction form of the high order approximations using multigrid techniques we also derive a fourth order approximation to the boundary conditions to be consistent with the fourth order discretization of the underlying differential equations the convergence analysis will be discussed for the parameterized form of a general second order correction difference scheme which includes a fourth order scheme as a special case

fourier regularization for a backward heat equation abstract in this paper a simple and convenient new regularization method for solving backward heat equation fourier regularization method is given meanwhile some quite sharp error estimates between the approximate solution and exact solution are provided a numerical example also shows that the method works effectively

a meshless method for solving an inverse spacewise dependent heat source problem in this paper an effective meshless and integration free numerical scheme for solving an inverse spacewise dependent heat source problem is proposed due to the use of the fundamental solution as basis functions the method leads to a global approximation scheme in both spatial and time domains the standard tikhonov regularization technique with the generalized cross validation criterion for choosing the regularization parameter is adopted for solving the resulting ill conditioned system of linear algebraic equations the effectiveness of the algorithm is illustrated by several numerical examples

a finite element variational multiscale method for incompressible flows based on two local gauss integrations in this article we present a finite element variational multiscale vms method for incompressible flows based on two local gauss integrations and compare it with common vms method which is defined by a low order finite element space l h on the same grid as x h for the velocity deformation tensor and a stabilization parameter a the best algorithmic feature of our method is using two local gauss integrations to replace projection operator we theoretically discuss the relationship between our method and common vms method for the taylor hood elements and show that the nonlinear system derived from our method by finite element discretization is much smaller than that of common vms method computationally additionally we present numerical simulations to demonstrate the effectiveness storage computational complexity of our method finally we give some numerical simulations of the nonlinear flow problems to show good stability and accuracy properties of the method

defect correction for the advection diffusion equation abstract a defect correction method for the advection diffusion equation is presented the discretization is performed by 2nd order finite difference schemes schemes where the 2nd order upstream scheme is combined with the standard central scheme for preconditioning the usual first order upstream scheme is employed the defect correction iteration is used for relaxation inside a multigrid procedure it is shown that the smoothing analysis yields rather pessimistic results with rates of about 0 7 0 8 in the practical computation the discretization error is reached in 2 v cycles numerical results are presented which demonstrate the high efficiency of our treatment

subgrid stabilized defect correction methods for the navier stokes equations we consider the synthesis of a recent subgrid stabilization method with defect correction methods the combination is particularly efficient and combines the best algorithmic features of each we prove convergence of the method for a fixed number of corrections as the mesh size goes to zero and derive parameter scalings from the analysis we also present some numerical tests which both verify the theoretical predictions and illustrate the method s promise

splitting techniques for the pseudospectral approximation of the unsteady stokes equations a pseudospectral or collocation approximation of the unsteady stokes equations is considered using the uzawa algorithm the spectral system is decoupled into a helmholtz equation and an equation with a pseudo laplace operator it is proven that the eigenvalues of the pseudo laplacian are real and negative except one zero eigenvalue belonging to the constant mode the pseudospectral method avoids spurious modes since the pressure is approximated with lower order two degrees lower polynomials than the velocity only one grid no staggered grids with the standard chebyshev gauss lobatto nodes is used for discretization for the iterative solution of the spectral system an effective preconditioner is necessary here effective finite difference preconditioners for the spectral pseudo laplacian are presented

a two level subgrid stabilized oseen iterative method for the steady navier stokes equations based on two grid finite element discretization and a recent subgrid scale model a two level subgrid stabilized oseen iterative method for the convection dominated navier stokes equations is proposed and analyzed this method combines the best algorithmic features of the two grid discretization and subgrid stabilization methods it first solves a subgrid stabilized nonlinear navier stokes problem by applying m oseen iterations on a coarse grid and then solves a linear problem on a finer grid where the nonlinear convection term is fixed by the coarse grid solution stability of the method and error estimates of the discrete solution are analyzed the algorithmic parameter scalings are also derived numerical results on an example with known analytical solution the lid driven cavity flow and the backward facing step flow are given to verify the theoretical predictions and demonstrate the method s promise

