on error estimates of the penalty method for unsteady navier stokes equations the penalty method has been widely used for numerical computations of the unsteady navier stokes equations however the best error estimates available to the author s knowledge were not optimal an

brezis gallouet wainger type inequalities and blow up criteria for navier stokes equations in unbounded domains we shall find the weakest norm that satisfies the brezis gallouet wainger type inequality under some conditions as an application of the brezis gallouet wainger type inequality we shall establish beale kato majda type blow up criteria of smooth solutions to the 3 d navier stokes equations in unbounded domains

low firing rates in a compartmental model neuron a nonlinear relationship is obtained between the membrane potential of a compartmental model neuron as a function of input stimulation this is used to show how shunting effects can lead to low output firing rates in associative memory networks

optimal error bounds for two grid schemes applied to the navier stokes equations abstract we consider two grid mixed finite element schemes for the spatial discretization of the incompressible navier stokes equations a standard mixed finite element method is applied over the coarse grid to approximate the nonlinear navier stokes equations while a linear evolutionary problem is solved over the fine grid the previously computed galerkin approximation to the velocity is used to linearize the convective term for the analysis we take into account the lack of regularity of the solutions of the navier stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data optimal error bounds are obtained

decay estimates of solutions for the equations of motion of compressible viscous and heat conductive gases in an exterior domain in ℝ3 we consider the equations of motion of compressible viscous and heat conductive gases in an exterior domain in 3 we give the l q l p estimates for solutions to the linearized equations and show an optimal decay estimate for solutions to the nonlinear problem

a multi level stabilized finite element method for the stationary navier stokes equations abstract this paper proposes and analyzes a multi level stabilized finite element method for the two dimensional stationary navier stokes equations approximated by the lowest equal order finite element pairs the method combines the new stabilized finite element method with the multi level discretization under the assumption of the uniqueness condition the multi level stabilized finite element method consists of solving the nonlinear problem on the coarsest mesh and then performs one newton correction step on each subsequent mesh thus only solving one large linear systems the error analysis shows that the multi level stabilized finite element method provides an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution solving the stationary navier stokes equations on a fine mesh for an appropriate choice of mesh widths h j h j 1 2 j 1 j moreover the numerical illustrations agree completely with the theoretical expectations therefore the multi level stabilized finite element method is more efficient than the standard one level stabilized finite element method

from two state jump to gaussian stochastic processes abstract a stochastic process which bridges the two state jump markoff process and the gaussian process is introduced and fundamental properties of the process are fully analyzed the resulting expressions are combined with the partial cumulant expansion formula to give an exact power spectrum in the form of a continued fraction a rigorous expression is also obtained for a relaxation function in an analytic form

on a class of stable steady flows to the exterior convection problem abstract a class of stable steady convection flows in an exterior domain is given it is proved that the class is l 3 w l m w with m m m 1 when the gravitational field is in l m w for some 1 m

the navier stokes equations in exterior lipschitz domains lp theory abstract we show that the stokes operator defined on l p for an exterior lipschitz domain r n n 3 admits maximal regularity provided that p satisfies 1 p 1 2 1 2 n e for some e 0 in particular we prove that the negative of the stokes operator generates a bounded analytic semigroup on l p for such p in addition l p l q mapping properties of the stokes semigroup and its gradient with optimal decay estimates are obtained this enables us to prove the existence of mild solutions to the navier stokes equations in the critical space l 0 t l 3 locally in time and globally in time for small initial data

global existence for elastic waves with memory we treat the cauchy problem for nonlinear systems of viscoelasticity with a memory term we study the existence and the time decay of the solution to this nonlinear problem the kernel of the memory term includes integrable singularity at zero and polynomial decay at infinity we prove the existence of a global solution for space dimensions n 3 and arbitrary quadratic nonlinearities

existence and uniqueness of time periodic physically reasonable navier stokes flow past a body let be a three dimensional exterior domain of class c2 0 1 assume that a navier stokes liquid is moving in under the action of a body force f that is time periodic of period t and that the velocity of the liquid is zero at spatial infinity in this paper we show that if f satisfies suitable conditions and its norm in appropriate function spaces is sufficiently small there is at least one time periodic strong solution furthermore the velocity field v of such a solution decays to zero for large x as x 1 and its spatial gradient decays as x 2 both uniformly in time in addition the pressure p decays like x 2 and its gradient like x 3 for almost all t 0 t in the special case where f is time independent these solutions are also time independent and coincide with finn s physically reasonable solutions 4 moreover we show that our time periodic solutions are unique in a very large class namely the class of time periodic weak solutions satisfying the energy inequality and with corresponding pressure fields verifying mild summability conditions in 0 t

existence and regularity of a class of weak solutions to the navier stokes equations abstract we construct a class of weak solutions to the navier stokes equations which have second order spatial derivatives and one order time derivatives of p power summability for 1 p 5 4 meanwhile we show that u l s 0 t w 2 r with 1 s 3 2 r 2 for 1 r 5 4 r can be relaxed not to exceed 3 2 if we consider only in the interior of in the end we extend the classical regularity theorem our results show that u is a regular solution if u l s 0 t l r with 1 s 3 2 r 1 for satisfying 1 3 with 1 s 1 r 5 6 for arbitrary domain in r 3 and 1 s 2 for r n with n 3 this result was previously obtained byh beirao da veiga chinese ann math ser b 16 1995 407 412

