on the finite volume element method for general self adjoint elliptic problems the finite volume element method fve is a discretization technique for partial differential equations this paper develops discretization energy error estimates for general self adjoint elliptic boundary value problems with fve based on triangulations on which there exist linear finite element spaces and a very general type of control volumes covolumes the energy error estimates of this paper are also optimal but the restriction conditions for the covolumes given in r e bank and d j rose siam j numer anal 24 1987 pp 777 787 z q cai numer math 58 1991 pp 713 735 are removed the authors finally provide a counterexample to show that an expected l2 error estimate does not exist in the usual sense it is conjectured that the optimal order of u u h 0 omega should be o h for the general case

a two level finite volume method for the unsteady navier stokes equations based on two local gauss integrations in this paper we consider a two level finite volume method for the two dimensional unsteady navier stokes equations by using two local gauss integrations this new stabilized finite volume method is based on the linear mixed finite element spaces some new a priori bounds for the approximate solution are derived moreover a two level stabilized finite volume method involves solving one small navier stokes problem on a coarse mesh with mesh size h a large general stokes problem on the fine mesh with mesh size h h the optimal error estimates of the h 1 norm for velocity approximation and the l 2 norm for pressure approximation are established if we choose h o h 2 the two level method gives the same order of approximation as the one level stabilized finite volume method however our method can save a large amount of computational time

higher order finite volume methods ii inf sup condition and uniform local ellipticity the main purpose of this paper is to study the construction of higher order finite volume methods fvms of triangle meshes we investigate the relationship of the three theoretical notions crucial in the construction of fvms the uniform ellipticity of the family of its discrete bilinear forms its inf sup condition and its uniform local ellipticity both the uniform ellipticity of the family of the discrete bilinear forms and its inf sup condition guarantee the unique solvability of the fvm equations and the optimal error estimate of the approximate solution we characterize the uniform ellipticity in terms of the inf sup condition and a geometric condition on the bijective operator mapping from the trial space onto the test space involved in the construction of fvms we present a geometric interpretation of the inf sup condition moreover since the uniform local ellipticity is a convenient sufficient condition for the uniform ellipticity we further provide sufficient conditions and necessary conditions of the uniform local ellipticity

a residual type a posteriori error estimate of finite volume element method for a quasi linear elliptic problem in this paper we analyze a residual type a posteriori error estimator of the finite volume element method for a quasi linear elliptic problem of nonmonotone type and derive computable upper and lower bounds on the error in the h 1 norm numerical experiments are provided to illustrate the performance of the proposed estimator

the finite volume element methods for poisson equation based on adini s element in this paper we construct and analyze the finite volume element scheme based on adini s element for poisson equation choosing trial and test spaces as the adini s finite element space and the piecewise linear function space respectively optimal order error estimates in h 1 and l 2 norms are derived in addition the global and locally pointwise superconvergence estimates are also discussed lastly a numerical experiment is carried out which confirms some results of theoretical analysis

an algorithm using the finite volume element method and its splitting extrapolation this paper is to present a new efficient algorithm by using the finite volume element method and its splitting extrapolation this method combines the local conservation property of the finite volume element method and the advantages of splitting extrapolation such as a high order of accuracy a high degree of parallelism less computational complexity and more flexibility than a richardson extrapolation because the splitting extrapolation formulas only require us to solve a set of smaller discrete subproblems on different coarser grids in parallel instead of on the globally fine grid a large scale multidimensional problem is turned into a set of smaller discrete subproblems additionally this method is efficient for solving interface problems if we regard the interfaces of the problems as the interfaces of the initial domain decomposition

a galerkin neural network based design of guaranteed cost control for nonlinear distributed parameter systems this paper presents a galerkin neural network based guaranteed cost control gcc design for a class of parabolic partial differential equation pde systems with unknown nonlinearities a parabolic pde system typically involves a spatial differential operator with eigenspectrum that can be partitioned into a finite dimensional slow one and an infinite dimensional stable fast complement motivated by this in the proposed control scheme galerkin method is initially applied to the pde system to derive an ordinary differential equation ode system with unknown nonlinearities which accurately describes the dynamics of the dominant slow modes of the pde system the resulting nonlinear ode system is subsequently parameterized by a multilayer neural network mnn with one hidden layer and zero bias terms then based on the neural model and a lure type lyapunov function a linear modal feedback controller is developed to stabilize the closed loop pde system and provide an upper bound for the quadratic cost function associated with the finite dimensional slow system for all admissible approximation errors of the network the outcome of the gcc problem is formulated as a linear matrix inequality lmi problem moreover by using the existing lmi optimization technique a suboptimal guaranteed cost controller in the sense of minimizing the cost bound is obtained finally the proposed design method is applied to the control of the temperature profile of a catalytic rod

