sub ode method and soliton solutions for the variable coefficient mkdv equation in this work an auxiliary equation is used for an analytic study on the time variable coefficient modified korteweg de vries mkdv equation five sets of new exact soliton like solutions are obtained the results show that the pulse parameters are time dependent variable coefficients moreover the basic conditions for the formation of derived solutions are presented

the number of small amplitude limit cycles in arbitrary polynomial systems abstract in this paper we study the number of small amplitude limit cycles in arbitrary polynomial systems it is found that almost all the results for the number of small amplitude limit cycles are obtained by calculating lyapunov constants and determining the order of the corresponding hopf bifurcation it is well known that the difficulty in calculating the lyapunov constants increases with the increasing of the degree of polynomial systems so it is necessary and valuable for us to achieve some general results about the number of small amplitude limit cycles in arbitrary polynomial systems with degree m which is denoted by m m in this paper by applying the method developed by c christopher and n lloyd in 1995 and m han and j li in 2012 we first obtain the lower bounds for m 6 m 14 and then prove that m m m 2 if m 23 finally we obtain that m m grows as least as rapidly as 18 25 1 2 ln 2 m 2 2 ln m 2 for all large m it is proved by m han j li lower bounds for the hilbert number of polynomial systems j differential equations 252 2012 3278 3304 that the number of all limit cycles in arbitrary polynomial systems with degree m grows as least as rapidly as 1 2 ln 2 m 2 2 ln m 2

quadratic systems with center and their perturbations abstract we study the quadratic systems on a plane z i z az2 b z 2 cz2 z x iy we give a simple algebraic proof of the center conditions 0 and b 0 or a 1 2 b 1 or a b 1 c c or a 2 b 1 c 1 and of the theorem of bautin that the number of small amplitude limit cycles bifurcating from a center or a focus is not greater than 3 we present the bifurcational diagrams and phase portraits of the systems with center and describe the cyclicity precisely in each case here one mistake in bautin s work is revealed and some questions concerning focus numbers are answered we study also global not small limit cycles for a small perturbation of the center case with two invariant lines the lotka volterra system using abelian integrals we show that the number of zeroes of the corresponding abelian integral is 0 1 or 2 some attention is devoted to the problem of limit cycles for a small perturbation of a system with a two fold center when two center conditions hold the problem turns out to be equivalent to the problem of zeroes of some integral with polynomial coefficients

analytic integrability and characterization of centers for generalized nilpotent singular points a method which provides necessary conditions for obtaining a local analytic integral in a neighborhood of a generalized nilpotent singular point is developed this method is applied to several classes of generalized nilpotent systems we obtain sufficient conditions for each case finding an inverse integrating factor or a local first integral defined at the origin we study which ones of the found cases possess a center at the origin

an unusual stabilized finite element method for a generalized stokes problem an unusual stabilized finite element is presented and analyzed herein for a generalized stokes problem with a dominating zeroth order term the method consists in subtracting a mesh dependent term from the formulation without compromising consistency the design of this mesh dependent term as well as the stabilization parameter involved are suggested by bubble condensation stability is proven for any combination of velocity and pressure spaces under the hypotheses of continuity for the pressure space optimal order error estimates are derived for the velocity and the pressure using the standard norms for these unknowns numerical experiments confirming these theoretical results and comparisons with previous methods are presented

implementation of a new algorithm of computation of the poincare liapunov constants in the last years many papers giving different methods to compute the poincare liapunov constants have been published in appl math warsaw 28 2002 17 a new method to compute recursively all the poincare liapunov constants as a function of the coefficients of the system for an arbitrary analytic system which has a perturbed linear center at the origin was given and thus a theoretical answer to the classical center problem was given the method also computes the coefficients of the poincare series as a function of the same coefficients we describe its implementation in two different ways by means of a computer algebra system and an algorithm in any computer language if this second alternative is used later it is necessary to translate the results so that they can be manipulated with a computer algebra system we describe also how the availability of symbolic manipulation procedures has recently led to a significant progress in the resolution of the different problems related with the poincare liapunov constants as they are the central problems like the small amplitude limit cycles

the number of limit cycles of certain polynomial differential equations two dimensional differential systems are considered where p and q are polynomials the question of interest is the maximum possible numberof limit cycles of such systems in terms of the degree of p and q an algorithm is described for determining a so called focal basis this can be implemented on a computer estimates can then be obtained for the number of small amplitude limit cycles the technique is applied to certain cubic systems a class of examples with exactly five small amplitude limit cycles is constructed quadratic systems are also considered

