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

chebyshev pseudospectral solution of second order elliptic equations with finite element preconditioning abstract a new chebyshev pseudospectral algorithm for second order elliptic equations using finite element preconditioning is proposed and tested on various problems bilinear and biquadratic lagrange elements are considered as well as bicubic hermite elements the numerical results show that bilinear elements produce spectral accuracy with the minimum computational work l shaped regions are treated by a subdomain approach

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

least squares spectral collocation for discontinuous and singular perturbation problems a least squares spectral collocation scheme for discontinuous problems is proposed for the first derivative operator the domain is decomposed in subintervals where the jumps are imposed at the discontinuities equal order polynomials are used on all subdomains for the discretization spectral collocation with chebyshev polynomials is employed fast fourier transforms are now available the collocation conditions and the interface conditions lead to an overdetermined system which can be efficiently solved by least squares the solution technique will only involve symmetric positive definite linear systems this approach is further extended to singular perturbation problems where least squares are used for stabilization by a suitable decomposition of the domain the boundary layer is well resolved numerical simulations confirm the high accuracy of our spectral least squares scheme

optimal control in fluid mechanics by finite elements with symmetric stabilization there are two main possibilities for the numerical computation of optimal control problems with constraints given by partial differential equations one may consider first the discretized problem and then build the optimality condition the other possibility is to formulate first the optimality condition on the continuous level and then discretize both approaches may lead to different discrete adjoint equations because discretization and building the adjoint do not commute in general this type of inconsistency takes place when conventional stabilized finite elements for flow problems as for instance streamline diffusion supg are used due to its nonsymmetry consequently the computed control is significantly affected by the way of defining the discrete optimality condition hence there is a need for symmetric stabilization so that discretization and building the adjoint commute we formulate the use of this kind of stabilization and give a quasi optimal a priori estimate in the context of optimal control problems for the oseen system in particular we show that local projection stabilization and edge oriented stabilization result to be quasi optimal for optimal control problems

superlinearly convergent cg methods via equivalent preconditioning for nonsymmetric elliptic operators the convergence of the conjugate gradient method is studied for preconditioned linear operator equations with nonsymmetric normal operators with focus on elliptic convection diffusion operators in sobolev space superlinear convergence is proved first for equations whose preconditioned form is a compact perturbation of the identity in a hilbert space then the same convergence result is verified for elliptic convection diffusion equations using different preconditioning operators the convergence factor involves the eigenvalues of the corresponding operators for which an estimate is also given the above results enable us to verify the mesh independence of the superlinear convergence estimates for suitable finite element discretizations of the convection diffusion problems

adaptive lagrange galerkin methods for unsteady convection diffusion problems in this paper we derive an a posteriori error bound for the lagrange galerkin discretisation of an unsteady linear convection diffusion problem assuming only that the underlying space time mesh is nondegenerate the proof of this error bound is based on strong stability estimates of an associated dual problem together with the galerkin orthogonality of the finite element method based on this a posteriori bound we design and implement the corresponding adaptive algorithm to ensure global control of the error with respect to a user defined tolerance

a direct solver for the least squares spectral collocation system on rectangular elements for the incompressible navier stokes equations a least squares spectral collocation scheme for the stokes and incompressible navier stokes equations is proposed the original domain is decomposed into quadrilateral subelements and on the element interfaces continuity of the functions is enforced in the least squares sense the collocation conditions and the interface conditions lead to overdetermined systems these systems are directly solved by qr decomposition of the underlying matrices by numerical simulations it is shown that the direct method leads to better results than the approach with normal equations furthermore it is shown that the condition numbers can be reduced by introducing the clenshaw curtis quadrature rule for imposing the average pressure to be zero finally our scheme is successfully applied to the regularized and lid driven cavity flow problems