source term identification in 1 d ihcp we propose stable numerical solutions for the simultaneous identification of temperature temperature gradient and general source terms in the one dimensional inverse heat conduction problem ihcp the numerical solution consists of a regularization procedure based on the mollification method and a marching scheme for the solution of the stabilized problem the stability error analysis and implementation of the algorithm are presented together with a set of numerical results

spectral multigrid methods for domain decomposition problems using patching techniques abstract we consider spectral domain decomposition problems that use patching techniques the resulting discrete systems are solved by the very efficient spectral multigrid methods for relaxation we employ a preconditioned richardson iteration effective preconditioning is achieved by one or more steps of a line relaxation method for the corresponding finite difference scheme numerical results are presented for rectangular and l shaped domains furthermore a cross point problem is considered

iterative schemes for the non homogeneous navier stokes equations based on the finite element approximation in this paper we consider the stability and convergence of three iterative schemes for the non homogeneous steady navier stokes equations as a nonlinear problem we will get a nonlinear discrete system if approximating the non homogeneous navier stokes equations after proving the stability and error estimates of the finite element method for the non homogeneous navier stokes equations three iterative schemes are investigated for solving the resulted nonlinear discrete system the stability and convergence conditions for these iterative methods are also analyzed respectively furthermore new results for the stop criterion are proved finally we show some numerical experiments to illustrate the theoretical prediction

convergent expansions and bounded operators in the umbral calculus on etudie des proprietes analytiques de certaines suites de fonctions speciales appelees suites de sheffer

a connection between subgrid scale eddy viscosity and mixed methods we consider a new mixed method related to the evss method in computational viscoelasticity for the convection dominated convection diffusion equation in which stabilization is added then removed through the extra mixed variables this consistent stabilization is equivalent to an artificial viscosity operator acting only on the fluctuations in u h by suitable choice of the mixed spaces a method of guermond is recovered exactly we show that for a different natural choice a new method results with global error estimates similar to both guermond s method and the streamline diffusion supg method

a spectral multigrid method for the stokes problem in streamfunction formulation abstract suitable spectral multigrid components for the stokes problem in streamfunction formulation are presented we split the stokes problem into a system of two equations with the laplace operator we discretize by means of a collocation or pseudo spectral method preconditioning with finite differences is employed numerical results for the spectral multigrid method are presented

postmodern logarithmo technia abstract in order to solve ordinary differential equations we use an equation with the so called logarithmic derivative similar equations with a linear operator permit us to define logarithmic and antilogarithmic mappings and obtain some results unknown before for the classical derivation operator this method and its applications are exposed in detail in the author s book 1 and may be treated as an introduction into the 21 st century logarithmo technia according to the original meaning of this word in the first section there are given basic notions and facts of algebraic analysis without proofs the second section consists of the most important definitions and theorems without proofs concerning logarithms and antilogarithms the third section is concerned with properties of multiplicative true shifts in the last section is given a generalization of the binomial theorem for harmonic logarithms

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

a finite element variational multiscale method based on two grid discretization for the steady incompressible navier stokes equations abstract by combining the best algorithmic features of two grid discretization method and a recent variational multiscale method a two level finite element variational multiscale method based on two local gauss integrations for convection dominated incompressible navier stokes equations is proposed and analyzed in this method a fully nonlinear navier stokes problem is first solved on a coarse grid and then a linear problem is solved on a fine grid to correct the coarse grid solution where the numerical forms of the navier stokes equations both on coarse and fine grids are stabilized by a stabilization term defined by the difference between a consistent and an under integrated matrix of the velocity gradient error bounds of the approximate solution are analyzed algorithmic parameter scalings of the method are derived numerical tests are also given to verify the theoretical predictions and demonstrate the efficiency and promise of the method

