about the regularizing properties of the non cut off kac equation we prove in this work that under suitable assumptions the solution of the spatially homogeneous non cut off kac equation or of the spatially homogeneous non cut off 2d boltzmann equation with maxwellian molecules in the radial case becomes very regular with respect to the velocity variable as soon as the time is strictly positive

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

entropy dissipation and long range interactions we study boltzmann s collision operator for long range interactions i e without grad s angular cut off assumption we establish a functional inequality showing that the entropy dissipation controls smoothness of the distribution function in a precise sense our estimate is optimal and gives a unified treatment of both the linear and the nonlinear cases we also give simple and self contained proofs of several useful results that were scattered in previous works as an application we obtain several helpful estimates for the cauchy problem and for the landau approximation in plasma physics

fractional hadamard powers of positive semidefinite matrices we consider the class sn of all real positive semidefinite n n matrices and the subclass sn of all a sn with non negative entries for a positive non integer number and some a sn when will the fractional hadamard power a again belong to sn it is known that for a specific this holds for all a sn if and only if n 2 now let a sn be of the form a t v where t sn has rank 1 and v sn has rank p 1 if the hadamard quotient of t and v is hadamard independent in general position and v has sufficently small entries then a complete answer is given depending on n p and special attention is given to the case that p 1

rank of a hadamard product abstract we provide a new proof of a sharp lower bound for the rank of the hadamard product of positive semidefinite matrices our proof uses standard matrix analysis tools and involves the rank of one factor and the kruskal rank of the other it leads to a new understanding of when the hadamard exponential of a positive semidefinite matrix can be singular

a two grid finite difference scheme for nonlinear parabolic equations we present a two level finite difference scheme for the approximation of nonlinear parabolic equations discrete inner products and the lowest order raviart thomas approximating space are used in the expanded mixed method in order to develop the finite difference scheme analysis of the scheme is given assuming an implicit time discretization in this two level scheme the full nonlinear problem is solved on a coarse grid of size h the nonlinearities are expanded about the coarse grid solution and an appropriate interpolation operator is used to provide values of the coarse grid solution on the fine grid in terms of superconvergent node points the resulting linear but nonsymmetric system is solved on a fine grid of size h some a priori error estimates are derived which show that the discrete l infty l2 and l2 h1 errors are o h 2 h 4 d 2 delta t where d geq 1 is the spatial dimension

time stepping along characteristics with incomplete iteration for a galerkin approximation of miscible displacement in porous media miscible displacement of one incompressible fluid by another in a porous medium is modeled by a nonlinear coupled system of two partial differential equations the pressure equation is elliptic while the concentration equation is parabolic but normally convection dominated a sequential backward difference time stepping scheme is defined it approximates the pressure by a standard galerkin procedure and the concentration by a combination of a galerkin method and the method of characteristics optimal convergence rates in l 2 and h 1 are demonstrated for this scheme and for a modified version in which the algebraic equations at each time step are solved approximately by a limited number of preconditioned conjugate gradient iterations time stepping along the characteristics of the hyperbolic part of the concentration equation is shown to result in smaller time truncation errors than those of standard methods numerical results published elsewhere have confirmed that larger time steps are appropriate

some errors estimates for the box method we define and analyze several variants of the box method for discretizing elliptic boundary value problems in the plane our estimates show the error to be comparable to a standard galerkin finite element method using piecewise linear polynomials

mixed finite elements for elliptic problems with tensor coefficients as cell centered finite differences we present an expanded mixed finite element approximation of second order elliptic problems containing a tensor coefficient the mixed method is expanded in the sense that three variables are explicitly approximated namely the scalar unknown the negative of its gradient and its flux the tensor coefficient times the negative gradient the resulting linear system is a saddle point problem in the case of the lowest order raviart thomas elements on rectangular parallelepipeds we approximate this expanded mixed method by incorporating certain quadrature rules this enables us to write the system as a simple cell centered finite difference method requiring the solution of a sparse positive semidefinite linear system for the scalar unknown for a general tensor coefficient the sparsity pattern for the scalar unknown is a 9 point stencil in two dimensions and 19 points in three dimensions existing theory shows that the expanded mixed method gives optimal order approximations in the l 2 and h s norms and superconvergence is obtained between the l 2 projection of the scalar variable and its approximation we show that these rates of convergence are retained for the finite difference method if h denotes the maximal mesh spacing then the optimal rate is o h the superconvergence rate o h 2 is obtained for the scalar unknown and rate o h 3 2 for its gradient and flux in certain discrete norms moreover the full o h 2 is obtained in the strict interior of the domain computational results illustrate these theoretical results