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

sensitivity based methods for convergence acceleration of iterative algorithms abstract a new method to accelerate the convergence of iterative schemes for the numerical integration of systems of partial differential equations has been developed the basic idea is that the residual at a grid point depends on the values of the solution vector at the neighboring grid points used in the local discretization approximation thus the new acceleration method is based on the sensitivity of the future residual to the change in the solution vector at the neighboring grid points with the objective to minimize the future residual the result is a set of optimum iterative relaxation parameters for the entire flow field or for each individual grid line the method is easy to implement in the existing codes we have applied it to a finite difference code for two dimensional incompressible navier stokes equations test cases involve laminar and turbulent flows with severe grid clustering and flow separation the results are compared with those of a basic explicit runge kutta rk time stepping iterative algorithm and with the implicit residual smoothing 1rs and the distributed minimal residual dmr acceleration techniques the new acceleration scheme is shown to be superior to these methods especially on highly clustered grids

sirt and cg type methods for the iterative solution of sparse linear least squares problems abstract sirt and cg type methods have been successfully employed for the approximate solution of least squares problems arising in tomography in this paper we study and compare their convergence and regularization properties it is pointed out that sirt methods apply an uncontrollable implicit rescaling which affects the statistical characteristics of the system whereas cg type methods do not for a large class of model problems it is shown that virtually the same solutions as obtained by sirt methods can be obtained by applying a cg type method to a properly rescaled system but with an amount of work proportional to the square root of the amount of work with sirt

krylov subspace methods for solving large unsymmetric linear systems some algorithms based upon a projection process onto the krylov subspace k sub m span r sub 0 ar sub 0 a sup m 1 r sub 0 are developed generalizing the method of conjugate gradients to unsymmetric systems these methods are extensions of arnoldi s algorithm for solving eigenvalue problems the convergence is analyzed in terms of the distance of the solution to the subspace k sub m and some error bounds are established showing in particular a similarity with the conjugate gradient method for symmetric matrices when the eigenvalues are real several numerical experiments are described and discussed

high order hierarchical a and l stable integration methods numerical integration of stiff first order systems of differential equations is considered it is shown how a c continuous polynomial time discretization in conjunction with a weighted residual method can be used to derive methods corresponding to the diagonal and first subdiagonal pade approximants in this manner a stable schemes of order 2k and l stable schemes of order 2k l are obtained at least for k 4 the methods are hierarchical in the sense that a scheme with a given accuracy embraces the equations of all lower order methods that have the same stability type

a characteristic mixed finite element method for time dependent convection diffusion optimal control problem abstract in this paper we examine the method of characteristic mixed finite element for the approximation of convex optimal control problem governed by time dependent convection diffusion equations with control constraints for the discretization of the state equation the characteristic finite element is used for the approximation of the material derivative term i e the time derivative term plus the convection term and the lowest order raviart thomas mixed element is applied for the approximation of the diffusion term we derive some a priori error estimates for both the state and control approximations

a new taxonomy of conjugate gradient methods abstract in this paper a framework is proposed for conjugate gradient and conjugate direction algorithms based on an indefinite quadratic form by the choice of the hessian g of this form and of another arbitrary matrix k no fewer than nineteen distinct cg algorithms may be succinctly defined four of these are believed to be new the choice of certain other vectors leads either to a two term or a three term recurrence formula and it is shown that to every two term formula there corresponds a three term formula although the converse is not generally true the two matrices g and k also determine the stability characteristics of the two term methods

quantitative domains via fuzzy sets part i continuity of fuzzy directed complete posets this paper deals with quantitative domain theory via fuzzy sets it examines the continuity of fuzzy directed complete posets dcpos for short based on complete residuated lattices first we show that a fuzzy partial order in the sense of fan and zhang and an l order in the sense of belohlavek are equivalent to each other then we redefine the concepts of fuzzy directed subsets and continuous fuzzy dcpos we also define and study fuzzy galois connections on fuzzy posets we investigate some properties of continuous fuzzy dcpos we show that a fuzzy dcpo is continuous if and only if the fuzzy double downward arrow operator has a right adjoint we define fuzzy auxiliary relations on fuzzy posets and approximating fuzzy auxiliary relations on fuzzy dcpos we show that a fuzzy dcpo is continuous if and only if the fuzzy way below relation is the smallest approximating fuzzy auxiliary relation