the method of simplified tikhonov regularization for dealing with the inverse time dependent heat source problem this paper investigates the inverse problem of determining a heat source using a parabolic equation where data are given at some fixed location the problem is ill posed i e the solution if it exists does not depend continuously on the data a simplified tikhonov regularization method is given and an order optimal stability estimate is obtained a numerical example shows that the regularization method is effective and stable

spectral collocation on triangular elements we consider the poisson problem on a segment of the unit disc and on triangles on the segment we transform the poisson problem by means of polar coordinates in these new coordinates we have a problem in a rectangle which can easily be mapped onto the square here standard chebyshev collocation techniques can be applied then the segment is mapped onto a triangle where the same spectral scheme may be used by numerical tests we observed the expected high spectral accuracy due to the corner singularity a singular behaviour of the solution can be expected here we improved the accuracy by auxiliary mapping techniques further it is shown that finite difference preconditioning can be successfully applied in order to construct an efficient iterative solver finally a domain decomposition technique is applied to the patching of a rectangular and a triangular element

conjugate gradient like algorithms for solving nonsymmetric linear systems this paper presents a unified formulation of a class of the conjugate gradient like algorithms for solving nonsymmetric linear systems the common framework is the petrov galerkin method on krylov subspaces we discuss some practical points concerning the methods and point out some of the interrelations between them 1 introduction in the recent few years a large number of generalizations of the conjugate gradient and conjugate residual methods which are very successful in solving symmetric positive definite linear systems have been proposed for solving nonsymmetric linear systems 3 6 5 7 12 in this paper we present an abstract framework which includes most of these methods and many new ones our goal is to understand the relationships among the methods and to synthesize consider the general linear system

improved spectral multigrid methods for periodic elliptic problems abstract the spectral multigrid method for periodic elliptic problems is examined several modifications are introduced including a midpoint pseudospectral discretization which eliminates the need for filtering the highest fourier mode and new relaxation schemes for isotropic and anisotropic problems numerical results are presented demonstrating substantial increases in efficiency and accuracy over previous methods

a discontinuous subgrid eddy viscosity method for the time dependent navier stokes equations in this paper we provide an error analysis of a subgrid scale eddy viscosity method using discontinuous polynomial approximations for the numerical solution of the incompressible navier stokes equations optimal continuous in time error estimates of the velocity are derived the analysis is completed with some error estimates for two fully discrete schemes which are first and second order in time respectively

spectral collocation methods and polar coordinate singularities this paper considers the numerical solution of elliptic differential equations on the unit disk using polar coordinates the disk is mapped onto a rectangle the resulting transformed problem is solved by a method related to collocation since the origin is a coordinate singularity some natural trial functions are singular there and a special technique is applied to use zero as a collocation point for poisson and helmholtz equations a fast algorithm with an operation count of o n2 log n is presented numerical results show the different stability and convergence properties of the algorithms

a fast spectral element solver combining static condensation and multigrid techniques we propose a spectral element multigrid method for the two dimensional helmholtz equation discretized on regular grids combining p multigrid with static condensation the method achieves nearly linear complexity with an order independent convergence rate for solving the condensed equations for smoothing we consider two groups of edge based relaxation schemes the best of which attains a multigrid convergence rate of 0 014 to 0 028 numerical experiments have been carried out that demonstrate the robustness of the approach for orders up to 32 and a total of 109 degrees of freedom in comparison with a fast finite difference solver the latter is clearly outperformed already for errors of one percent or lower

analysis of generalized negative binomial distributions attached to hyperbolic landau levels to each hyperbolic landau level of the poincare disc is attached a generalized negative binomial distribution in this paper we compute the moment generating function of this distribution and supply its atomic decomposition as a perturbation of the negative binomial distribution by a finitely supported measure using the mandel parameter we also discuss the nonclassical nature of the associated coherent states next we derive a levy khintchine type representation of its characteristic function when the latter does not vanish and deduce that it is quasi infinitely divisible except for the lowest hyperbolic landau level corresponding to the negative binomial distribution by considering the total variation of the obtained quasi levy measure we introduce a new infinitely divisible distribution for which we derive the characteristic function