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

on the finite volume element method the finite volume element method fve is a discretization technique for partial differential equations it uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations then restricts the admissible functions to a finite element space to discretize the solution this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with fve based on triangulations with linear finite element spaces and a general type of control volume we establisho h estimates of the error in a discreteh 1 semi norm under an additional assumption of local uniformity of the triangulation the estimate is improved too h 2 results on the effects of numerical integration are also included

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

the modified method of characteristics with adjusted advection the mmoc procedure for approximating the solutions of transport dominated diffusion problems does not automatically preserve integral conservation laws leading to mass balance errors in many kinds of flow problems the variant called the mmocaa discussed herein preserves the conservation law at a minor additional computational cost it is shown that its solution in either galerkin or finite difference form converges at the same rates as were proved earlier by dougl as and russell for the standard mmoc procedure

two grid methods for characteristic finite volume element solution of semilinear convection diffusion equations abstract two grid methods for characteristic finite volume element solutions are presented for a kind of semilinear convection dominated diffusion equations the methods are based on the method of characteristics two grid method and the finite volume element method the nonsymmetric and nonlinear iterations are only executed on the coarse grid with grid size h and the fine grid solution with grid size h can be obtained by a single symmetric and linear step it is proved that the coarse grid can be much coarser than the fine grid the two grid methods achieve asymptotically optimal approximation as long as the mesh sizes satisfy h o h 1 3

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

lp error estimates of two grid schemes of expanded mixed finite element methods abstract we present a scheme for solving two dimensional semi linear reaction diffusion equations using an expanded mixed finite element method to linearize the mixed method equations we use a two grid scheme based on the newton iteration method we give the l p error estimate to the variable p in the two grid solution it is shown that the algorithm achieve asymptotically optimal approximation applying the two grid methods as long as the mesh sizes satisfy h o h 1 3 for semi linear problems

two grid method for miscible displacement problem by mixed finite element methods and finite element method of characteristics abstract the miscible displacement of one incompressible fluid by another in a porous medium is governed by a system of two equations one is elliptic form equation for the pressure and the other is parabolic form equation for the concentration of one of the fluids since only the velocity and not the pressure appears explicitly in the concentration equation we use a mixed finite element method for the approximation of the pressure equation and finite element method with characteristics for the concentration equation to linearize this full discrete scheme problems we use two newton iterations on the fine grid in our methods with the initial guess coming from the coarse grid solution then we get the error estimates for the three step two grid algorithms it is showed that coarse space can be extremely coarse and we achieve asymptotically optimal approximation as long as the mesh sizes satisfy h o h 1 4 in the three step algorithm finally numerical experiment indicates that two grid algorithm is very effective

a new expanded mixed method for parabolic integro differential equations a new expanded mixed scheme is studied and analyzed for linear parabolic integro differential equations the proposed method s gradient belongs to the simple square integrable space replacing the classical h div space the new expanded mixed projection is introduced the existence and uniqueness of solution for semi discrete scheme are proved and the fully discrete error estimates based on both backward euler scheme are obtained moreover the optimal a priori error estimates in l 2 and h 1 norm for the scalar unknown u and the error results in l 2 norm for its gradient and its flux the coefficients times the negative gradient are derived finally some numerical results are calculated to verify our theoretical analysis

eigenvalues of hadamard powers of large symmetric pascal matrices abstract let s n be the positive real symmetric matrix of order n with i j entry equal to i j 2 j 1 and let x be a positive real number eigenvalues of the hadamard or entry wise power s n x are considered in particular for k a positive integer it is shown that both the perron root and the trace of s n k are approximately equal to 4 k 4 k 1 2 n 2 n 1 k

on the accuracy of the finite volume element method for diffusion equations on composite grids this paper develops the finite volume element fve method which is similar to the so called control volume finite element method but tailored to composite grid applications fve is described for the general two dimensional diffusion equation in divergence form o h 3 2 discretization error estimates are then developed for the case of triangular elements these results are improved to o h 2 using a special modification involving rectilinear elements at the grid interface