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

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

l ordered and l lattice ordered groups this paper pursues an investigation on groups equipped with an l ordered relation where l is a fixed complete heyting algebra first by the concept of join and meet on an l ordered set the notion of an l lattice a weak l lattice is introduced and some related results are obtained then we applied them to define an l lattice ordered group we also introduce convex l subgroups to construct a quotient l ordered group at last a relation between the positive cone of an l ordered group and special type of elements of l g is found where g is a group

fuzzy rough sets based on generalized residuated lattices this paper is devoted to propose generalized l fuzzy rough sets as a further generalization of the notion of l fuzzy rough sets a quadruple of approximation operators are defined to suit the situation when generalized residuated lattices are non commutative generalized l fuzzy rough sets are characterized from both constructive and axiomatic approaches in the constructive approach various classes of generalized l fuzzy rough sets are investigated moreover the relationship between generalized l fuzzy rough sets and l topologies on an arbitrary universe is discussed with generalized lower and upper sets as an application of generalized l fuzzy rough sets fuzzy rough sets are proposed and studied on the unit interval 0 1 which are based on generalized residuated lattices induced by left continuous pseudo t norms

a generalized conjugate gradient least square method a generalizeds term truncated conjugate gradient method of least square type proposed in 1a b is extended to a form more suitable for proving when the truncated version is identical to the full term version advantages with keeping a control term in the truncated version is pointed out a computationally efficient new algorithm based on a special inner product with a small demand of storage is also presented we also give simplified and slightly extended proofs of termination of the iterative sequence and of existence of ans term recursion identical to the full term version important earlier results on this latter topic are found in 15 16 8 and 11

vector sequence transformations methodology and applications to linear systems in this paper a methodology for the construction of various vector sequence transformations is formulated leading to a unified presentation of the subject and to new results the connections to the general interpolation problem and to projections are discussed various particular cases are examined in more details applications to the solution of systems of linear equations will end the paper and in particular their relation with lanczos method will be studied some numerical examples will be given

the essence of ideal completion in quantitative form if a poset lacks joins of directed subsets one can pass to its ideal completion but doing this means also changing the setting the universal property of ideal completion of posets suggests that it should be regarded as a functor from the category of posets with monotone maps to the category of dcpos with scott continuous functions as morphisms the same applies for the quantitative version of ideal completion suggested in the literature as in the case of posets it seems advantageous to consider a different topology with the completed spaces we introduce topological v continuity spaces and their smyth completion and show that this is an adequate setting to consider ideal completion of quantitative domains performing the smyth completion of a v continuity space regarded as topological v continuity space gives the ideal completion of the original space together with its scott topology

global estimates for mixed methods for second order elliptic equations global error estimates in l2 92 l 9 and h s q 2 in r2 or a re derived for a mixed finite element method for the dirichlet problem for the elliptic operator lp div a grad p bp cp based on the raviart thomas nedelec space vh x wh c h div 92 x l2 g2 optimal order estimates are obtained for the approximation of p and the associated velocity field u a grad p bp in l2 g2 and h s 92 0 6 s k 1 and if q c r2 forp in lv2

a priori error estimates for space time finite element discretization of parabolic optimal control problems part i problems without control constraints in this paper we develop a priori error analysis for galerkin finite element discretizations of optimal control problems governed by linear parabolic equations the space discretization of the state variable is done using usual conforming finite elements whereas the time discretization is based on discontinuous galerkin methods for different types of control discretizations we provide error estimates of optimal order with respect to both space and time discretization parameters the paper is divided into two parts in the first part we develop some stability and error estimates for space time discretization of the state equation and provide error estimates for optimal control problems without control constraints in the second part of the paper the techniques and results of the first part are used to develop a priori error analysis for optimal control problems with pointwise inequality constraints on the control variable