spectral methods and a maximum principle etude de diverses approximations spectrales de tchebychev d un probleme de couche limite pour un operateur d helmholtz et d advection diffusion comportement des solutions spectrales dans l espace des frequences et dans l espace physique estimations d erreur

chebyshev 3 d spectral and 2 d pseudospectral solvers for the helmholtz equation two chebyshev solvers are presented for the linear helmholtz equation the first algorithm is a 3 d direct spectral solver based on a diagonalization technique whilst the second performs an iterative pseudospectral 2 d calculation with finite difference preconditioning both techniques handle general nonhomogeneous boundary conditions computing times and accuracies of the two methods are compared

a parallel two level finite element variational multiscale method for the navier stokes equations abstract a combination method of two grid discretization approach with a recent finite element variational multiscale algorithm for simulation of the incompressible navier stokes equations is proposed and analyzed the method consists of a global small scale nonlinear navier stokes problem on a coarse grid and local linearized residual problems in overlapped fine grid subdomains where the numerical form of the navier stokes equations on the coarse grid is stabilized by a stabilization term based on two local gauss integrations at element level and defined by the difference between a consistent and an under integrated matrix involving the gradient of velocity by the technical tool of local a priori estimate for the finite element solution error bounds of the discrete solution are estimated algorithmic parameter scalings are derived numerical tests are also given to verify the theoretical predictions and demonstrate the effectiveness of the method

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

assessment of subgrid scale models for the incompressible navier stokes equations in this paper we assess two kinds of subgrid finite element methods for the two dimensional 2d incompressible naver stokes equations nses these methods introduce subgrid scale sgs eddy viscosity terms which do not act on the large flow structures the eddy viscous terms consist of the fluid flow fluctuation strain rate stress tensors the fluctuation tensor can be calculated by a elliptic projection or a simple l 2 projection projective filter in finite element spaces the finite element pair p 2 p 1 is adopted to numerically implement analysis and computation we give a complete error analysis based on the assumptions of some regularity conditions on the part of numerical tests the numerical computations for the stationary flows show that the numerical results agree with some benchmark solutions and theoretical analysis very well furthermore the given sgs models are applied to the non stationary fluid flows

spectral multigrid techniques for the navier stokes equations abstract we present a spectral multigrid method for the effective solution of the navier stokes equations for incompressible fluids here we consider the vorticity streamfunction formulation discretization in time is performed by a second order semi implicit scheme where the diffusive term is treated in an implicit manner while the convective term is evaluated explicitly for the discretization of the resulting stokes type problem we introduce a new spectral collocation method this scheme is highly accurate and allows efficient solution by spectral multigrid methods we propose an efficient finite difference preconditioner numerical results are presented which substantiate the usefulness of our concept

spectral schemes on triangular elements abstract the poisson problem with homogeneous dirichlet boundary conditions is considered on a triangle the transformation from square to triangle is realized by mapping an edge of the square onto a corner of the triangle then standard chebyshev collocation techniques can be implemented numerical experiments demonstrate the expected high spectral accuracy for smooth solutions furthermore it is shown that finite difference preconditioning can be successfully employed to construct an efficient iterative solver then the convection diffusion equation is considered here finite difference preconditioning with central differences leads to instability however using the first order upstream scheme we obtain a stable method finally a domain decomposition technique is applied to the patching of rectangular and triangular elements

