postprocessing the galerkin method the finite element case a postprocessing technique developed earlier for spectral methods is extended here to galerkin finite element methods for dissipative evolution partial differential equations the postprocessing amounts to solving a linear elliptic problem on a finer grid or higher order space once the time integration on the coarser mesh is completed this technique increases the convergence rate of the finite element method to which it is applied and this is done at almost no additional computational cost the numerical experiments presented here show that the resulting postprocessed method is computationally more efficient than the method to which it is applied say quadratic finite elements as well as standard methods of similar order of convergence as the postprocessed one say cubic finite elements the error analysis of the new method is performed in l2 and in l infty norms

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

the maximal determinant and subdeterminants of 1 matrices abstract in this paper we study the maximal absolute values of determinants and subdeterminants of 1 matrices especially hadamard matrices it is conjectured that the determinants of 1 matrices of order n can have only the values k p where p is specified from an appropriate procedure this conjecture is verified for small values of n the question of what principal minors can occur in a completely pivoted 1 matrix is also studied an algorithm to compute the n j n j minors j 1 2 of hadamard matrices of order n is presented and these minors are determined for j 1 4

two level method and some a priori estimates in unsteady navier stokes calculations a two level method proposed for quasielliptic problems is adapted in this paper to the simulation of unsteady incompressible navier stokes flows the method requires a solution of a nonlinear problem on a coarse grid and a solution of linear symmetric problem on a fine grid the scaling between these two grids is superlinear approximation stability and convergence aspects of a fully discrete scheme are considered stability properties of the two level scheme are compared with those for a commonly used semi implicit scheme some new estimates are also proved for the latter

local and parallel finite element algorithms based on the partition of unity for the stokes problem by combining the techniques of the two grid method and the partition of unity we derive two local and parallel finite element algorithms for the stokes problem the most interesting features of these algorithms are 1 the partition of unity technique introduces a framework for domain decomposition 2 only a series of local residual problems need to be solved on these subdomains in parallel meanwhile requiring very little communication 3 a globally continuous finite element solution is constructed by combining all the local solutions via the partition of unity functions the optimal error estimates in l 2 and energy norms are proved under some assumptions also several numerical simulations are presented to demonstrate the effectiveness and flexibility of the new algorithms

nonlinear galerkin methods this article presents a new method of integrating evolution differential equations the non linear galerkin method that is well adapted to the long term integration of such equations while the usual galerkin method can be interpreted as a projection of the considered equation on a linear space the methods considered here are related to the projection of the equation on a nonlinear manifold from the practical point of view some terms have been identified as small and sometimes but not always disregarded

an exact two dimensional non guillotine cutting tree search procedure we consider the two dimensional cutting problem of cutting a number of rectangular pieces from a single large rectangle so as to maximize the value of the pieces cut we develop a lagrangean relaxation of a zero one integer programming formulation of the problem and use it as a bound in a tree search procedure subgradient optimization is used to optimize the bound derived from the lagrangean relaxation problem reduction tests derived from both the original problem and the lagrangean relaxation are given incorporating the bound and the reduction tests into a tree search procedure enables moderately sized problems to be solved

a mathematical model and a solution method for the problem of placing various sized circles into a strip abstract a mathematical model of the problem of placing different radii circles into a strip is constructed several peculiarities of the mathematical model are considered on the ground of these peculiarities an original method of transition from one local minimum to another one to provide a decrease of the objective function value is suggested the method is based on the idea of increasing the problem dimension and a reduced gradient method as well as on the concept of active inequalities and the newton method an algorithm to solve the problem is described in detail numerical examples which demonstrate the possibilities of the method are also given

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

an expandable local and parallel two grid finite element scheme an expandable local and parallel two grid finite element scheme based on superposition principle for elliptic problems is proposed and analyzed in this paper by taking example of poisson equation compared with the usual local and parallel finite element schemes the scheme proposed in this paper can be easily implemented in a large parallel computer system that has a lot of cpus convergence results based on h 1 and l 2 a priori error estimation of the scheme are obtained which show that the scheme can reach the optimal convergence orders within ln h 2 or ln h two grid iterations if the coarse mesh size h and the fine mesh size h are properly configured in 2 d or 3 d case respectively some numerical results are presented at the end of the paper to support our analysis