multiple soliton solutions for the boussinesq equation abstract in this work we use the hirota s direct method combined with the simplified version of this method to determine the n soliton solutions for the boussinesq equation the one soliton solutions will be handled by the tanh coth method the work highlights the significant features of the employed methods

the center problem for a 1 4 resonant quadratic system abstract the main objective of this paper is to find necessary and sufficient conditions for a 1 4 resonant system of the form x x a 10 x 2 a 01 x y a 12 y 2 y 4 y b 2 1 x 2 b 10 x y b 01 y 2 to have a center at the origin since applying a linear change of variables any system of this form can be transformed either to system with a 10 1 or a 10 0 these are the two cases considered here when a 10 1 there appear 46 resonant center conditions and for a 10 0 there are 9 center conditions to obtain necessary conditions for integrability the computation of the resonant saddle quantities focus quantities and the decomposition of the variety of the ideal generated by an initial string of them were used the theory of darboux first integrals and some other methods as the monodromy arguments for instance are used to show the sufficiency since decomposition of the variety mentioned above was performed using modular computations the obtained conditions of integrability represent the complete list of the integrability conditions only with very high probability and there remains an open problem to verify this statement

analysis of mixed finite elements methods for the stokes problem a unified approach we develop a method for the analysis of mixed finite element methods for the stokes problem in the velocity pressure formulation a technical macroelement condition which is sufficient for the classical babuska brezzi inequality to be valid is introduced using this condition we are able to verify the stability and optimal order of convergence of several known mixed finite element methods

error analysis of some finite element methods for the stokes problem we prove the optimal order of convergence for some two dimensional finite element methods for the stokes equations first we consider methods of the taylor hood type the triangular p3 p2 element and the qk qk 1 i k 2 family of quadrilateral elements then we introduce two new low order methods with piecewise constant approximations for the pressure the analysis is performed using our macroelement technique which is reviewed in a slightly altered form

parallelization of the lyapunov constants and cyclicity for centers of planar polynomial vector fields christopher in 2006 proved that under some assumptions the linear parts of the lyapunov constants with respect to the parameters give the cyclicity of an elementary center this paper is devote to establish a new approach namely parallelization to compute the linear parts of the lyapunov constants more concretely it is showed that parallelization computes these linear parts in a shorter quantity of time than other traditional mechanisms to show the power of this approach we study the cyclicity of the holomorphic center iz z 2 z 3 z n under general polynomial perturbations of degree n for n 13 we also exhibit that from the point of view of computation among the hamiltonian time reversible and darboux centers the holomorphic center is the best candidate to obtain high cyclicity examples of any degree for n 4 5 13 we prove that the cyclicity of the holomorphic center is at least n 2 n 2 this result give the highest lower bound for m 6 m 7 m 13 among the existing results where m n is the maximum number of limit cycles bifurcating from an elementary monodromic singularity of polynomial systems of degree n as a direct corollary we also obtain the highest lower bound for the hilbert numbers h 6 40 h 8 70 and h 10 108 because until now the best result was h 6 39 h 8 67 and h 10 100

multigrid methods for pde optimization research on multigrid methods for optimization problems is reviewed optimization problems considered include shape design parameter optimization and optimal control problems governed by partial differential equations of elliptic parabolic and hyperbolic type

mixed hp finite element methods for stokes and non newtonian flow abstract we analyze the stability of hp finite elements for viscous incompressible flow for the classical velocity pressure formulation we give new estimates for the discrete inf sup constants on geometric meshes which are explicit in the polynomial degree k of the elements in particular we obtain new bounds for p elements on triangles for the three field stokes problem describing linearized non newtonian flow we estimate discrete inf sup constants explicit in both h and k for various subspace choices continuous and discontinuous for the extra stress we also give a stability analysis of the hp version of an elastic viscous split stress evss method and present elements that are stable and optimal in h and k finally we present numerical results that show the exponential convergence of the hp version for stokes flow over unsmooth domains

the hirota s direct method and the tanh coth method for multiple soliton solutions of the sawada kotera ito seventh order equation abstract in this work the sawada kotera ito seventh order equation is studied the tanh coth method is applied to obtain soliton solution of this equation the hirota s direct method combined with the simplified hereman s method are applied to determine the n soliton solutions for this equation the study highlights the significant features of the employed methods and its capability of handling completely integrable equations