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

an absolutely stabilized finite element method for the stokes problem on presente une formulation a elements finis absolument stabilisee pour le probleme de stokes on etablit des estimations d erreurs optimales dans la norme l 2 pour l approximation des champs de vitesses et de pressions

stabilized mixed methods for the stokes problem the solution of the stokes problem is approximated by three stabilized mixed methods one due to hughes balestra and franca and the other two being variants of this procedure in each case the bilinear form associated with the saddle point problem of the standard mixed formulation is modified to become coercive over the finite element space error estimates are derived for each procedure

weighted estimates for the incompressible fluid in exterior domains for a strong solution u x t of the navier stokes equations in exterior domain in rn where n 2 3 we study the time decay of x u t lp for n when a domain has a boundary pressure term makes an obstacle since we do not have enough information on the pressure term near the boundary to overcome the difficulty we adopt the ideas in h o bae b j jin temporal and spatial decay rates of navier stokes solutions in exterior domains bull korean math soc 44 3 2007 547 567 h o bae b j jin asymptotic behavior for the navier stokes solutions in 2d exterior domains j funct anal 240 2006 508 529 and we will extend bae and jin s results by modifying their methods

on a larger class of stable solutions to the navier stokes equations in exterior domains where w w x w1 x wn x and x denote the unknown velocity vector and the unknown pressure of the fluid at point x respectively while f f x f i j x i j 1 n is the given tensor with div f n j 1 f 1 j xj nj 1 f n j xj denoting the external force in our previous paper 23 we established the theory of existence uniqueness and regularity for solutions to n s0 in the class w ln

asymptotic stability of large solutions with large perturbation to the navier stokes equations abstract consider weak solutions w of the navier stokes equations in serrin s class w l 0 l q for 2 3 q 1 with 3 q where is a general unbounded domain in r 3 we shall show that although the initial and external disturbances from w are large every perturbed flow v with the energy inequality converges asymptotically to w as v t w t l 2 0 v t w t l 2 o t 1 2 as t

spatial stability of 3d exterior stationary navier stokes flows abstract in this paper we study the stability of stationary solutions w for the navier stokes flows in an exterior domain with zero velocity at infinity with suitable assumptions of w by the works of chen 1993 kozono ogawa 1994 and borchers miyakawa 1995 if u 0 w l r l 3 then one can obtain u t w p o t 3 2 1 r 1 p for 1 r p u t w p o t 3 2 1 r 1 p 1 2 for 1 r p 3 where u x t is a solution of the navier stokes equations with the initial condition u 0 in this paper we will prove that for any 0 3 if x u 0 w belongs to l r then one has x u t w l p o t 3 2 1 r 1 p 2 for p 3 r 3 r

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

on weighted norm estimates for nonstationary incompressible navier stokes flows in a 3d exterior domain abstract we study the time decay of weighted norms of weak and strong solutions to the navier stokes equations in a 3d exterior domain moment estimates for weak solutions and weighted l q estimates for strong solutions are deduced both of which seem to be optimal the relation is discussed between the space time decay and the vanishing of the total net force exerted by the fluid to the body a class of initial data is given so that the total net force associated to the corresponding fluid flows does not vanish

a multilevel mesh independence principle for the navier stokes equations multilevel finite element discretization methods for the navier stokes equations are considered in contrast to usual multilevel methods a superlinear scaling of the consecutive meshwidths h j 1 mathcal o h j alpha j is used on the coarsest mesh the discretized system is solved which is small and nonlinear on each subsequent mesh only one or two newton correction steps are performed so that only one or two larger linear systems are solved the scalings of the meshwidths that lead to optimal accuracy of the approximate solution in both the h 1 and l 2 norm are investigated an error analysis of independent interest is also presented for the basic finite element method using the conservation form of the nonlinear term this formulation leads to a nonlinear system whose nonlinearity is easier to resolve than the systems arising when either the convective or explicitly skew symmetric corms are used for the nonlinear term

on the uniqueness of time periodic solutions to the navier stokes equations in unbounded domains we present a uniqueness theorem for time periodic solutions to the navier stokes equations in unbounded domains thus far results on the uniqueness of time periodic solutions to the navier stokes equations in unbounded domain roughly speaking have only found that a small time periodic ln solution is unique within the class of solutions which have sufficiently small l ln norm in this paper we show that a small time periodic ln solution is unique within the class of all time periodic ln solutions which contains large solutions we also consider the uniqueness of solutions in weak ln space the proof of the present uniqueness theorem is based on the method of dual equations

two level mixed finite element methods for the navier stokes equations with damping abstract in this paper the mixed finite element methods are considered for the navier stokes equations with damping optimal error estimates of the h 1 norm and l 2 norm for the velocity and the l 2 norm for the pressure are obtained moreover two kinds of two level methods based on the oseen correction and newton correction are presented finally two numerical examples are implemented to confirm the analysis