the locally stabilized finite volume method for incompressible flow a locally stabilized finite volume method for the two dimensional incompressible flow governed by stationary stokes and navier stokes equations is investigated in this work a macroelement condition is introduced for constructing the local stabilized formulation for the problem the optimal l 2 error estimate of u h p h determined by stokes equations is also deduced some numerical examples are provided to check the efficiency of this method

two grid methods for finite volume element approximations of nonlinear parabolic equations two grid methods are studied for solving a two dimensional nonlinear parabolic equation using finite volume element method the methods are based on one coarse grid space and one fine grid space 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 the two grid methods achieve asymptotically optimal approximation as long as the mesh sizes satisfy h o h 3 lnh as a result solving such a large class of nonlinear parabolic equations will not be much more difficult than solving one single linearized equation

superconvergence of mixed covolume method for elliptic problems on triangular grids in this paper we consider the superconvergence of a mixed covolume method on the quasi uniform triangular grids for the variable coefficient matrix poisson equations the superconvergence estimates between the solution of the mixed covolume method and that of the mixed finite element method have been obtained with these superconvergence estimates we establish the superconvergence estimates and the l error estimates for the mixed covolume method for the elliptic problems based on the superconvergence of the mixed covolume method under the condition that the triangulation is uniform we construct a post processing method for the approximate velocity which improves the order of approximation of the approximate velocity

a posteriori error estimates of two grid finite volume element methods for nonlinear elliptic problems abstract in this paper we study the a posteriori error estimates of two grid finite volume element method for second order nonlinear elliptic equations we derive the residual based a posteriori error estimator and prove the computable upper and lower bounds on the error in h 1 norm the a posteriori error estimator can be used to assess the accuracy of the two grid finite volume element solutions in practical applications numerical examples are provided to illustrate the performance of the proposed estimator

analysis of a finite volume element method for the stokes problem in this paper we propose a stabilized conforming finite volume element method for the stokes equations on stating the convergence of the method optimal a priori error estimates in different norms are obtained by establishing the adequate connection between the finite volume and stabilized finite element formulations a superconvergence result is also derived by using a postprocessing projection method in particular the stabilization of the continuous lowest equal order pair finite volume element discretization is achieved by enriching the velocity space with local functions that do not necessarily vanish on the element boundaries finally some numerical experiments that confirm the predicted behavior of the method are provided

a stabilized multi level method for non singular finite volume solutions of the stationary 3d navier stokes equations this paper proposes and analyzes a stabilized multi level finite volume method fvm for solving the stationary 3d navier stokes equations by using the lowest equal order finite element pair without relying on any solution uniqueness condition this multi level stabilized fvm consists of solving the nonlinear problem on the coarsest mesh and then performing one newton correction step on each subsequent mesh thus only solving a large linear system an optimal convergence rate for the finite volume approximations of nonsingular solutions is first obtained with the same order as that for the usual finite element solution by using a relationship between the stabilized fvm and a stabilized finite element method then the multi level finite volume approximate solution is shown to have a convergence rate of the same order as that of the stabilized finite volume solution of the stationary navier stokes equations on a fine mesh with an appropriate choice of the mesh size h j h j 1 2 j 1 ldots j finally numerical results presented validate our theoretical findings

the finite volume method based on stabilized finite element for the stationary navier stokes problem a finite volume method based on stabilized finite element for the two dimensional stationary navier stokes equations is investigated in this work a macroelement condition is introduced for constructing the local stabilized formulation for the problem we obtain the well posedness of the fvm based on stabilized finite element for the stationary navier stokes equations moreover for quadrilateral and triangular partition the optimal h 1 error estimate of the finite volume solution u h and l 2 error estimate for p h are introduced finally we provide a numerical example to confirm the efficiency of the fvm