bifurcation of limit cycles from centers and separatrix cycles of planar analytic systems this paper presents a survey of results on the bifurcation of limit cycles from centers and separatrix cycles of perturbed planar analytic systems and contributes some new results on the bifurcation of multiple limit cycles from centers and on the multiplicity of separatrix cycles of such systems the basic theme throughout the paper is that the number positions and multiplicities of the limit cycles that bifurcate under perturbations are related to the number positions and multiplicities of the zeros of the melnikov function for the system the general theory is illustrated by a number of examples from the literature some of which are extended to include new results

multiple front solutions for the burgers kadomtsev petviashvili equation abstract in this work we study a completely integrable dissipative equation the burgers equation is extended by using the sense of the kadomtsev petviashvili kp equation the new established burgers kp equation is studied by using the tanh coth method to obtain kink solutions and periodic solutions we also apply the powerful hirota s bilinear method to establish exact n soliton solutions for the derived integrable equation

polynomial first integrals via the poincare series in appl math 28 2001 17 a method to compute the poincare liapunov constants for an arbitrary analytic differential system which has a linear center at the origin in function of the coefficients of the system was given this method also computes the coefficients of the poincare series in function of the same coefficients in this work we use this method to determine polynomial differential systems which have a polynomial first integral

limit cycle bifurcations from a non degenerate center in this work we discuss the computational problems which appear in the computation of the poincare liapunov constants and the determination of their functionally independent number moreover we calculate the minimum number of bautin ideal generators which give the number of small limit cycles under certain hypothesis about the generators in particular we consider polynomial systems of the form x y pn x y y x qn x y where pn and qn are a homogeneous polynomial of degree n we use center bifurcation rather than multiple hopf bifurcations used a previous work 19 to estimate the cyclicity of a unique singular point of focus center type for n 4 5 6 7 and compare with the results given by the conjecture presented in 18

stability of higher order hood taylor methods the stability of a higher order hood taylor method for the approximation of the stationary stokes equations using continuous piecewise polynomials of degree 3 to approximate velocities and continuo

isochronous centers of a linear center perturbed by fifth degree homogeneous polynomials in this work we study isochronous centers of two dimensional autonomous system in the plane with linear part of center type and nonlinear part given by homogeneous polynomials of fifth degree a complete classification of the necessary conditions for the time reversible systems of this class is given in order to have an isochronous center at the origin an open problem is stated for the sufficient conditions moreover we find two nonreversible isochronous families from the center cases known all the computations in order to obtain necessary conditions for such isochronous centers are given in polar coordinates and we give a proof of the isochronicity of these systems by using different methods

multiple front solutions for the burgers equation and the coupled burgers equations abstract in this work multiple front solutions for the burgers equation and the coupled burgers equations are examined the tanh coth method and the cole hopf transformation are used the work highlights the power of the proposed schemes and the structures of the obtained multiple front solutions

bifurcation of limit cycles from quadratic isochrones abstract for a one parameter family of plane quadratic vector fields x e depending analytically on a small real parameter e we determine the number and position of the local families of limit cycles which emerge from the periodic trajectories surrounding an isochronous or linearizable center techniques are developed for treating the bifurcations of all orders and these are applied to prove the following results for the linear isochrone the maximum number of continuous families of limit cycles which can emerge is three for one class of nonlinear isochrones at most one continuous family of limit cycles can emerge whereas for all other nonlinear isochrones at most two continuous families of limit cycles can emerge moreover for each isochrone in one of these classes there are small perturbations such that the indicated maximum number of continuous families of limit cycles can be made to emerge from a corresponding number of arbitrarily prescribed periodic orbits within the period annulus of the isochronous center

isochronicity into a family of time reversible cubic vector fields in this work we study necessary and sufficient conditions for the existence of isochronous centers into a family of cubic time reversible systems this class of reversible systems is characterized by the existence of an inverse integrating factor which is a certain power of an invariant straight line