finite element approximation of the nonstationary navier stokes problem part iii smoothing property and higher order error estimates for spatial discretization this paper continues our error analysis of finite element galerkin approximation of the nonstationary navier stokes equations optimal order error estimates both local and global are derived for higher order finite elements under appropriate assumptions about the smoothness and stability of the solution these assumptions take into account the loss of regularity at t 0 that one generally has to expect in the absence of higher order nonlocal compatibility conditions for the data of the problem

stabilized finite element method based on the crank nicolson extrapolation scheme for the time dependent navier stokes equations this paper provides an error analysis for the crank nicolson extrapolation scheme of time discretization applied to the spatially discrete stabilized finite element approximation of the two dimensional time dependent navier stokes problem where the finite element space pair x h m h for the approximation u n h p n h of the velocity u and the pressure p is constructed by the low order finite element the q 1 p 0 quadrilateral element or the p 1 p 0 triangle element with mesh size h error estimates of the numerical solution u n h p n h to the exact solution u t n p t n with t n 0 t are derived

postprocessing fourier galerkin method for the navier stokes equations a full discrete two level postprocessing fourier galerkin scheme for the unsteady navier stokes equations with a periodic boundary condition is proposed in this paper by defining a new projection the interaction between the large and small eddies is reflected by the associated space splitting to some extent therefore a weakly coupled system of the large and small eddies is obtained stability and error estimates for the weakly coupled two level scheme are established in the paper the proposed scheme is an effective algorithm because nonlinearity is treated only in the coarse level subspace by solving the coarse level standard galerkin equation and the matrix of the linear algebraic equations arising in each fine level time stepping is a sparse matrix especially for large scale computations as a result the new scheme saves a lot of cpu time and memory in comparison with the standard fourier galerkin method for deriving an approximation of prescribed accuracy

nonlinear galerkin methods the finite elements case with the increase in the computing power and the advent of supercomputers the approximation of evolution equations on large intervals of time is emerging as a new type of numerical problem in this article we consider the approximation of evolution equations on large intervals of time when the space discretization is accomplished by finite elements the algorithm that we propose called the nonlinear galerkin method stems from the theory of dynamical systems and amounts to some approximation of the attractor in the discrete finite elements space essential here is the utilization of incremental unknown which is accomplished in finite elements by using hierarchical bases beside a detailed description of the algorithm the article includes some technical results on finite elements spaces and a full study of the stability and convergence of the method

d optimal designs embedded in hadamard matrices and their effect on the pivot patterns abstract in this paper we develop a new approach for detecting if specific d optimal designs exist embedded in sylvester hadamard matrices specifically we investigate the existence of the d optimal designs of orders 5 6 7 and 8 the problem is motivated to explaining why specific values appear as pivot elements when gaussian elimination with complete pivoting is applied to hadamard matrices using this method and a complete search algorithm we explain for the first time the appearance of concrete pivot values for equivalence classes of hadamard matrices of orders n 12 16 and 20

a two level stochastic collocation method for semilinear elliptic equations with random coefficients in this work we propose a novel two level discretization for solving semilinear elliptic equations with random coefficients motivated by the two grid method for deterministic partial differential equations pdes introduced by xu 1994 our two level stochastic collocation method utilizes a two grid finite element discretization in the physical space and a two level collocation method in the random domain in particular we solve semilinear equations on a coarse mesh t h with a low level stochastic collocation corresponding to the polynomial space p p and solve linearized equations on a fine mesh t h using high level stochastic collocation corresponding to the polynomial space p p we prove that the approximated solution obtained from this method achieves the same order of accuracy as that from solving the original semilinear problem directly by stochastic collocation method with t h and p p the two level method is computationally more efficient than the standard stochastic collocation method for solving nonlinear problems with random coefficients numerical experiments are provided to verify the theoretical results

local and parallel finite element post processing scheme for the stokes problem abstract a local and parallel finite element post processing scheme based on partition of unity method is proposed and analyzed in this paper for the stokes problem firstly a standard galerkin finite element method on a relatively coarse grid is used to obtain the approximation of the lower frequency components secondly the higher frequency components are computed on fine grids by some local and parallel procedure to post process the standard galerkin approximation the motivation of the proposed local and parallel finite element post processing scheme is based on the superposition principle finally to eliminate the effect of the dirichlet boundary conditions which are imposed on the internal artificial boundaries a global coarse grid correction is done to improve the l 2 accuracy of the approximation