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

some optimal error estimates for piecewise linear finite element approximations it is shown that the ritz projection onto spaces of piecewise linear finite elements is bounded in the sobolev space wl for 2 p oc this implies that for functions in wf n w2 the error in approximation behaves like 0 h in wp for 2 p oo and like 0 h2 in lp for 2 p oo in all these cases the additional logarithmic factor previously included in error estimates for linear finite elements does not occur

from the hu washizu formulation to the average nodal strain formulation abstract we present a stabilized finite element method for the hu washizu formulation of linear elasticity based on simplicial meshes leading to the stabilized nodal strain formulation or node based uniform strain elements we show that the finite element approximation converges uniformly to the exact solution for the nearly incompressible case

lur e systems with multilayer perceptron and recurrent neural networks absolute stability and dissipativity sufficient conditions for absolute stability and dissipativity of continuous time recurrent neural networks with two hidden layers are presented in the autonomous case this is related to a lur e system with multilayer perceptron nonlinearity such models are obtained after parametrizing general nonlinear models and controllers by a multilayer perceptron with one hidden layer and representing the control scheme in standard plant form the conditions are expressed as matrix inequalities and can be employed for nonlinear h sub spl infin control and imposing closed loop stability in dynamic backpropagation

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 finite volume method based on the crouzeix raviart element for elliptic pde s in two dimensions we introduce and analyse a finite volume method for the discretization of elliptic boundary value problems in mathbb r 2 the method is based on nonuniform triangulations with piecewise linear nonconforming spaces we prove optimal order error estimates in the l 2 norm and a mesh dependent h 1 norm

some new error estimates of a semidiscrete finite volume element method for a parabolic integro differential equation with nonsmooth initial data a semidiscrete finite volume element fve approximation to a parabolic integro differential equation pide is analyzed in a two dimensional convex polygonal domain an optimal order l 2 error estimate for smooth initial data and nearly the same optimal order l 2 error estimate for nonsmooth initial data are obtained more precisely for homogeneous equations an elementary energy technique and a duality argument are used to derive an error estimate of order o left t 1 h 2 ln h right in the l 2 norm for positive time when the given initial function is only in l 2

rational krylov methods for functions of matrices with applications to fractional partial differential equations abstract in this paper we propose a new choice of poles to define reliable rational krylov methods these methods are used for approximating function of positive definite matrices in particular the fractional power and the fractional resolvent are considered because of their importance in the numerical solution of fractional partial differential equations the numerical experiments on some fractional partial differential equation models confirm that the proposed approach is promising

an immersed finite volume element method for 2d pdes with discontinuous coefficients and non homogeneous jump conditions an immersed finite volume element method is developed to solve 2d elliptic interface problems with a variable coefficient that has a finite jump across an interface the solution and the flux may also have a finite jump across the interface using the source removal technique an equivalent elliptic interface problem with homogeneous jump conditions is obtained the nodal basis functions are constructed to satisfy the homogeneous jump conditions near the interface and the usual finite element nodal basis functions are applied away from the interface the resulting linear problem is simple and easy to solve a proof of the error estimate in the energy norm is given numerical experiments demonstrate the convergence rates of the proposed method with the usual o h 2 in the l 2 the l norms and o h in the h 1 norm

a finite volume method on general surfaces and its error estimates abstract in this paper we study a finite volume method and its error estimates for the numerical solution of some model second order elliptic partial differential equations defined on a smooth surface the discretization is defined via a surface mesh consisting of piecewise planar triangles and piecewise polygons the optimal error estimates of the approximate solution are proved in both the h 1 and l 2 norms which are of first order and second order respectively under mesh regularity assumptions some numerical tests are also carried out to experimentally verify our theoretical analysis

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

piecewise linear petrov galerkin error estimates for the box method the perpendicular bisector box method pbm discretization of the laplacian on a simplicial element is equivalent to a piecewise linear petrov galerkin method with piecewise constant test functions this pbm element matrix is expressed in terms of galerkin s piecewise linear plg element matrix and the average values assumed by test functions over element faces only these results are shown to be applicable to the accuracy analysis of pbm discretizations of extremum stable elliptic gradient equations subject to regularity conditions on variable coefficients in these equations these gradient equations are shown to satisfy sufficient conditions to validate a discrete extremum principle on n dimensional delaunay triangulations this conclusion need not hold for the plg equations the dependence of the discretized laplacian on only the element face averages of test functions enables the substitution of piecewise polynomial test functions for the piecewise constant functions the latter are used to establish piecewise linear accuracy for the pbm by an adaptation of an error estimate for petrov galerkin methods by strang

convergence of finite volume schemes for poisson s equation on nonuniform meshes an error analysis of a family of cell center finite volume schemes for poisson s equation on cartesian product nonuniform meshes in two dimensions is presented optimal order error estimates are derived in the discrete h 1 norm under minimum smoothness requirements on the exact solution and without any additional assumption on the regularity of the mesh on quasi uniform meshes analogous estimates are obtained in the maximum norm