iterative least squares solutions of coupled sylvester matrix equations abstract in this paper we present a general family of iterative methods to solve linear equations which includes the well known jacobi and gauss seidel iterations as its special cases the methods are extended to solve coupled sylvester matrix equations in our approach we regard the unknown matrices to be solved as the system parameters to be identified and propose a least squares iterative algorithm by applying a hierarchical identification principle and by introducing the block matrix inner product the star product for short we prove that the iterative solution consistently converges to the exact solution for any initial value the algorithms proposed require less storage capacity than the existing numerical ones finally the algorithms are tested on computer and the results verify the theoretical findings

regularity of solutions to the spatially homogeneous boltzmann equation for non maxwellian molecules without angular cutoff abstract in this paper we use the pseudo differential calculus to analyze the smoothing property of weak solutions to the spatially homogeneous boltzmann equation precisely we show that for the non maxwellian molecules with debye yukawa potential if the positive weak solution is lipschitz continuous in the velocity variable then it lies in the sobolev space h loc r 3 and hence it is automatically smooth

error estimates in l 2 h 1 and l in covolume methods for elliptic and parabolic p roblems a unified approach in this paper we consider covolume or finite volume element methods for variable coefficient elliptic and parabolic problems on convex smooth domains in the plane we introduce a general approach for connecting these methods with finite element method analysis this unified approach is used to prove known convergence results in the h 1 l 2 norms and new results in the max norm for the elliptic problems we demonstrate that the error u u h between the exact solution u and the approximate solution u h in the maximum norm is o h 2 ln h in the linear element case furthermore the maximur norm error in the gradient is shown to be of first order similar results hold for the parabolic problems

on the accuracy of the finite volume element method based on piecewise linear polynomials we present a general error estimation framework for a finite volume element fve method based on linear polynomials for solving second order elliptic boundary value problems this framework treats the fve method as a perturbation of the galerkin finite element method and reveals that regularities in both the exact solution and the source term can affect the accuracy of fve methods in particular the error estimates and counterexamples in this paper will confirm that the fve method cannot have the standard o h2 convergence rate in the l2 norm when the source term has the minimum regularity only being in l2 even if the exact solution is in h2

a two grid method for finite volume element approximations of second order nonlinear hyperbolic equations the two grid method is studied for solving a two dimensional second order nonlinear hyperbolic equation using finite volume element method the method is based on two different finite element spaces defined on one coarse grid with grid size h and one fine grid with grid size h respectively the nonsymmetric and nonlinear iterations are only executed on the coarse grid and the fine grid solution can be obtained in a single symmetric and linear step it is proved that the coarse grid can be much coarser than the fine grid a prior error estimate in the h 1 norm is proved to be o h h 3 lnh for the two grid semidiscrete finite volume element method with these proposed techniques solving such a large class of second order nonlinear hyperbolic equations will not be much more difficult than solving one single linearized equation finally a numerical example is presented to validate the usefulness and efficiency of the method

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

regularity theory for the spatially homogeneous boltzmann equation with cut off we develop the regularity theory of the spatially homogeneous boltzmann equation with cut off and hard potentials for instance hard spheres by i revisiting the lp theory to obtain constructive bounds ii establishing propagation of smoothness and singularities iii obtaining estimates on the decay of the singularities of the initial datum our proofs are based on a detailed study of the regularity of the gain operator an application to the long time behavior is presented

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

analysis of expanded mixed finite element methods for a nonlinear parabolic equation modeling flow into variably saturated porous media we present an analysis of expanded mixed finite element methods applied to richards equation a nonlinear parabolic partial differential equation modeling the flow of water into a variably saturated porous medium we consider the full range of saturated to completely unsaturated media in the case of the lowest order raviart thomas spaces and the range of all possible saturations we bound the h 1 norm of the error in capacity in terms of approximation error this estimate uses a time integrated scheme and the kirchhoff transformation to handle a degeneracy in the case of completely unsaturated flow optimal convergence is then shown for a nonlinear form related to the error in the capacity for the case of saturated to partially saturated flow convergence rates depending on the holder continuity of the capacity term are derived last optimal convergence of pressures and fluxes is stated for the case of strictly partially saturated flow

a two grid method for expanded mixed finite element solution of semilinear reaction diffusion equations we present a scheme for solving two dimensional semilinear reaction diffusion equations using an expanded mixed finite element method to linearize the mixed method equations we use a two grid algorithm based on the newton iteration method the solution of a non linear system on the fine space is reduced to the solution of two small one linear and one non linear systems on the coarse space and a linear system on the fine space 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 h1 3 as a result solving such a large class of non linear equation will not be much more difficult than solving one single linearized equation copyright 2003 john wiley sons ltd