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 grid method for the two dimensional time dependent schrodinger equation by the finite element method abstract in this paper we construct a backward euler full discrete two grid finite element scheme for the two dimensional time dependent schrodinger equation with this method the solution of the original problem on the fine grid is reduced to the solution of same problem on a much coarser grid together with the solution of two poisson equations on the same fine grid we analyze the error estimate of the standard finite element solution and the two grid solution in the h 1 norm it is shown that the two grid algorithm can achieve asymptotically optimal approximation as long as the mesh sizes satisfy h o h k k 1 finally a numerical experiment indicates that our two grid algorithm is more efficient than the standard finite element method

a multi level discontinuous galerkin method for solving the stationary navier stokes equations abstract in this paper we consider the multi level discontinuous finite element method for solving the stationary incompressible navier stokes equations on the coarsest mesh the discrete nonlinear navier stokes equations are solved by using piecewise polynomial functions which are totally discontinuous across inter element boundaries and are pointwise divergence free on each element for the velocity and are continuous functions for the pressure subsequent approximations are generated on a succession of refined grids by solving the newton linearized navier stokes equations using piecewise polynomial functions which are similar to that on the coarsest mesh finally the well posedness and the optimal error estimate for the multi level discontinuous galerkin method are provided the error analysis shows that when the mesh scales k j 1 o k j 2 h j 1 o h j 2 with j 0 1 j 1 are chosen the multi level finite element method can save a large amount of computational time compared with the one level finite element method

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

on the constructions of constant weight codes two methods of constructing binary constant weight codes from 1 codes over gf q and 2 constant weight codes over gf q are presented several classes of binary optimum constant weight codes are derived from these methods in general we show that binary optimum constant weight codes which achieve the johnson bound can be constructed from optimum codes over gf q which achieve the plotkin bound finally several classes of optimum constant weight codes over gf q are constructed

on the first and second variations of a nonlocal isoperimetric problem we consider a nonlocal perturbation of an isoperimetric variational prob lem the problem may be viewed as a mathematical paradigm for the ubiquitous phenom enon of energy driven pattern formation associated with competing short and long range interactions in particular it arises as a g limit of a model for microphase separation of diblock copolymers in this article we establish precise conditions for criticality and stability i e we explicitly compute the first and second variations we also present some applications

a two grid block centered finite difference method for darcy forchheimer flow in porous media a two grid block centered finite difference method is proposed for solving the two dimensional darcy forchheimer model describing non darcy flow in porous media to construct the two grid method we modify the original nonlinear elliptic operator of darcy forchheimer flow to a twice continuously differentiable one by introducing a small and positive parameter varepsilon by using the two grid method solving a nonlinear equation on a fine grid is reduced to solving a nonlinear equation on a coarse grid together with solving a linear equation on a fine grid optimal order error estimates for pressure and velocity in discrete l 2 norms are obtained some numerical examples are given to show the accuracy and efficiency of the presented method

equilibrium configurations of epitaxially strained elastic films second order minimality conditions and qualitative properties of solutions we consider a variational model introduced in the physical literature to describe the epitaxial growth of an elastic film over a thick flat substrate when a lattice mismatch between the two materials is present we study quantitative and qualitative properties of equilibrium configurations that is of local and global minimizers of the free energy functional more precisely we determine analytically the critical threshold for the local minimality of the flat configuration and we also prove several results concerning its global minimality the non occurrence of singularities in non flat global minimizers is also addressed one of the main results of the paper is a new sufficient condition for local minimality which provides the first extension of the classical criteria based on the positivity of the second variation to the context of functionals with bulk and surface energies

constructions for q ary constant weight codes this paper introduces a new combinatorial construction for q ary constant weight codes which yields several families of optimal codes and asymptotically optimal codes the construction reveals intimate connection between q ary constant weight codes and sets of pairwise disjoint combinatorial designs of various types

bounds and constructions for ternary constant composition codes the problem of determining the maximum size of a ternary code is considered under the restriction that each symbol should appear a given number of times in each codeword upper and lower bounds on the size of such codes under the hamming metric are discussed where the lower bounds follow from constructions of good codes some of the results are obtained by explicitly finding codes by computer search a table of exact values and best known bounds on the maximum size for codes of length at most 10 is presented