comments on lur e systems with multilayer perceptron and recurrent neural networks absolute stability and dissipativity the authors point out that some of the claims and expressions in the paper mentioned in the title by suykens et al ibid vol 44 1999 are incorrect

optimal l 2 h 1 l analysis of finite volume methods for the stationary navier stokes equations with large data previous work on the stability and convergence analysis of numerical methods for the stationary navier stokes equations was carried out under the uniqueness condition of the solution which required that the data be small enough in certain norms in this paper an optimal analysis for the finite volume methods is performed for the stationary navier stokes equations which relaxes the solution uniqueness condition and thus the data requirement in particular optimal order error estimates in the h 1 h 1 norm for velocity and the l 2 l 2 norm for pressure are obtained with large data and a new residual technique for the stationary navier stokes equations is introduced for the first time to obtain a convergence rate of optimal order in the l 2 l 2 norm for the velocity in addition after proving a number of additional technical lemmas including weighted l 2 l 2 norm estimates for regularized green s functions associated with the stokes problem optimal error estimates in the l infty l norm are derived for the first time for the velocity gradient and pressure without a logarithmic factor o log h o log h for the stationary naiver stokes equations

convergence of the discontinuous finite volume method for elliptic problems with minimal regularity this paper investigates convergence of the discontinuous finite volume method dfvm under minimal regularity assumptions on solutions of second order elliptic boundary value problems conventional analysis requires the solutions to be in sobolev spaces h 1 s s 12 here we assume the solutions are in h 1 s s 0 and employ the techniques developed in gudi 2010 18 20 to derive error estimates in a mesh dependent energy norm and the l 2 norm for dfvm the theoretical estimates are illustrated by numerical results which include problems with corner singularity and intersecting interfaces

some new error estimates for ritz galerkin methods with minimal regularity assumptions new uniform error estimates are established for finite element approximations u h of solutions u of second order elliptic equations lu f using only the regularity assumption u 1 c f 1 using an aubin nitsche type duality argument we show for example that for arbitrary fixed e sufficiently small there exists an h 0 such that for 0 h h 0 u u h 0 e u u h 1 here s denotes the norm on the sobolev space h s other related results are established

superconvergence of finite volume methods for the second order elliptic problem abstract this paper derives a superconvergence result for finite volume approximations of the second order elliptic problem by using a l2 projection post processing technique the superconvergence result is applicable to different kind of finite volume methods and to general quasi uniform meshes

unified analysis of finite volume methods for the stokes equations we establish a general framework for analyzing the class of finite volume methods for the stokes equations under the framework optimal order convergence for velocity and pressure in various sobolev norms are obtained in a natural and systematic way specifically optimal l 2 error estimates for velocity are obtained for the first time for several different finite volume methods

error estimation of a quadratic finite volume method on right quadrangular prism grids in this paper we develop a finite volume element method with affine quadratic bases on right quadrangular prism meshes for three dimensional elliptic boundary value problems the optimal h 1 norm error estimate of second order accuracy is proved under certain assumptions about the meshes numerical results are presented to illustrate the theoretical analysis

is 2k conjecture valid for finite volume methods this paper is concerned with superconvergence properties of a class of finite volume methods of arbitrary order over rectangular meshes our main result is to prove the 2k conjecture at each vertex of the underlying rectangular mesh the bi k degree finite volume solution approximates the exact solution with an order o h 2k where h is the mesh size as byproducts superconvergence properties for finite volume discretization errors at lobatto and gauss points are also obtained all theoretical findings are confirmed by numerical experiments

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

a posteriori error estimates for finite volume method based on bilinear trial functions for the elliptic equation this paper is concerned with a residual type a posteriori error estimators for a finite volume method for the second order elliptic equation bilinear functions are used as trial functions in this finite volume method reliability and efficiency bounds for the error estimator are established numerical examples are provided to confirm the theory