multigrid preconditioning of linear systems for interior point methods applied to a class of box constrained optimal control problems in this article we construct and analyze multigrid preconditioners for discretizations of operators of the form mathcal d lambda mathcal k mathcal k where d lambda is the multiplication with a relatively smooth function lambda 0 and mathcal k is a compact linear operator these systems arise when applying interior point methods to the minimization problem min u frac 1 2 mathcal k u f 2 beta u 2 with box constraints underline u leqslant u leqslant overline u on the controls the presented preconditioning technique is closely related to the one developed by draganescu and dupont math comp 77 2008 pp 2001 2038 for the associated unconstrained problem and is intended for large scale problems as in that work the quality of the resulting preconditioners is shown to increase as h downarrow 0 but it decreases as the smoothness of lambda declines we test this algorithm on a tikhonov regularized backward parabolic equation with box constraints on the control and on a standard elliptic constrained optimization problem in both cases it is shown that the number of linear iterations per optimization step as well as the total number of finest scale matrix vector multiplications is decreasing with increasing resolution thus showing the method to be potentially very efficient for truly large scale problems

multiple soliton solutions for the sixth order ramani equation and a coupled ramani equation in this work the completely integrable sixth order nonlinear ramani equation and a coupled ramani equation are studied multiple soliton solutions and multiple singular soliton solutions are formally derived for these two equations the hirota s bilinear method is used to determine the two distinct structures of solutions the resonance relations for the three cases are investigated

the hirota s direct method for multiple soliton solutions for three model equations of shallow water waves abstract multiple soliton solutions for three model equations for shallow water waves are determined the three models are completely integrable the hirota bilinear method is used to determine multiple soliton solutions of sech squared type for these equations the tanh coth method is used to obtain single soliton solutions and other solutions for these three models the three models have different linear dispersion relations but possess the same coefficients for the polynomials of exponentials

the 1 q resonant center problem for certain cubic lotka volterra systems abstract necessary conditions and distinct sufficient conditions are derived for the system x x 1 a 20 x 2 a 11 xy a 02 y 2 y y q b 20 x 2 b 11 xy b 02 y 2 to admit a first integral of the form x y x q y in a neighborhood of the origin in which case the origin is termed a 1 q resonant center necessary and sufficient conditions are obtained for odd q q 9 necessary conditions most of which are also sufficient are obtained for even q q 8 key ideas in the proofs are computation of focus quantities for the complexified systems and decomposition of the variety of the ideal generated by an initial string of them to obtain necessary conditions and the theory of darboux first integrals to show sufficiency

multiple soliton solutions for the lax kadomtsev petviashvili lax kp equation a new completely integrable dispersive equation is derived the new equation is obtained by extending the lax fifth order equation using the sense of the kadomtsev petviashvili kp equation in extending the kdv equation the newly derived lax kadomtsev petviashvili lax kp equation is investigated by using the tanh coth method and the hirota bilinear method to derive single soliton solution and n soliton solutions respectively the study highlights the multiple soliton solutions of the derived completely integrable equation

a method of constructing cycles without contact around a weak focus has a weak focus then limit cycles can be made to appear in its neighborhood by varying slightly the right sides 11 frommer 2 bautin 3 chin et al 4 and tung 5 are examples of excellent expositions of poincare s idea it is necessary to go from numerical examples to general theory in the first part of this paper we present a method of constructing cycles without contact around a weak focus which makes clear poincart s idea used in the above work 2 51 in the second part another method to analyze global phase portraits is used quadratic systems with at least four limit cycles are produced by means of both the above methods this contradicts the results of petrovskii and landis 6

analysis of some mixed finite element methods for plane elasticity equations we analyze some mixed finite element methods based on rectangular elements for solving the two dimensional elasticity equations we prove error estimates for a method proposed by taylor and zienkiewicz and for some new variants of the known equilibrium methods a numerical example is given demonstrating the performance of the various algorithms considered 1 introduction in the numerical solution of problems of continuum mechanics the stresses are normally of primary interest in the elastic region it is therefore natural to design the numerical algorithms so that the stresses can be obtained directly without first computing the displacements such methods can be derived from the dual variational formulation of the elasticity problem the corresponding finite element algorithms are usually formulated as mixed methods where both the displacements and the stresses are first approximated and the displacements are then eliminated from the discrete equations in many cases the elimination can be rather effectively done using penalty perturbation techniques or their iterative variants cf 3 11 12 the best known finite element methods of the above type are the so called equilibrium methods first proposed by fraejis de veubeke 17 cf also 14 16 18 and analyzed theoretically by johnson and mercier 9 cf also 8 in these methods one uses specific composite elements which allow the equilibrium condi tion between the stresses and the volume load to be satisfied exactly in the case where the volume load is zero