simplified tikhonov and fourier regularization methods on a general sideways parabolic equation the inverse heat conduction problem ihcp can be considered to be a sideways parabolic equation in the quarter plane and now the results available in the literature on ihcp mainly devoted to the standard sideways heat equation numerical methods have been developed also for more general equations but in most cases the stability theory and convergence proofs have not been generalized accordingly this paper remedies this by a simplified tikhonov and a new fourier regularization methods on a general sideways parabolic equation some known results for sideways heat equation are only the special case of the conclusions in this paper the numerical example shows that the computation effect is satisfactory

nonhomogeneous heat equation identification and regularization for the inhomogeneous term we study the nonhomogeneous heat equation under the form ut uxx t f x where the unknown is the pair of functions u f under various assumptions about the function and the final value in t 1 i e g x we propose different regularizations on this ill posed problem based on the fourier transform associated with a lebesgue measure for 0 the solution is unique

error correction method for navier stokes equations at high reynolds numbers in this paper we present a new iterative method for solving the stationary navier stokes equations nses at high reynolds numbers the method consists of first solving the nses by the oseen iterative scheme and then an error correction strategy is implemented to control the error arising from the linearization of the nonlinear nses the new method retains the advantage of the classical oseen scheme but it leads to a rapid rate of convergence and also enhances the capability for solving problems with higher reynolds numbers it will be shown that under the uniqueness condition the proposed method accelerates up to a factor of three in the convergence rate the stability analysis and error estimate are presented furthermore numerical simulations using the new method and other classical schemes are reported to verify the superior performance of the proposed method

semicirculant preconditioning of elliptic operators in this work we consider the semicirculant preconditioning of elliptic differential operators of the form lu epsilon delta u au x bu y cu in two cases 0 epsilon ll 1 and epsilon equiv 1 the paper numer math 81 1998 pp 211 249 provided extremely interesting and useful results in the first case on the other hand those appear to contradict basic results on preconditioning given in siam j numer anal 27 1990 pp 656 694 we reobtain the results of numer math 81 1998 pp 211 249 by a new approach which we believe to be more transparent we also clarify the situation regarding the apparent contradiction with siam j numer anal 27 1990 pp 656 694 finally we describe the distribution of the preconditioned eigenvalues in the uniformly elliptic case epsilon equiv 1

adaptive variational multiscale methods for incompressible flow based on two local gauss integrations we consider variational multiscale vms methods with h adaptive technique for the stationary incompressible navier stokes equations the natural combination of vms with adaptive strategy retains the best features of both methods and overcomes many of their deficits a reliable a posteriori projection error estimator is derived which can be computed by two local gauss integrations at the element level finally some numerical tests are presented to illustrate the method s efficiency

banach algebras logarithms and polynomials of convolution type abstract using elementary banach algebra techniques it is determined which elements of banach algebras like l1 b k and a have logarithms the wiener levy theorem will be used to answer the same question for more complicated banach algebras like the wiener algebra these results will be applied to polynomials of convolution type and a generalization thereof

splitting techniques for the unsteady stokes equations a pseudospectral or collocation approximation of the unsteady stokes equations is presented using the uzawa algorithm the spectral system is decoupled into helmholtz equations for the velocity components and an equation with the pseudo laplacian for the pressure both the velocity components and the pressure are approximated by polynomials of the same degree n compatible velocity and pressure approximations are constructed to ensure that the discrete problem is well posed the resulting discrete velocity field is identically divergence free only one grid no staggered grids with the standard chebyshev gauss lobatto nodes is used here we further compare our treatment with an inhomogeneous neumann condition for the pressure this scheme was proposed in g karniadakis m israeli and s a orszag j comp phys 97 1991 pp 414 443 the comparison shows that the neumann condition yields somewhat better approximations for the pressure in the time discretization a high order backward differentiation scheme for the intermediate velocity is combined with a high order extrapolant for the pressure it is numerically shown that a stable third order method in time can be achieved in the case of periodic boundary conditions this result is confirmed by fourier analysis