optimal ternary constant weight codes of weight four and distance six recently chee and ling constructions for q ary constant weight codes ieee trans inf theory vol 53 no 1 135 146 jan 2007 introduced a new combinatorial construction for q ary constant weight codes which reveals a close connection between q ary constant weight codes and sets of pairwise disjoint combinatorial designs in this paper we study the problem of constructing optimal ternary constant weight codes with hamming weight four and minimum distance six using this approach the construction here exploits completely reducible super simple designs and group divisible codes the problem is solved leaving only two cases undetermined previously the sizes of constant weight codes of weight four and distance six were known only for those of length no greater than 10

iterative two grid methods for semilinear elliptic equations abstract we design and analyze a new finite element discretization technique based on iterative two grid methods for semilinear elliptic partial differential equations pdes we first present an iterative two grid method that just applies the classical two grid for semilinear elliptic pdes in a successive fashion namely to solve a semilinear problem on the coarse space and then to solve a symmetric positive definite problem on the fine space thus we only need to deal with semilinear problems in the coarse space in the first algorithm secondly we replace the semilinear term by using the corresponding first order taylor expansion in the second iterative two grid we provide the error estimates for the second iterative algorithm and present numerical experiments to show the efficiency of our methods

the sizes of optimal q ary codes of weight three and distance four a complete solution this correspondence introduces two new constructive techniques to complete the determination of the sizes of optimal q ary codes of constant weight three and distance four

trust region methods for nonlinear elliptic equations with radial basis functions we consider the numerical solution of nonlinear elliptic boundary value problems with kansas method we derive analytic formulas for the jacobian and hessian of the resulting nonlinear collocation system and exploit them within the framework of the trust region algorithm this ansatz is tested on semilinear quasilinear and fully nonlinear elliptic pdes including plateaus problem heleshaw flow and the mongeampre equation with excellent results the new approach distinctly outperforms previous ones based on linearization or finite difference jacobians

investigations on two kinds of two level stabilized finite element methods for the stationary navier stokes equations abstract this article considers two kinds of two level stabilized finite element methods based on local gauss integral technique for the two dimensional stationary navier stokes equations approximated by the lowest equal order elements which do not satisfy the inf sup condition the two level methods consist of solving a small non linear system on the coarse mesh and then solving a linear system on the fine mesh the error analysis shows that the two level stabilized finite element methods provide an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution solving the navier stokes equations on a fine mesh for a related choice of mesh widths therefore the two level methods are of practical importance in scientific computation finally the performance of two kinds of two level stabilized methods are compared in efficiency and precision aspects by a series of numerical experiments the conclusion is that simple two level stabilized method is a viable choice for the lowest equal order approximations of the stationary navier stokes problem

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

two grid scheme for characteristics finite element solution of nonlinear convection diffusion problems abstract a two grid scheme for characteristics finite element solution of nonlinear convection dominated diffusion equations was constructed the l 2 and h 1 error estimates of the scheme were derived a numerical example was also presented the scheme involves solving one small nonlinear problem on the coarse grid system one linear problem on the fine grid system the error estimates and the numerical example show that the new scheme is efficient to the nonlinear convection dominated diffusion equations

a two grid method for incompressible miscible displacement problems by mixed finite element and eulerian lagrangian localized adjoint methods abstract in this paper we present a scheme for solving two dimensional miscible displacement problems using eulerian lagrangian localized adjoint methods and mixed finite element methods since only the velocity and not the pressure appears explicitly in the concentration equation an eulerian lagrangian localized adjoint method is used to solve the concentration equation and a mixed finite element method is used for the pressure equation to linearize and decouple the mixed method equations we use a two grid algorithm based on the newton iteration method for this fully discrete problems first we solve the original nonlinear equations on the coarse grid then we solve the linearized problem on the fine grid using newton iteration once it is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy h o h 1 2 in this paper

a two grid eulerian lagrangian localized adjoint method to miscible displacement problems with dispersion term abstract in this paper we propose a novel two grid method of eulerian lagrangian localized adjoint method ellam mixed finite element method mfem solutions for nonlinear and coupled miscible displacement problems which describe complex fluid flow processes in porous media an ellam is used to solve the nonlinear convection diffusion equation for concentration and we present a mfem to compute the pressure equation for the pressure and darcy velocity a two grid scheme is investigated to linearize and decouple the mixed method equations it is illustrated that the coarse space can be selected as coarse as h o h 1 2 and the asymptotically optimal approximation as the nonlinear schemes can be still achieved as a result a large class of such nonlinear and coupled problems can be solved efficiently with large time steps