superconvergence of triangular mixed finite elements for optimal control problems with an integral constraint abstract in this paper we shall investigate the superconvergence property of quadratic elliptical optimal control problems by triangular mixed finite element methods the state and co state are approximated by the order k 1 raviart thomas mixed finite elements and the control is discretized by piecewise constant functions we prove the superconvergence error estimate of h 2 in l 2 norm between the approximated solution and the interpolation of the exact control variable moreover by postprocessing technique we find that the projection of the discrete adjoint state is superclose in order h 2 to the exact control variable

a splitting positive definite mixed finite element method for elliptic optimal control problem in this paper we propose a splitting positive definite mixed finite element method mfem for the approximation of convex optimal control problem governed by elliptic equations with control constraints by selecting the variation functional properly the presented procedure can be split into two independent symmetric and positive definite weak formula for the unknown state variable y and for the unknown flux variable s it then follows from the first order necessary and sufficient optimality condition we deduce another two corresponding adjoint state equations for z and w which are also independent symmetric and positive definite also a variational inequality for the control variable u is involved convergence analysis shows that the method yields the approximate solutions with optimal accuracy in l 2 w u norm for the control u l 2 w norm for the original state y and adjoint state z h div w norm for the flux state s and adjoint state w without requiring the lbb consistency condition finally some numerical examples are presented to confirm our theoretical results

optimal control of the convection diffusion equation using stabilized finite element methods in this paper we analyze the discretization of optimal control problems governed by convection diffusion equations which are subject to pointwise control constraints we present a stabilization scheme which leads to improved approximate solutions even on corse meshes in the convection dominated case moreover the in general different approaches optimize then discretize and discretize then optimize coincide for the proposed discretization scheme this allows for a symmetric optimality system at the discrete level and optimal order of convergence

continuity in quantitative domains based on the notion of an l fuzzy partially ordered set constructed in an l fuzzy approach to quantitative domain i generalized ordered set valued in frame and adjunction theory fuzzy systems and mathematics 14 2000 6 7 and by introducing the concepts of an l fuzzy directed set and the join of an l fuzzy set w r t the l fuzzy partial order l fuzzy domains are defined and the generalized scott topology on an l fuzzy domain is built this approach is similar to flagg s logic approach to quantitative domain theory a logical approach to quantitative domain theory elsevier amsterdam 1996 submitted for publication in addition the concepts of stratified approximation and a basis for an l fuzzy domain are proposed and a notion of a continuous l fuzzy domain is developed it is proved that if l is a completely distributive lattice in which 1 is irreducible and the well below relation is multiplicative then the stratified interpolation property holds in a continuous l fuzzy domain x e and ax 0 a 1 x x is a base for the generalized scott topology on x e

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 characteristic finite element method for optimal control problems governed by convection diffusion equations in this paper we analyze a characteristic finite element approximation of convex optimal control problems governed by linear convection dominated diffusion equations with pointwise inequality constraints on the control variable where the state and co state variables are discretized by piecewise linear continuous functions and the control variable is approximated by either piecewise constant functions or piecewise linear discontinuous functions a priori error estimates are derived for the state co state and the control numerical examples are given to show the efficiency of the characteristic finite element method

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

two conjugate gradient type methods for unsymmetric linear equations we propose two new conjugate gradient type methods for the solution of sparse unsymmetric linear systems we present a new tridiagonalization process for unsymmetric matrices that is closely related to the lanczos process we use orthogonal factorizations of the tridiagonal matrix to derive the new algorithms usymlq and usymqr in the same fashion as symmlq and minres c c paige and m a saunders solution of sparse indefinite systems of linear equations siam j numer anal 12 1975 pp 617 629 for symmetric matrices some numerical results for the new methods and comparisons with other methods are presented

an hdg method for distributed control of convection diffusion pdes abstract we propose a hybridizable discontinuous galerkin hdg method to approximate the solution of a distributed optimal control problem governed by an elliptic linear convection diffusion pde we use degree k polynomials to approximate the state adjoint state their fluxes and the optimal control and we show the approximations converge with order k 1 in the l 2 norm finally we use a simple element by element postprocessing scheme to obtain new superconvergent approximations of the state dual state and the control we show the postprocessed variables converge with order k 2 in the l 2 norm we present 2d and 3d numerical experiments to illustrate our theoretical results