minimum number of ideal generators for a linear center perturbed by homogeneous polynomials abstract using the algorithm presented in j gine x santallusia on the poincare liapunov constants and the poincare series appl math warsaw 28 1 2001 17 30 the poincare liapunov constants are calculated for polynomial systems of the form x y p n x y y x q n x y where p n and q n are homogeneous polynomials of degree n the objective of this work is to calculate the minimum number of ideal generators i e the number of functionally independent poincare liapunov constants through the study of the highest fine focus order for n 4 and n 5 and compare it with the results that give the conjecture presented in j gine on the number of algebraically independent poincare liapunov constants appl math comput 188 2 2007 1870 1877 moreover the computational problems which appear in the computation of the poincare liapunov constants and the determination of the number of functionally independent ones are also discussed

stability of finite elements under divergence constraints a finite dimensional stability test for checking velocity pressure finite element trial spaces is presented applications are made to a new class of element pairs proposed in this paper as well as to existing spaces

optimal order multilevel preconditioners for regularized ill posed problems in this article we design and analyze multilevel preconditioners for linear systems arising from regularized inverse problems using a scale independent distance function that measures spectral equivalence of operators it is shown that these preconditioners approximate the inverse of the operator to optimal order with respect to the spatial discretization parameter h as a consequence the number of preconditioned conjugate gradient iterations needed for solving the system will decrease when increasing the number of levels with the possibility of performing only one fine level residual computation if h is small enough the results are based on the previously known two level preconditioners of rieder 1997 see also hanke and vogel 1999 and on applying newton like methods to the operator equation x 1 a 0 we require that the associated forward problem has certain smoothing properties however only natural stability and approximation properties are assumed for the discrete operators the algorithm is applied to a reverse time parabolic equation that is the problem of finding the initial value leading to a given final state we also present some results on constructing restriction operators with preassigned approximating properties that are of independent interest

preconditioning iterative methods for the optimal control of the stokes equations solving problems regarding the optimal control of partial differential equations pdes also known as pde constrained optimization is a frontier area of numerical analysis of particular interest is the problem of flow control where one would like to effect some desired flow by exerting for example an external force the bottleneck in many current algorithms is the solution of the optimality system a system of equations in saddle point form that is usually very large and ill conditioned in this paper we describe two preconditioners a block diagonal preconditioner for the minimal residual method and a block lower triangular preconditioner for a nonstandard conjugate gradient method which can be effective when applied to such problems where the pdes are the stokes equations we consider only distributed control here although we believe other problems could be treated in the same way we give numerical results and we compare these with those obtained by solving the equivalent forward problem using similar techniques

multiple soliton solutions of two extended model equations for shallow water waves abstract multiple soliton solutions for two extensions of model equations for shallow water waves presented in m j ablowitz d j kaup a c newell h segur the inverse scattering transform fourier analysis for nonlinear problems stud appl math 53 1974 249 315 r hirota j satsuma n soliton solutions of model equations for shallow water waves j phys soc jpn 40 2 1976 611 612 are obtained the two extensions possess identical dispersion relations but with different structures for the n soliton solutions for n 1 the hirota bilinear method is used to determine multiple soliton solutions of sech squared type for these extended equations the tanh coth method is used to obtain single soliton solutions

on the number of algebraically independent poincare liapunov constants abstract in this paper an upper bound for the number of algebraically independent poincare liapunov constants in a certain basis for planar polynomial differential systems is given finally it is conjectured that an upper bound for the number of functionally independent poincare liapunov quantities would be m 2 3 m 7 where m is the degree of the polynomial differential system moreover the computational problems which appear in the computation of the poincare liapunov constants and in the determination of the center cases are also discussed

on the structure of poincare lyapunov constants for the weak focus of polynomial vector fields in this paper generalized poincare lyapunov constants vi i 1 2 are defined and an explicit formula is given for the lyapunov functions it is shown that vi is determined modulo v1 vi 1 this is used for the investigation and creation of limit cycles

stable nonconforming quadrilateral finite elements for the stokes problem stability result is obtained for the approximation of the stationary stokes problem with nonconforming elements proposed by douglas et al j douglas jr j e santos d s xiu ye nonconforming galerkin methods based on quadrilateral elements for second order elliptic problems preprints added by conforming bubbles to the velocity and discontinuous piecewise linear to the pressure on quadrilateral elements optimal order error estimates derived