an effective explicit pressure gradient scheme implemented in the two level non staggered grids for incompressible navier stokes equations in this paper an improved two level method is presented for effectively solving the incompressible navier stokes equations this proposed method solves a smaller system of nonlinear navier stokes equations on the coarse mesh and needs to solve the oseen type linearized equations of motion only once on the fine mesh level within the proposed two level framework a prolongation operator which is required to linearize the convective terms at the fine mesh level using the convergent navier stokes solutions computed at the coarse mesh level is rigorously derived to increase the prediction accuracy this indispensable prolongation operator can properly communicate the flow velocities between the two mesh levels because it is locally analytic solution convergence can therefore be accelerated for the sake of numerical accuracy momentum equations are discretized by employing the general solution for the two dimensional convection diffusion reaction model equation the convective instability problem can be simultaneously eliminated thanks to the proper treatment of convective terms the converged solution is thus very high in accuracy as well as in yielding a quadratic spatial rate of convergence for the sake of programming simplicity and computational efficiency pressure gradient terms are rigorously discretized within the explicit framework in the non staggered grid system the proposed analytical prolongation operator for the mapping of solutions from the coarse to fine meshes and the explicit pressure gradient discretization scheme which accommodates the dispersion relation preserving property have been both rigorously justified from the predicted navier stokes solutions

optimal quaternary constant weight codes with weight four and distance five constant weight codes play an important role in coding theory the problem of determining the sizes for optimal quaternary constant weight codes with length n weight 4 and minimum hamming distance 5 n 5 4 4 codes has been investigated in several papers although some constructions and several infinite families for such codes with length n 0 1 mod 4 have been given the problem is still far from complete in this paper we determine the size of an optimal n 5 4 4 code for each integer n 4 leaving 55 lengths unsolved especially for length n 0 1 mod 4 the existence problem of the equivalent combinatorial object namely the generalized steiner system is solved leaving only seven values undetermined

a new two grid nonconforming mixed finite element method for nonlinear benjamin bona mahoney equation abstract a new low order two grid mixed finite element method fem is developed for the nonlinear benjamin bona mahoney bbm equation in which the famous nonconforming rectangular c n q 1 r o t element and q0 q0 constant element are used to approximate the exact solution u and the variable p u t respectively then based on the special properties of these two elements and interpolation post processing technique the superconvergence results for u in broken h1 norm and p in l2 norm are obtained for the semi discrete and crank nicolson fully discrete schemes without the restriction between the time step and coarse mesh size h or the fine mesh size h which improve the results of the existing literature finally some numerical results are provided to confirm the theoretical analysis

minimality via second variation for a nonlocal isoperimetric problem we discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts we show that critical configurations with positive second variation are local minimizers of the nonlocal area functional and in fact satisfy a quantitative isoperimetric inequality with respect to sets that are l 1 close the link with local minimizers for the diffuse interface ohta kawasaki energy is also discussed as a byproduct of the quantitative estimate we get new results concerning periodic local minimizers of the area functional and a proof via second variation of the sharp quantitative isoperimetric inequality in the standard euclidean case as a further application we address the global and local minimality of certain lamellar configurations

a two grid discretization scheme for eigenvalue problems a two grid discretization scheme is proposed for solving eigenvalue problems including both partial differential equations and integral equations with this new scheme the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy

error estimates of two level finite element method for smagorinsky model two level finite element method for simulating smagorinsky model in large eddy simulation is investigated in this two level algorithm a linearized discrete problem is solved by using the variational multiscale method on the coarse mesh corresponding to newton linearization method the linearized smagorinsky model is solved on the fine mesh the error estimates derived in this paper implies that by choosing appropriate mesh sizes and the radius of the spatial filter used in smagorinsky model the two level method provides the optimal convergence rates for the velocity in h1 norm and the pressure in l2 norm meanwhile the two different numerical experiments are given to support the optimal convergence rates and the high efficiency of two level algorithm