numerical solution of time dependent component with sparse structure of source term for a time fractional diffusion equation abstract we consider an inverse time dependent component of source term with sparse structure for the time fractional diffusion equation in the present paper we prove the uniqueness of the inverse problem with nonlocal observation data by laplace transform technique concerning the sparsity of the source term we transform the inverse source problem into an elastic net regularization optimization problem the semi smooth newton method is adopted to solve the optimization problem and the superconvergence of the semi smooth newton algorithm is proven several numerical examples are tested to verify the efficiency of the algorithm

analysis of the combined finite element and fourier interpolation we consider lagrange interpolation involving trigonometric polynomials of degree n in one space direction and piecewise polynomials over a finite element decomposition of mesh size h in the other space directions we provide error estimates in non isotropic sobolev norms depending additively on the parametersh andn an application to the convergence analysis of an elliptic problem with some numerical results is given

error analysis of a fully discrete finite element variational multiscale method for the natural convection problem abstract for the natural convection problem we propose a new projection based finite element variational multiscale method by defining the stabilization terms via two local gauss integrations at the element level based on the implicit backward euler and implicit crank nicolson schemes for temporal discretization and stabilized mixed finite element spatial discretization we establish two numerical schemes for the natural convection problem unconditional stabilities of the two numerical schemes are proved we derive error bounds of the fully discrete solution which are first and second order in time respectively the optimal error estimates in space could be achieved for velocity and temperature in the h 1 semi norm and for pressure in the l 2 norm with the proper choosing of stabilized parameters however the error estimates in space are suboptimal for velocity and temperature in the l 2 norm the derived theoretical results are supported by two numerical examples

on a backward problem for inhomogeneous time fractional diffusion equations abstract in this paper we consider a final value problem for time fractional diffusion equation with inhomogeneous source the main goal of our paper is to determine an approximated initial data from the observation data at final time by constructing a regularized solution using a mollification method under appropriate regularity assumptions of the exact solution we give convergence rate between the reconstructed solution and the exact one we also provide a numerical example to illustrate the main results

optimal error bound and fourier regularization for identifying an unknown source in the heat equation in this paper we investigate a problem of the identification of an unknown source from one supplementary temperature measurement at a given instant of time for the transient heat equation under an a priori condition we answer the question concerning the best possible accuracy for the problem the fourier regularization method is utilized for solving the problem and its convergent rate is analyzed numerical results are presented to illustrate the accuracy and efficiency of the method

wavelet and fourier methods for solving the sideways heat equation we consider an inverse heat conduction problem the sideways heat equation which is a model of a problem where one wants to determine the temperature on both sides of a thick wall but where one side is inaccessible to measurements mathematically it is formulated as a cauchy problem for the heat equation in a quarter plane with data given along the line x 1 where the solution is wanted for 0 leq x 1 the problem is ill posed in the sense that the solution if it exists does not depend continuously on the data we consider stabilizations based on replacing the time derivative in the heat equation by wavelet based approximations or a fourier based approximation the resulting problem is an initial value problem for an ordinary differential equation which can be solved by standard numerical methods e g a runge kutta method we discuss the numerical implementation of fourier and wavelet methods for solving the sideways heat equation theory predicts that the fourier method and a method based on meyer wavelets will give equally good results our numerical experiments indicate that also a method based on daubechies wavelets gives comparable accuracy as test problems we take model equations with constant and variable coefficients we also solve a problem from an industrial application with actual measured data

a mollification method for ill posed problems a mollification method for a class of ill posed problems is suggested the idea of the method is very simple and natural if the data are given inexactly then we try to find a sequence of mollification operators which map the improper data into well posedness classes of the problem mollify the improper data within these mollified data our problem becomes well posed and when these facts are in hand we try to obtain error estimates and optimal or quasi optimal mollification parameters the method is working not only for problems in hilbert spaces but also for problems in banach spaces applications of the method to concrete problems like numerical differentiation parabolic equations backwards in time the cauchy problem for the laplace equation one and multidimensional non characteristic cauchy problems for parabolic equations in infinite or finite domains give us very sharp stability estimates of h older continuous type in these cases the method is optimal in the sense that it gives the same order of h older continuous dependence on the data as for the regularized problems furthermore the method may be implemented numerically using fast fourier transforms for the first time a uniform stability estimate of h older continuous type of the solution of the heat equation backwards in time in the space l p bbb r for all p in 1 infty could be established by our mollification method a new simple sharp pointwise estimate of h older type for the weak solution of a non characteristic cauchy problem for parabolic equations in a finite domain is established