multigrid solution of a distributed optimal control problem constrained by the stokes equations in this work we construct multigrid preconditioners to accelerate the solution process of a linear quadratic optimal control problem constrained by the stokes system the first order optimality conditions of the control problem form a linear system the kkt system connecting the state adjoint and control variables our approach is to eliminate the state and adjoint variables by essentially solving two stokes systems and to construct efficient multigrid preconditioners for the schur complement of the block associated with the state and adjoint variables these multigrid preconditioners are shown to be of optimal order with respect to the convergence properties of the discrete methods used to solve the stokes system in particular the number of conjugate gradient iterations is shown to decrease as the resolution increases a feature shared by similar multigrid preconditioners for elliptic constrained optimal control problems

optimal control of the stokes equations a priori error analysis for finite element discretization with postprocessing an optimal control problem for 2d and 3d stokes equations is investigated with pointwise control constraints this paper is concerned with the discretization of the control by piecewise constant functions the state and the adjoint state are discretized by finite element schemes in the paper a postprocessing strategy is suggested which allows for significant improvement of the accuracy

multiple soliton solutions for the kp equation by hirota s bilinear method and by the tanh coth method abstract in this work we use the hirota s direct method combined with the simplified version of this method to determine the n soliton solutions n 1 2 3 4 5 for the kadomtsev petviashvili kp equation the one soliton solutions will be handled by the tanh coth method the work emphasizes the power of the two methods that will be used

the center problem for a 2 3 resonant cubic lotka volterra system in this paper we obtain conditions on the coefficients of a cubic lotka volterra system of the form 1 x x 2 a 2 0x 2 a 1 1xy a 0 2y 2 y y 3 b 2 0x 2 b 1 1xy b 0 2y 2 which fulfillment yields the existence in a neighborhood of the origin of a first integral of the form f x y x 3y 2 h o t in which case the origin is termed a 2 3 resonant center this system was studied in 13 where due to computational constrains the consideration was limited to the cases where either one or both coefficients a 1 1 b 1 1 in system 1 were equal to zero or both coefficients were equal to 1 here we are studying the case where the coefficient a 1 1 is equal to 1 and b 1 1 is arbitrary the obtained results represent the study of the center problem for general system 1 since by a linear substitution any system of the form 1 can be transformed either to system 1 with a 1 1 1 or to one of systems studied in 13 computation of the resonant saddle quantities focus quantities and the decomposition of the variety of the ideal generated by an initial string of them were used to obtain necessary conditions of integrability and the theory of darboux first integrals and some other methods to show the sufficiency since the decompositions of the variety mentioned above was performed using modular computations the obtained 19 conditions of integrability represent the complete list of the integrability conditions only with very high probability and there remains an open problem to verify this statement

a robust all at once multigrid method for the stokes control problem in this paper we present an all at once multigrid method for a distributed stokes control problem velocity tracking problem for solving such a problem we use the fact that the solution is characterized by the optimality system karush kuhn tucker system the discretized optimality system is a large scale linear system whose condition number depends on the grid size and on the choice of the regularization parameter forming a part of the problem recently block diagonal preconditioners have been proposed which allow to solve the problem using a krylov space method with convergence rates that are robust in both the grid size and the regularization parameter or cost parameter in the present paper we develop an all at once multigrid method for a stokes control problem and show robust convergence more precisely we show that the method converges with rates which are bounded away from one by a constant which is independent of the grid size and the choice of the regularization or cost parameter

higher order limit cycle bifurcations from non degenerate centers abstract the computational problems which appear in the computation of the poincare liapunov constants and the determination of their functionally independent number has led in recent works to consider only the lowest terms of these constants in this work we improve the results obtained in this direction for polynomials systems of the form x y p n x y y x q n x y where p n and q n are a homogeneous polynomial of degree n we use center bifurcation to estimate the cyclicity of a unique singular point of focus center type for different values of n and compare with the results given by the conjecture presented in 15

analysis of some mixed elements for the stokes problem abstract in this paper we discuss some mixed finite element methods related to the reduced integration penalty method for solving the stokes problem we prove optimal order error estimates for bilinear constant and biquadratic bilinear velocity pressure finite element solutions the result for the biquadratic bilinear element is new while that for the bilinear constant element improves the convergence analysis of johnson and pitkaranta 1982 in the degenerate case when the penalty parameter is set to be zero our results reduce to some related known results proved in by brezzi and fortin 1991 for the bilinear constant element and bercovier and pironneau 1979 for the biquadratic bilinear element our theoretical results are consistent with the numerical results reported by carey and krishnan 1982 and oden et al 1982