group divisible codes and their application in the construction of optimal constant composition codes of weight three the concept of group divisible codes a generalization of group divisible designs with constant block size is introduced in this paper this new class of codes is shown to be useful in recursive constructions for constant weight and constant composition codes large classes of group divisible codes are constructed which enabled the determination of the sizes of optimal constant composition codes of weight three and specified distance leaving only four cases undetermined previously the sizes of constant composition codes of weight three were known only for those of sufficiently large length

a two level method with backtracking for the navier stokes equations we consider a two level method for resolving the nonlinearity in finite element approximations of the equilibrium navier stokes equations the method yields l2 and h1 optimal velocity approximations and an l2 optimal pressure approximation the two level method involves solving one small nonlinear coarse mesh system one oseen problem hence linear with positive definite symmetric part on the fine mesh and one linear correction problem on the coarse mesh the algorithm we study produces an approximate solution with the optimal asymptotic in h accuracy for any fixed reynolds number we do not consider the behavior of the error for fixed h as re rightarrow infty i e for flows in transition to turbulence

a multi level correction scheme for eigenvalue problems abstractin this paper a new type of multi level correction scheme is proposed forsolving eigenvalue problems by nite element method with this new scheme the accuracy of eigenpair approximations can be improved after each correc tion step which only needs to solve a source problem on ner nite elementspace and an eigenvalue problem on the coarsest nite element space thiscorrection scheme can improve the e ciency of solving eigenvalue problemsby nite element method keywords eigenvalue problem multi level correction scheme nite ele ment method multi space multi grid ams subject classi cations 65n30 65b99 65n25 65l15 1 introduction the purpose of this paper is to propose a new type of multi level correction schemebased on nite element discretization to solve eigenvalue problems the two gridmethod for solving eigenvalue problems has been proposed and analyzed by xu andzhou in 21 the idea of the two grid comes from 19 20 for nonsymmetric orinde nite problems and nonlinear elliptic equations since then there have existedmany numerical methods for solving eigenvalue problems based on the idea of two grid method 1 6 17

optimal ternary constant weight codes with weight 4 and distance 5 constant weight codes cwcs play an important role in coding theory the problem of determining the sizes for optimal ternary cwcs with length n weight 4 and minimum hamming distance 5 n 5 4 3 code has been settled for all positive integers n 10 or n 10 and n 1 mod 3 with n 13 52 58 undetermined in this paper we investigate the problem of constructing optimal n 5 4 3 codes for all lengths n with the tool of group divisible codes we determine the size of an optimal n 5 4 3 code for each integer n 4 leaving the lengths n 12 13 21 27 33 39 45 52 unsolved

equilibria and dislocations in epitaxial growth abstract an overview of recent analytical developments in epitaxial growth is presented the energy release via the onset of dislocations is addressed regularity of quasistatic equilibria is studied both in the absence and in the presence of dislocations morphological evolution of anisotropic epitaxially strained films is considered under the assumption that the surface evolves by surface diffusion under the influence of a chemical potential

a free boundary approach to two dimensional steady capillary gravity water waves we prove that suitable weak solutions of the two dimensional steady capillary gravity water wave problem even in the presence of vorticity are smooth on a technical level solutions are closely related to critical points of the mumford shah functional so that our main task is to exclude cusps pointing into the water phase we prove that cusps do not exist and that the free surface is smooth

stability and convergence of a finite element method for a semilinear elliptical problem with small viscosity abstract in this paper we propose and analyze a new finite element method for a semilinear elliptical problem with small viscosity firstly we prove the existence and uniqueness of solution for its variational problem then we obtain the stability and convergence of the corresponding finite element galerkin approximation furthermore we derive the optimal error estimates in the l 2 and h 1 norms for the finite element approximations respectively finally several numerical experiments are provided to confirm the above theoretical results

a novel two grid method for semilinear elliptic equations a new finite element discretization technique based on two coarse and fine subspaces is presented for a semilinear elliptic boundary value problem the solution of a nonlinear system on the fine space is reduced to the solution of two small one linear and one nonlinear systems on the coarse space and a linear system on the fine space it is shown both theoretically and numerically that the coarse space can be extremely coarse and still achieve asymptotically optimal approximation as a result the numerical solution of such a nonlinear equation is not significantly more expensive than the solution of one single linearized equation