a new defect correction method for the navier stokes equations at high reynolds numbers abstract in this paper a new defect correction method for the navier stokes equations is presented with solving an artificial viscosity stabilized nonlinear problem in the defect step and correcting the residual by linearized equations in the correction step for a few steps this combination is particularly efficient for the navier stokes equations at high reynolds numbers in both the defect and correction steps we use the oseen iterative scheme to solve the discrete nonlinear equations furthermore the stability and convergence of this new method are deduced which are better than that of the classical ones finally some numerical experiments are performed to verify the theoretical predictions and show the efficiency of the new combination

negative binomial and multinomial states probability distributions and coherent states following the relationship between probability distribution and coherent states for example the well known poisson distribution and the ordinary coherent states and relatively less known one of the binomial distribution and the su 2 coherent states we propose interpretation of su 1 1 and su r 1 coherent states in terms of probability theory they will be called the negative binomial multinomial states which correspond to the negative binomial multinomial distribution the non compact counterpart of the well known binomial multinomial distribution explicit forms of the negative binomial multinomial states are given in terms of various boson representations which are naturally related to the probability theory interpretation here we show fruitful interplay of probability theory group theory and quantum theory

line relaxation for spectral multigrid methods abstract spectral multigrid methods for elliptic problems with dirichlet and periodic boundary conditions are examined the multigrid convergence is improved by certain line relaxation techniques which are suggested by a smoothing analysis furthermore several choices of relaxation parameters are compared numerical results are presented showing the gain in efficiency and accuracy over previous methods

a finite element variational multiscale method for the navier stokes equations this paper presents a variational multiscale method vms for the incompressible navier stokes equations which is defined by a large scale space lh for the velocity deformation tensor and a turbulent viscosity nu t the connection of this method to the standard formulation of a vms is explained the conditions on lh under which the vms can be implemented easily and efficiently into an existing finite element code for solving the navier stokes equations are studied numerical tests with the smagorinsky large eddy simulation model for nu t are presented

spectral methods with sparse matrices spectral methods employ global polynomials for approximation hence they give very accurate approximations for smooth solutions unfortunately for dirichlet problems the matrices involved are dense and have condition numbers growing aso n 4 for polynomials of degree n in each variable we propose a new spectral method for the helmholtz equation with a symmetric and sparse matrix whose condition number grows only aso n 2 certain algebraic spectral multigrid methods can be efficiently used for solving the resulting system numerical results are presented which show that we have probably found the most effective solver for spectral systems

a new method of stabilization for singular perturbation problems with spectral methods a stabilized treatment of singular perturbation problems with spectral methods is presented the stabilization is achieved by introducing one additional collocation point the resulting overdetermined system leads to the normal equations which yield a stabilized system numerical results in one and two dimensions are presented which show the improved condition number of the stabilized spectral system

convergence analysis of the defect correction iteration for hyperbolic problems this paper explains some of the convergence behaviour of iterative implicit and defect correction schemes for the solution of the discrete steady euler equations such equations are also commonly solved by pseudo time integration the steady solution being achieved as the limit for t to infty of the solution of a time dependent problem implicit schemes are then often chosen for their favourable stability properties permitting large timesteps for efficiency an important class of implicit schemes involving first and second order accurate upwind discretisations is considered in the limit of an infinite timestep these implicit schemes approach defect correction algorithms thus our analysis is informative for both types of construction simple scalar linear model problems are introduced for the one dimensional and two dimensional cases these model problems are analyzed in detail by both fourier and matrix analyses the convergence behaviour appears to be strongly dependent on a parameter beta