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

the geometry of the gauss picard modular group we give a construction of a fundamental domain for rm pu 2 1 mathbb z i that is the group of holomorphic isometries of complex hyperbolic space with coefficients in the gaussian ring of integers mathbb z i we obtain from that construction a presentation of that lattice and relate it in particular to lattices constructed by mostow

a finite volume method for stokes problems on quadrilateral meshes abstract we present a finite volume method for stokes problems using the isoparametric q 1 q 0 element pair on quadrilateral meshes to offset the lack of the i n f s u p condition a jump term of discrete pressure stabilizing term is added to the continuity approximation equation thus we establish a stabilized finite volume scheme on quadrilateral meshes then based on some superclose estimates we derive the optimal error estimates in the h 1 and l 2 norms for velocity and in the l 2 norm for pressure respectively numerical examples are provided to illustrate our theoretical analysis we emphasize that our work is the first time to propose and analyze a finite volume method for stoke problems using isoparametric elements on quadrilateral meshes

on the accuracy of the mortar finite volume element method with lagrange multipliers abstract in this paper the mortar finite volume element mfve method with lagrange multipliers is considered for second order elliptic boundary value problems we prove l 2 error estimates which are affected by regularities in both the exact solution and the source term meanwhile the uniform convergence of the solution is obtained for the mfve method under minimal regularity assumption

additive average schwarz method for a crouzeix raviart finite volume element discretization of elliptic problems with heterogeneous coefficients in this paper we introduce an additive schwarz method for a crouzeix raviart finite volume element discretization of a second order elliptic problem with discontinuous coefficients where the discontinuities are both inside the subdomains and across and along the subdomain boundaries we show that depending on the distribution of the coefficient in the model problem the parameters describing the generalized minimal residual method gmres convergence rate of the proposed method depend linearly on the mesh parameters inlineequation not available see fulltext also under certain restrictions on the distribution of the coefficient the convergence of the gmres method is independent of jumps in the coefficient

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

two grid finite volume element method for linear and nonlinear elliptic problems two grid finite volume element discretization techniques based on two linear conforming finite element spaces on one coarse and one fine grid are presented for the two dimensional second order non selfadjoint and indefinite linear elliptic problems and the two dimensional second order nonlinear elliptic problems with the proposed techniques solving the non selfadjoint and indefinite elliptic problem on the fine space is reduced into solving a symmetric and positive definite elliptic problem on the fine space and solving the non selfadjoint and indefinite elliptic problem on a much smaller space solving a nonlinear elliptic problem on the fine space is reduced into solving a linear problem on the fine space and solving the nonlinear elliptic problem on a much smaller space convergence estimates are derived to justify the efficiency of the proposed two grid algorithms a set of numerical examples are presented to confirm the estimates

a finite volume method for solving navier stokes problems abstract we develop a finite volume method for solving the navier stokes equations on a triangular mesh we prove that the unique solution of the finite volume method converges to the true solution with optimal order for velocity and for pressure in discrete h 1 norm and l 2 norm respectively

on the construction and analysis of high order locally conservative finite volume type methods for one dimensional elliptic problems locally conservative finite volume type methods based on continuous piecewise polynomial functions of degree r ge 2 are introduced and analyzed in the context of indefinite elliptic problems in one space dimension the new methods extend and generalize the classical finite volume method based on piecewise linear functions we derive a priori error estimates in the l2 h1 and l infty norm and discuss superconvergence effects for the error and its derivative explicit residual based a posteriori error bounds in the l2 and energy norm are also derived we compute the experimental order of convergence and show the results of an adaptive algorithm based on the a posteriori error estimates

the mmoc and mmocaa schemes for the finite volume element method of convection diffusion problems we present and analyze the modified method of characteristics mmoc and the modified method of characteristics with adjusted advection mmocaa for the finite volume element fve method of convection diffusion problems these two schemes maintain the advantages of both the mmoc and the fve method and the mmocaa scheme discussed herein conserves the conservation law globally at a minor additional computational cost optimal order error estimates in the h 1 norm are proved for these schemes a numerical example is presented to confirm the estimates

simplified weak galerkin and new finite difference schemes for the stokes equation abstract this article presents a simplified formulation for the weak galerkin finite element method for the stokes equations without using the degrees of freedom associated with the unknowns in the interior of each element as formulated in the original weak galerkin algorithm the simplified formulation preserves the important mass conservation property locally on each element and allows the use of general polygonal partitions a particular application of the simplified weak galerkin on rectangular partitions yields a new class of 5 and 7 point finite difference schemes for the stokes equations an explicit formula is presented for the computation of the element stiffness matrices on arbitrary polygonal elements error estimates of optimal order are established for the simplified weak galerkin finite element method in the h 1 and l 2 norms furthermore a superconvergence of order o h 3 2 is established on rectangular partitions for the velocity approximation in the h 1 norm at cell centers and a similar superconvergence is derived for the pressure approximation in the l 2 norm at cell centers some numerical results are reported to confirm the convergence and superconvergence theory