on optimal regularity of free boundary problems and a conjecture of de giorgi we present a general theory to study optimal regularity for a large class of nonlinear elliptic systems satisfying general boundary conditions and in the presence of a geometric transmission condition on the free boundary as an application we give a full positive answer to a conjecture of de giorgi on the analyticity of local minimizers of the mumford shah functional 2004 wiley periodicals inc

a lower bound for ternary constant weight codes we derive a lower bound for ternary constant weight codes with minimum hamming distance three the bound is similar to a bound for binary constant weight codes with minimum distance four described by graham and sloane 1980 it improves upon the gilbert 1952 bound and coincides asymptotically with the johnson 1962 bound for fixed weight as the codeword length tends to infinity

local and parallel finite element algorithms based on two grid discretizations a number of new local and parallel discretization and adaptive nite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems these algorithms are motivated by the observation that for a solution to some elliptic problems low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a ne grid by some local and parallel procedure the theoretical tools for analyzing these methods are some local a priori and a posteriori estimates that are also obtained in this paper for nite element solutions on general shape regular grids some numerical experiments are also presented to support the theory in this paper we will propose some new parallel techniques for nite element computation these techniques are based on our understanding of the local and global properties of a nite element solution to some elliptic problems simply speaking the global behavior of a solution is mostly governed by low frequency components while the local behavior is mostly governed by high frequency compo nents the main idea of our new algorithms is to use a coarse grid to approximate the low frequencies and then to use a ne grid to correct the resulted residue which contains mostly high frequencies by some local parallel procedures let us now give a somewhat more detailed but informal and hopefully infor mative description of the main ideas and results in this paper we consider the following very simple model problem posed on a convex polygonal domain r 2 u bru f in

an adaptive multigrid scheme for bose einstein condensates in a periodic potential we present a novel multigrid continuation method for treating parameter dependent problems the proposed algorithm which can be flexibly implemented is a generalization of the two grid discretization schemes c s chien b w jeng a two grid discretization scheme for semilinear elliptic eigenvalue problems siam j sci comput 27 2006 1287 1304 that is approximating points on a solution curve do not necessarily lie on the same fine grid we apply the algorithm to compute energy levels and superfluid densities of bose einstein condensates bec in a periodic potential both positive and negative scattering lengths are considered in our numerical experiments for positive scattering length if the chemical potential is large enough and the domain is properly chosen the results show that the number of peaks of the first few energy states of the 2d bec in a periodic potential depends on the wave number of the periodic potential moreover for bright solitons the number of peaks of the ground state solutions is 1d 1 2 and 1d 2 where the periodic potential is expressed in terms of the sine or the cosine functions respectively however these formulae do not hold if the scattering length is negative the numerical study is extended to the two component 1d and 2d bec in a periodic potential

long time behavior of the two grid finite element method for fully discrete semilinear evolution equations with positive memory abstract based on two grid discretizations two fully discrete finite element algorithms for semilinear parabolic integro differential equations with positive memory are proposed with the backward euler scheme for the temporal discretization the basic idea of the space two grid finite element algorithms is to approximate the semilinear equations on a coarse space grid and to solve the linearized equations on a finer space grid at each time step to further decreases the amount of computational work a space time two grid algorithm based on a coarse space grid with large time stepsize t and a finer space grid with small time stepsize t for the evolutional equations is proposed in this paper the sharp long time stability and error estimates for the standard finite element method the space two grid finite element method and the space time two grid finite element method are derived it is showed that the two grid algorithms long time stability and error estimates are similar to those of the direct resolution of the semilinear problem on a fine grid

motion of elastic thin films by anisotropic surface diffusion with curvature regularization in this paper we prove short time existence uniqueness and regularity for a surface diffusion evolution equation with curvature regularization in the context of epitaxially strained two dimensional films this is achieved by using the h 1 gradient flow structure of the evolution law via de giorgi s minimizing movements this seems to be the first short time existence result for a surface diffusion type geometric evolution equation in the presence of elasticity

study of island formation in epitaxially strained films on unbounded domains we consider a variational model related to the formation of islands in heteroepitaxial growth on unbounded domains we first derive the scaling regimes of the minimal energy in terms of the volume of the film and the amplitude of the crystallographic misfit for small volumes non existence of minimizers is then proven this corresponds to the experimentally observed wetting effect on the other hand we show the existence of minimizers for large volumes we finally study the asymptotic behavior of the optimal shapes