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

unconditional convergence and optimal l2 error estimates of the crank nicolson extrapolation fem for the nonstationary navier stokes equations abstract in this paper we study stability and convergence of fully discrete finite element method on large timestep which used crank nicolson extrapolation scheme for the nonstationary navier stokes equations this approach bases on a finite element approximation for the space discretization and the crank nicolson extrapolation scheme for the time discretization it reduces nonlinear equations to linear equations thus can greatly increase the computational efficiency we prove that this method is unconditionally stable and unconditionally convergent moreover taking the negative norm technique we derive the l 2 h 1 unconditionally optimal error estimates for the velocity and the l 2 unconditionally optimal error estimate for the pressure also numerical simulations on unconditional l 2 stability and convergent rates of this method are shown

stability of galerkin and inertial algorithms with variable time step size this paper provides galerkin and inertial algorithms for solving a class of nonlinear evolution equations spatial discretization can be performed by either spectral or finite element methods time discretization is done by euler explicit or euler semi implicit difference schemes with variable time step size moreover the boundedness and stability of these algorithms are studied by comparison we find that the boundedness and stability of inertial algorithm are superior to the ones of galerkin algorithm in the case of explicit scheme and the boundedness and stability of two algorithms are same in the case of semi implicit scheme

the euler implicit explicit scheme for the 2d time dependent navier stokes equations with smooth or non smooth initial data this paper considers the stability and convergence results for the euler implicit explicit scheme applied to the spatially discretized two dimensional 2d time dependent navier stokes equations a galerkin finite element spatial discretization is assumed and the temporal treatment is implicit explict scheme which is implicit for the linear terms and explicit for the nonlinear term here the stability condition depends on the smoothness of the initial data u 0 h i e the time step condition is c 0 in the case of a 2 log h c 0 in the case of a 1 and th 2 c 0 in the case of a 0 for mesh size h and some positive constant c 0 we provide the h 2 stability of the scheme under the stability condition with a 0 1 2 and obtain the optimal h 1 l 2 error estimate of the numerical velocity and the optimal l 2 error estimate of the numerical pressure under the stability condition with a 1 2

finite element methods for the viscous incompressible fluid abstract we consider the numerical implementations of the standard one level finite element method and two level finite element method for the steady and the unsteady navier stokes equations respectively two numerical examples are presented first we solve the steady navier stokes equations in a bounded multiple connected domain by the standard one level finite element method while we calculate the unsteady navier stokes equations by full discrete two level finite element method the numerical results show that the two methods are feasible and the new two level finite element method is significantly effective

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

iterative penalty methods for the steady navier stokes equations abstract this paper presents one level and two level iterative penalty finite element methods to approximate the solutions of steady navier stokes equations first one level iterative penalty finite element method is applied to solve the steady navier stokes equations numerically and its h 1 and l 2 error estimates are derived then two level iterative penalty scheme is given and its error estimates are obtained for velocity and pressure finally the numerical results are displayed to verify the theoretical analysis

nonlinear galerkin methods and mixed finite elements two grid algorithms for the navier stokes equations a nonlinear galerkin method using mixed finite elements is presented for the two dimensional incompressible navier stokes equations the scheme is based on two finite element spaces x h and x h for the approximation of the velocity defined respectively on one coarse grid with grid size h and one fine grid with grid size h ll h and one finite element space m h for the approximation of the pressure nonlinearity and time dependence are both treated on the coarse space we prove that the difference between the new nonlinear galerkin method and the standard galerkin solution is of the order of h 2 both in velocity h 1 omega and pressure l 2 omega norm we also discuss a penalized version of our algorithm which enjoys similar properties

a new second order nonconforming mixed finite element scheme for the stationary stokes and navier stokes equations abstract in this paper a new stable nonconforming mixed finite element scheme with second order accuracy is proposed for the stationary stokes and navier stokes equations in which a new nonconforming rectangular element is taken for approximating space the velocity and the bilinear element for the pressure the optimal error estimates for the approximation of both the velocity and the pressure in l 2 norm are established as well as one in a broken h 1 norm for the velocity numerical experiments are given which are consistent with our theoretical analysis

convergence of three iterative methods based on the finite element discretization for the stationary navier stokes equations abstract this paper considers three iterative methods for solving the stationary navier stokes equations iterative method i consists in solving the stationary stokes equations iterative method ii consists in solving the stationary linearized navier stokes equations and iterative method iii consists in solving the stationary oseen equations under the finite element discretization respectively at each iterative step also we discuss the stability and convergence of three iterative methods the iterative methods i and ii are stability and convergence under the strong uniqueness conditions where the iterative method ii is the second order convergence furthermore the iterative method iii is uncondition stability and convergence under the uniqueness condition finally some numerical tests show that the efficiency of the theoretical analysis