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

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

generalized conjugate directions abstract this paper presents a simple unifying framework for a wide class of conjugate directions algorithms whose iterates minimize some quadratic functional over a subspace our approach is motivated by its advantages for nonlinear minimization but the purpose of this paper is to present the greatly simplified convergence analysis that results for the linear case

contraction numbers for additive correction methods abstract generic contraction numbers are derived for a class of additive correction methods based on orthogonal projection the only assumption on the range of the projector is that it contains the residual this generalizes previous work of the authors on positive definite operators and provides convergence results for restarted generalized conjugate gradient methods under a variety of conditions

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

the block grade of a block krylov space abstract the aim of the paper is to compile and compare basic theoretical facts on krylov subspaces and block krylov subspaces many krylov sub space methods for solving a linear system ax b have the property that in exact computer arithmetic the true solution is found after iterations where is the dimension of the largest krylov subspace generated by a from r 0 the residual of the initial approximation x 0 this dimension is called the grade of r 0 with respect to a though the structure of block krylov subspaces is more complicated than that of ordinary krylov subspaces we introduce here a block grade for which an analogous statement holds when block krylov space methods are applied to linear systems with multiple say s right hand sides in this case the s initial residuals are bundled into a matrix r 0 with s columns the possibility of linear dependence among columns of the block krylov matrix r 0 ar 0 a 1 r 0 which in practical algorithms calls for the deletion or deflation of some columns requires extra care relations between grade and block grade are also established as well as relations to the corresponding notions of a minimal polynomial and its companion matrix

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

a survey of numerical techniques for solving singularly perturbed ordinary differential equations this survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984 2000 the work done in this area during the period 1905 1984 has already been surveyed by the first author of this paper see appl math comput 30 1989 223 for details due to the space constraints we have covered only singularly perturbed one dimensional problems

error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problem abstract in this paper we study the fully discrete mixed finite element methods for quadratic convex optimal control problem governed by semilinear parabolic equations the space discretization of the state variable is done using usual mixed finite elements whereas the time discretization is based on difference methods the state and the co state are approximated by the lowest order raviart thomas mixed finite element spaces and the control is approximated by piecewise constant elements by applying some error estimates techniques of mixed finite element methods we derive a priori error estimates both for the coupled state and the control approximation finally we present a numerical example which confirms our theoretical results

a comparison of direct and preconditioned iterative techniques for sparse unsymmetric systems of linear equations in this paper we compare direct and preconditioned iterative methods for the solution of nonsymmetric sparse systems of linear algebraic equations these problems occur in finite difference and finite element simulations of semiconductor devices and fluid flow problems we consider five iterative methods that appear to be the most promising for this class of problems the biconjugate gradient method the conjugate gradient squared method the generalized minimal residual method the generalized conjugate residual method and the method of orthogonal minimization each of these methods was tested using similar preconditioning incomplete lu factorization on a set of large sparse matrices arising from finite element simulation of semiconductor devices results are shown where we compare the computation time and memory requirements for each of these methods against one another as well as against a direct method that uses lu factorization to solve these problems the results of our numerical experiments show that preconditioned iterative methods are a practical alternative to direct methods in the solution of large sparse systems of equations and can offer significant savings in storage and cpu time

a stabilized mixed finite element approximation of bilinear state optimal control problems abstract we propose a stabilized mixed finite element approximation for optimal control problems governed by bilinear state equations it is proved that the resulting mixed bilinear formulation is coercive and also continuous which avoids the difficulty in choosing the mixed finite element spaces i e the ladyzhenkaya babuska brezzi matching condition for the mixed finite element spaces is unnecessary under pointwise bilateral constraint on the control variable we deduce the optimality conditions at both continuous and discrete levels for the optimal control problems under consideration then an a priori error analysis in a weighted norm is discussed with relatively low regularity requirements for the solutions to the optimal control problems finally numerical experiments are given to confirm the efficiency and reliability of the proposed stabilized mixed finite element method