adaptive defect correction methods for convection dominated convection diffusion problems abstract we present a posteriori error estimates for a defect correction method for approximating solutions of convection diffusion problems the algorithms and estimators include the possibility of using in the discretization a nonlinear selection mechanism which we find improves solution quality in and near layers energy norm and l2 a posteriori error estimates are proven for the full algorithm two examples of fully adaptive finite element defect correction calculations are presented these examples illustrate the scheme and demonstrate the effectiveness of the method

iterative finite element variational multiscale method for the incompressible navier stokes equations abstract three iterative finite element variational multiscale methods are proposed and applied to the numerical simulation of the navier stokes equations the main idea of these methods is to combine the finite element variational multiscale method based on two local gauss integrations with three different iterative schemes the existence and uniqueness of approximate solutions of these iterative finite element variational multiscale methods are proved firstly and then the convergence and error estimates of them are deduced finally some numerical examples are given to support the theoretical analysis the numerical results show that the iterative finite element variational multiscale method has a wider range of reynolds numbers than standard galerkin iterative finite element method and the oseen iterative scheme is much more efficient than the other two under high reynolds numbers

spectral multigrid methods for elliptic equations ii a detailed description of spectral multigrid methods is provided this includes the interpolation and coarse grid operators for both periodic and dirichlet problems the spectral methods for periodic problems use fourier series and those for dirichlet problems are based upon chebyshev polynomials an improved preconditioning for dirichlet problems is given numerical examples and practical advice are included

boundary conditions in chebyshev and legendre methods we discuss two different ways of treating non dirichlet boundary conditions in chebyshev and legendre collocation methods for second order differential problems an error analysis is provided the effect of preconditioning the corresponding spectral operators by finite difference matrices is also investigated

analysis of the spectral vanishing viscosity method for periodic conservation laws the convergence of the spectral vanishing method for both the spectral and pseudospectral discretizations of the inviscid burgers equation is analyzed it is proved that this kind of vanishing viscosity is responsible for spectral decay of those fourier coefficients located toward the end of the computed spectrum consequently the discretization error is shown to be spectrally small independently of whether or not the underlying solution is smooth this in turn implies that the numerical solution remains uniformly bounded and convergence follows by compensated compactness arguments

defect correction for convection dominated flow a defect correction scheme with the first order upwind preconditioner is considered by fourier analysis the preconditioning properties for the second order upwind scheme the central scheme and spectral methods are examined since the eigenvalues of the preconditioned operator are complex the generalized minimal residual iteration is used for the iterative solution this procedure is applied to the boussinesq flow problem in vorticity streamfunction formulation numerical results are presented for increasing rayleigh numbers

a 3d spectral multigrid method we consider a 3d spectral multigrid method for 3d elliptic problems with arbitrary anisotropies the smoothing method is a preconditioned richardson iteration where a pseudospectral discretization is used on the right hand side and a finite difference discretization on the left hand side in each step of this iteration a finite difference problem is solved approximately by plane relaxation sweeps the plane relaxation requires the solution of 2d finite difference problems these problems are solved approximately using 2d multigrid cycles

on tikhonov s method and optimal error bound for inverse source problem for a time fractional diffusion equation abstract we investigate the linear but ill posed inverse problem of determining a multi dimensional space dependent heat source in a time fractional diffusion equation we show that the problem is ill posed in the hilbert scale h r r n and establish global order optimal lower bound for the worst case error next we use the tikhonov regularization method to deal with this problem in the hilbert scale h r r n locally optimal choices of parameters for the family of regularization operator in the hilbert scales h r r n are analyzed by a priori and a posteriori methods numerical implementations are presented to illustrate our theoretical findings