a formulation space search heuristic for packing unequal circles in a fixed size circular container abstract in this paper we consider the problem of packing unequal circles in a fixed size circular container where the objective is to maximise the value of the circles packed we consider two different objectives maximise the number of circles packed maximise the area of the circles packed for the particular case when the objective is to maximise the number of circles packed we prove that the optimal solution is of a particular form we present a heuristic for the problem based upon formulation space search computational results are given for a number of publicly available test problems involving the packing of up to 40 circles we also present computational results for test problems taken from the literature relating to packing both equal and unequal circles

tangent space correction method for the galerkin approximation based on two grid finite element abstract a novel fully discrete two grid finite element method for certain semi linear parabolic equations is presented in this paper the new scheme is based on two different finite element subspaces defined respectively on one coarse grid with grid size h and one fine grid with grid size h h nonlinearity is treated in the coarse grid subspace by solving the standard galerkin equation while in the fine grid subspace only a linear equation has to be solved differing from the usual two grid method the splitting of the coarse grid subspace and its related fine grid incremental subspace is based on a new projection in which sense the incremental subspace is closely identified with the tangent space of certain operator on each time step with linear finite element discretization the stability and error estimate results for such new scheme are derived the results show that the difference between this new approximation and the fine grid standard galerkin approximation in h 1 norm is of the order h 4 d 2 where d 1 2 3 is the space dimension

error estimates on a new nonlinear galerkin method based on two grid finite elements a new nonlinear galerkin method based on finite element discretization is presented in this paper for semilinear parabolic equations the new scheme is based on two different finite element spaces defined respectively on one coarse grid with grid size h and one fine grid with grid size h ll h nonlinearity and time dependence are both treated on the coarse space and only a fixed stationary equation needs to be solved on the fine space at each time with linear finite element discretizations it is proved that the difference between the new nonlinear galerkin solution and the standard galerkin solution in h 1 omega norm is of the order of h 3

long term stability estimates and existence of a global attractor in a finite element approximation of the navier stokes equations with numerical subgrid scale modeling variational multiscale methods lead to stable finite element approximations of the navier stokes equations dealing with both the indefinite nature of the system pressure stability and the velocity stability loss for high reynolds numbers these methods enrich the galerkin formulation with a subgrid component that is modeled in fact the effect of the subgrid scale on the captured scales has been proved to dissipate the proper amount of energy needed to approximate the correct energy spectrum thus they also act as effective large eddy simulation turbulence models and allow one to compute flows without the need to capture all the scales in the system in this article we consider a dynamic subgrid model that enforces the subgrid component to be orthogonal to the finite element space in the l 2 sense we analyze the long term behavior of the algorithm proving the existence of appropriate absorbing sets and a compact global attractor the improvements with respect to a finite element galerkin approximation are the long term estimates for the subgrid component which are translated to effective pressure and velocity stability thus the stabilization introduced by the subgrid model into the finite element problem does not deteriorate for infinite time intervals of computation

solving circle packing problems by global optimization numerical results and industrial applications abstract a general circle packing is an optimized arrangement of n arbitrary sized circles inside a container e g a rectangle or a circle such that no two circles overlap in this paper we present several circle packing problems review their industrial applications and some exact and heuristic strategies for their solution we also present illustrative numerical results using generic global optimization software packages our work highlights the relevance of global optimization in solving circle packing problems and points towards the necessary advancements in both theory and numerical practice

iterated tabu search for the circular open dimension problem abstract this paper investigates the circular open dimension problem codp which consists of packing a set of circles of known radii into a strip of fixed width and unlimited length without overlapping the objective is to minimize the length of the strip in this paper codp is solved by a series of sub problems each corresponding to a fixed strip length for each sub problem an iterated tabu search approach named its is proposed its starts from a randomly generated solution and attempts to gain improvements by a tabu search procedure after that if the obtained solution is not feasible a perturbation operator is subsequently employed to reconstruct the incumbent solution and an acceptance criterion is implemented to determine whether or not accept the perturbed solution as a supplementary method the length of the strip is determined in monotonously decreasing way with the aid of some post processing techniques the search terminates and returns the best found solution after the allowed computation time has been elapsed computational experiments based on numerous well known benchmark instances show that its produces quite competitive results with respect to the best known results while the computational time remains reasonable for each instance