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

the lanczos biorthogonalization algorithm and other oblique projection methods for solving large unsymmetric systems many powerful methods for solving systems of equations can be regarded as projection methods most of the projection methods known for solving linear systems are orthogonal projection methods but little attention has been given to the class of nonorthogonal or oblique projection methods which is particularly attractive for large nonsymmetric systems the purpose of the paper is to present some methods in the general setting of oblique projection methods and to give some theoretical results some experiments comparing the various algorithms are reported

numerical experiments of some krylov subspace methods for black oil model this paper discusses preconditioned krylov subspace methods for solving large scale linear systems that originate from oil reservoir numerical simulations two types of preconditioners one being based on an incomplete lu decomposition and the other being based on iterative algorithms are used together in a combination strategy in order to achieve an adaptive and efficient preconditioner numerical tests show that different krylov subspace methods combining with appropriate preconditioners are able to achieve optimal performance

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

on the lanczos method for solving symmetric linear systems with several right hand sides this paper analyzes a few methods based on the lanczos algorithm for solving large sparse symmetric linear systems with several right hand sides the methods examined are suitable for the situation when the right sides are not too different from one another as is often the case in time dependent or parameter dependent problems we propose a theoretical error bound for the approximation obtained from a projection process onto a krylov subspace generated from processing a previous right hand side

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

poisson equation solver with fourth order accuracy by using interpolated differential operator scheme abstract a poisson equation solver with fourth order spatial accuracy has been developed by using the interpolated differential operator ido scheme the number of grids required for obtaining the results of numerical computation with the same accuracy as that by the second order center finite difference method is drastically reduced the consistency is confirmed to the use of the multigrid method and the red black algorithm the decrease of the cpu time by the multigrid method and by the parallel computing indicates the applicability of the ido poisson solver to large scale problems

a priori and a posteriori error analysis of edge stabilization galerkin method for the optimal control problem governed by convection dominated diffusion equation in this paper we study an edge stabilization galerkin approximation scheme for the constrained optimal control problem governed by convection dominated diffusion equation the method uses least square stabilization of the gradient jumps across element edges a priori and a posteriori error estimates are derived for both the state co state and the control the theoretical results are illustrated by two numerical experiments

breakdowns in the implementation of the lanczos method for solving linear systems abstract the lanczos method for solving systems of linear equations is based on formal orthogonal polynomials its implementation is realized via some recurrence relationships between polynomials of a family of orthogonal polynomials or between those of two adjacent families of orthogonal polynomials a division by zero can occur in such recurrence relations thus causing a breakdown in the algorithm which has to be stopped in this paper two types of breakdowns are discussed the true breakdowns which are due to the nonexistence of some polynomials and the ghost breakdowns which are due to the recurrence relationship used among all the recurrence relationships which can be used and all the algorithms for implementing the lanczos method which came out from them the only reliable algorithm is lanczos orthodir which can only suffer from true breakdowns it is shown how to avoid true breakdowns in this algorithm other algorithms are also discussed and the case of near breakdown is treated the same treatment applies to other methods related to lanczos

boundary and interface conditions within a finite element preconditioner for spectral methods abstract the performances of a finite element preconditioner in the iterative solution of spectral collocation schemes for elliptic boundary value problems is investigated it is shown how to make the preconditioner cheap by adi iterations and how to take advantage of the finite element properties in enforcing neumann and interface conditions in the spectral schemes

on conjugate gradient type methods and polynomial preconditioners for a class of complex non hermitian matrices we consider conjugate gradient type methods for the solution of large linear systemsa x b with complex coefficient matrices of the typea t i i wheret is hermitian and a real scalar three different conjugate gradient type approaches with iterates defined by a minimal residual property a galerkin type condition and an euclidean error minimization respectively are investigated in particular we propose numerically stable implementations based on the ideas behind paige and saunder s symmlq and minres for real symmetric matrices and derive error bounds for all three methods it is shown how the special shift structure ofa can be preserved by using polynomial preconditioning and results on the optimal choice of the polynomial preconditioner are given also we report on some numerical experiments for matrices arising from finite difference approximations to the complex helmholtz equation

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 second order characteristic finite element scheme for convection diffusion problems a new characteristic finite element scheme is presented for it is of second order accuracy in time increment symmetric and unconditionally stable optimal error estimates are proved in the framework of l 2 theory numerical results are presented for two examples which show the advantage of the scheme

section retraction pairs between fuzzy domains in this paper we give the concepts of fuzzy monotone and fuzzy continuous section retraction pairs between fuzzy domains defined in q y zhang c y zheng l fuzzy domain and its related properties fuzzy systems and math 18 2004 1 7 in chinese q y zhang l fan continuity in quantitative domains fuzzy sets and systems 154 2005 118 131 discuss some properties about them and prove that the fuzzy monotone retract of a fuzzy domain is also a fuzzy domain the fuzzy continuous retract of a continuous fuzzy domain is also a continuous fuzzy domain when l is a completely distributive lattice in which is multiplicative and every continuous fuzzy domain is a fuzzy continuous retract of an algebraic fuzzy domain when l is a completely distributive lattice in which 1 is irreducible and is multiplicative

performance of certain iterative methods in solving implicit difference schemes for 2 d navier stokes equations the performance of three iterative methods local sor preconditioned generalized conjugate gradient methods such as orthomin 1 and the gcg ls method with preconditioning matrices derived by incomplete modified and unmodified pointwise factorizations of the matrix corresponding to implicit finite difference schemes for the 2 d navier stokes equations is compared the relative merits of these methods in terms of cpu time average number of iterations with respect to the reynolds number and discretization parameters are emphasized

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

experiments with conjugate gradient algorithms for homotopy curvetracking there are algorithms for finding zeros or fixed points of nonlinear systems of equations that are globally convergent for almost all starting points i e with probability one the essence of all such algorithms is the construction of an appropriate homotopy map and then tracking some smooth curve in the zero set of this homotopy map hompack is a mathematical software package implementing globally convergent homotopy algorithms with three different techniques for tracking a homotopy zero curve and has separate routines for dense and sparse jacobian matrices the hompack algorithms for sparse jacobian matrices use a preconditioned conjugate gradient algorithm for the computation of the kernal of the homotopy jacobian matrix a required linear algebra step for homotopy curve tracking here variants of the conjugate gradient algorithm are implemented in the context of homotopy curve tracking and compared with craig s preconditioned conjugate gradient method used in hompack the test problems used include actual large scale sparse structural mechanics problems

convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics abstract a nonlinear system of two coupled partial differential equations models miscible displacement of one incompressible fluid by another in a porous medium conservation of mass for the mixture leads to an elliptic equation for pressure and conservation for the displacing fluid yields a convectiondominated parabolic equation for the concentration of that fluid a sequential implicit time stepping procedure is defined in which the pressure and darcy velocity of the mixture are approximated simultaneously by a mixed finite element method and the concentration is approximated by a combination of a galerkin finite element method and the method of characteristics optimal order convergence in l2 is proved time truncation errors of standard procedures are reduced by time stepping along the characteristics of the hyperbolic part of the concentration equation temporal and spatial errors are lessened by direct computation of the velocity in the mixed method as opposed to differentiation of the pressure several extensions of these results are outlined