a new nonlinear galerkin finite element method for the computation of reaction diffusion equations abstract to get the stationary patterns in the approximation of reaction diffusion systems the computation of evolution equations on large intervals of time is required in this paper a new nonlinear galerkin method based on finite element discretization is presented for the approximation of reaction diffusion equations on large intervals the new scheme is based on two different finite element spaces defined respectively on one subspace of lower degree shape function and one of higher degree shape function nonlinearity and time dependence are both treated on the lower degree space and only a fixed stationary equation needs to be solved on the higher degree space at each time level the proof of the stability and convergence of the method is presented in one dimension numerical solutions of some well studied reaction diffusion systems are presented to demonstrate the effectiveness of the nonlinear galerkin method

a small eddy correction method for nonlinear dissipative evolutionary equations considering the interaction between the large and small eddy components of solutions and using the idea of the newton iteration a small eddy correction method is proposed for approximating and numerically solving nonlinear dissipative pdes of parabolic type in particular the navier stokes equations nse we assume that the large eddy approximation to the solution is known formally applying the newton iterative procedure to the small eddy equation we then generate approximate systems it is shown that the first two steps in fact lead to the standard galerkin method sgm and the so called optimum nonlinear galerkin method ong and therefore the small eddy correction method is actually a certain generalization of sgm and ong the boundedness and convergence analysis are presented in the framework of the two dimensional nse the results show that the small eddy correction method can greatly improve the accuracy of sgm approximate solutions

packing different sized circles into a rectangular container abstract this paper is motivated by the problem of fitting pipes of different diameters into a shipping container here we study the subproblem of fitting circles of different sizes into a rectangle since that problem is a central part of the larger problem we formulate this situation as a nonlinear mixed integer programming problem and develop a number of heuristic procedures for approximately solving this problem the heuristics are based on a variety of solution building rules that emulate the process of packing a container some of these methods including a genetic algorithm were based on a more structured design intended to provide solutions which are stable from a stowage viewpoint these heuristics are described in detail and their relative performances are compared for a sample set of 66 randomly generated problems based on this sample the best performing heuristic methods were a quasi random technique and a genetic algorithm of the stable solution structure

a second order decoupled implicit explicit method of the 3d primitive equations of ocean ii finite element spatial discretization summary a fully discrete second order decoupled implicit explicit method is proposed for solving 3d primitive equations of ocean in the case of dirichlet boundary conditions on the side where a second order decoupled implicit explicit scheme is used for time discretization and a finite element method based on the p1 p1 p1 p1 p1 elements for velocity pressure and density is used for spatial discretization of these primitive equations optimal h1 l2 h1 error estimates for numerical solution uhn phn hn and an optimal l2 error estimate for uhn hn are established under the convergence condition of 0 h 1 0 2 and 3h for some positive constants 1 2 and 3 furthermore numerical computations show that the h1 l2 h1 convergence rate for numerical solution uhn phn hn is of o h 2 and an l2 convergence rate for uhn hn is o h2 2 with the assumed convergence condition where h is a mesh size and is a time step size more practical calculations are performed as a further validation of the numerical method copyright 2016 john wiley sons ltd

evaluation of minors associated to weighing matrices in the present paper we focus our research on calculating minors of weighing matrices of order n and weight n k denoted by w n n k we provide analytical determinant computations counting techniques for specifying the existence of certain submatrices inside a w n n k and an algorithm for computing the n j n j minors of a w n n k which is realized with the notion of symbolic manipulation these results are valid of general n the ideas presented in this work can be used as the fundamental basis on which the calculation of minors of other weighing matrices and in general of orthogonal matrices can be developed an application of the derived formulas to an interesting problem of numerical analysis the growth problem is also presented

a penalty finite element method based on the euler implicit explicit scheme for the time dependent navier stokes equations a fully discrete penalty finite element method is presented for the two dimensional time dependent navier stokes equations where the time discretization is based on the euler implicit explicit scheme with some implicit linear terms and an explicit nonlinear term and the finite element spatial discretization is based on the p 1b p 1 element pair which satisfies the discrete inf sup condition this method allows us to separate the computation of the velocity from the computation of the pressure with a larger time step size dt so that the numerical velocity u e h n and the pressure p e h n are easily computed an optimal error estimate of the numerical velocity and the pressure is provided for the fully discrete penalty finite element method when the penalty parameter e the time step size dt and the mesh size h satisfy the following stability conditions ec 1 1 dt k 1 1 and h 2 b 1 dt respectively for some positive constants c 1 k 1 and b 1 finally some numerical tests to confirm the theoretical results of the penalty finite element method are provided

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

stabilized finite element approximation of transient incompressible flows using orthogonal subscales abstract in this paper we present a stabilized finite element method to solve the transient navier stokes equations based on the decomposition of the unknowns into resolvable and subgrid scales the latter are approximately accounted for so as to end up with a stable finite element problem which in particular allows to deal with convection dominated flows and the use of equal velocity pressure interpolations three main issues are addressed the first is a method to estimate the behavior of the stabilization parameters based on a fourier analysis of the problem for the subscales secondly the way to deal with transient problems discretized using a finite difference scheme is discussed finally the treatment of the nonlinear term is also analyzed a very important feature of this work is that the subgrid scales are taken as orthogonal to the finite element space in the transient case this simplifies considerably the numerical scheme

an efficient two step algorithm for the stationary incompressible magnetohydrodynamic equations a new highly efficient two step algorithm for the stationary incompressible magnetohydrodynamic equations is studied in this paper the algorithm uses a lower order finite element pair i e p 1 b p 1 p 1 to compute an initial approximation that is using the mini element i e p 1 b p 1 to approximate the velocity and pressure and p1 element to approximate the magnetic field then applies a higher order finite element pair i e p 2 p 1 p 2 to solve a linear system on the same mesh furthermore the convergence analyses of standard galerkin finite element method and the two step algorithm are addressed lastly numerical experiments are presented to verify both the theory and the efficiency of the algorithm

long time unconditional stability of a two level hybrid method for nonstationary incompressible navier stokes equations abstract this paper presents a two level maccormack rapid solver method for solving a 2d time dependent navier stokes problem that models an incompressible fluid flow a decoupling approach based on the coupling term via spatial extrapolation is proposed for devising decoupled marching algorithms for the nonstationary incompressible model in this hybrid method an explicit maccormack scheme provides a h 1 optimal velocity approximation and a l 2 optimal pressure approximation by solving a nonlinear navier stokes problem on a coarse grid with mesh size h while an implicit crank nicolson algorithm consists in dealing with the fully discrete linear generalized stokes problem on a fine grid with mesh size h h the theoretical result suggests that our method is unconditionally stable over long time intervals while the numerical evidences both confirm the theoretic analysis and show that the algorithm is convergent the tests also suggest that our method is both cheaper and faster than the two level finite element galerkin scheme

the coupling of boundary integral and finite element methods for the exterior steady oseen problem abstract in this article we present a new numerical method for solving the steady oseen equations in an unbounded plane domain the technique consists in coupling the boundary integral and the finite element methods an artificial smooth boundary is introduced separating an interior inhomogeneous region from an exterior homogeneous one the solution in exterior region is represented by an integral equation over the artificial boundary this integral equation is incorporated into a velocity pressure formulation for the interior region and a finite element method is used to approximate the resulting variational problem finally the optimal error estimates of the numerical solution are derived computer results will be discussed in a forthcoming paper

nonlinear galerkin finite element method applied to the system of reactiondiffusion equations in one space dimension we study the finite element nonlinear galerkin method in one spatial dimension and its application to the numerical solution of nontrivial dynamics in selected reactiondiffusion systems this method was suggested as well adapted for the long term integration of evolution equations and is studied as an alternative to the commonly used numerical approaches the proof of the convergence of the method applied to a particular class of reactiondiffusion systems is presented computational properties are illustrated by results of numerical simulations we performed the measurement of the experimental order of convergence and the computational efficiency in comparison to the usual finite difference method

a pressure poisson stabilized finite element method for the non stationary stokes equations to circumvent the inf sup condition in this article stabilized finite element methods are considered for the non stationary stokes equations based on some lowest equal order finite elements space pair x h m h which do not satisfy the discrete inf sup condition the stability of two kinds of methods is derived under some regularity assumptions then the convergence of the penalty method and the pressure poisson stabilized method is compared the result shows that the former error limits the order of approximation to o e h e and the latter yields the optimal error estimate o h

nonconforming mixed finite element approximation to the stationary navier stokes equations on anisotropic meshes abstract the main aim of this paper is to study the error estimates of a rectangular nonconforming finite element for the stationary navier stokes equations under anisotropic meshes that is the nonconforming rectangular element is taken as approximation space for the velocity and the piecewise constant element for the pressure the convergence analysis is presented and the optimal error estimates both in a broken h 1 norm for the velocity and in an l 2 norm for the pressure are derived on anisotropic meshes