a finite element variational multiscale method for incompressible flow in this paper we present a numerical scheme prove stability existence of solutions uniqueness and convergence of the incompressible navier stokes equations it has the advantage of being defined from strictly algebraic considerations a significant feature of the present method is that the structure of the stabilization term based on the multiscale enrichment and derived from the navier stokes problem itself ample numerical experiments are carried out to confirm the theory and illustrate the effectiveness of the scheme on incompressible fluid

lp lq estimate of the stokes operator and navier stokes flows in the exterior of a rotating obstacle we consider the motion of a viscous fluid filling the whole three dimensional space exterior to a rotating obstacle with constant angular velocity we develop the l p l q estimates and the similar estimates in the lorentz spaces of the stokes semigroup with rotation effect we next apply them to the navier stokes equation to prove the global existence of a unique solution which goes to a stationary flow as t with some definite rates when both the stationary flow and the initial disturbance are sufficiently small in l3 weak l3 space

optimal weighted estimates of the flows in exterior domains abstract in this paper we prove some decay properties of global solutions for the navier stokes equations in an exterior domain r n n 2 3 when a domain has a boundary the pressure term is troublesome since we do not have enough information on the pressure near the boundary to overcome this difficulty by multiplying a special form of test functions we obtain an integral equation he xin 2000 12 first introduced this method and then bae jin 2006 2007 1 13 modified their method to obtain better decay rates also bae roh 2009 11 improved bae jin s results unfortunately their results were not optimal because there exists an unpleasant positive small in their rates in this paper we obtain the following optimal rate without x u t l p c t n 2 1 r 1 p 2 for the initial condition u 0 with x u 0 l r and for any n

a two grid method of the non conforming crouzeix raviart element for the steklov eigenvalue problem abstract this paper discusses a high efficient scheme for the steklov eigenvalue problem a two grid discretization scheme of nonconforming crouzeix raviart element is established with this scheme the solution of a steklov eigenvalue problem on a fine grid h is reduced to the solution of the eigenvalue problem on a much coarser grid h and the solution of a linear algebraic system on the fine grid h by using spectral approximation theory and nitsche lascaux lesaint technique in space h 1 2 we prove that the resulting solution obtained by our scheme can maintain an asymptotically optimal accuracy by taking h h and the numerical experiments indicate that when the eigenvalues k h of nonconforming crouzeix raviart element approximate the exact eigenvalues from below the approximate eigenvalues k h obtained by the two grid discretization scheme also approximate the exact ones from below and the accuracy of k h is higher than that of k h

a critical functional framework for the inhomogeneous navier stokes equations in the half space abstract this paper is devoted to solving globally the boundary value problem for the incompressible inhomogeneous navier stokes equations in the half space in the case of small data with critical regularity in dimension n 3 we state that if the initial density 0 is close to a positive constant in l w n 1 r n and the initial velocity u 0 is small with respect to the viscosity in the homogeneous besov space b n 1 0 r n then the equations have a unique global solution the proof strongly relies on new maximal regularity estimates for the stokes system in the half space in l 1 0 t b p 1 0 r n interesting for their own sake

on time periodic solutions to the boussinesq equations in exterior domains abstract we construct time periodic solutions to the boussinesq equations in a three dimensional exterior domain to this end we estimate the l 1 norm of buoyancy and give some estimates of the heat semigroup in the lorentz space

average firing rate of a neural network with dynamical disorder we analyse the effects of time varying synaptic background activity on the steady state firing rate of a compartmental model neural network with shunting the background is taken to be a multi component dichotomous coloured noise process distributed randomly across the compartments of each neuron we exploit the formal similarity between the neural network model and a model of excitons moving on a lattice with random modulations of the local energy at each site in particular we use a dynamical coherent potential approximation and the method of partial cumulants to evaluate the single neuron green s function averaged over the stochastic background this is then used to determine the firing rate it is found that the firing rate increases with the variance and correlation time of the coloured noise process

decay estimates of gradient of a generalized oseen evolution operator arising from time dependent rigid motions in exterior domains let us consider the motion of a viscous incompressible fluid past a rotating rigid body in three dimensions where the translational and angular velocities of the body are prescribed but time dependent in a reference frame attached to the body we have the navier stokes system with the drift and one half of the coriolis terms in a fixed exterior domain the existence of the evolution operator t t s in the space l q generated by the linearized non autonomous system was proved by hansel and rhandi j reine angew math 694 1 26 2014 and the large time behavior of t t s f in l r for t s rightarrow infty was then developed by hishida math ann 372 915 949 2018 when f is taken from l q with q leqq r the contribution of the present paper concerns such l q l r decay estimates of nabla t t s with optimal rates which must be useful for the study of stability attainability of the navier stokes flow in several physically relevant situations our main theorem completely recovers the l q l r estimates for the autonomous case stokes and oseen semigroups those semigroups with rotating effect in three dimensional exterior domains which were established by hishida and shibata arch ration mech anal 193 339 421 2009 iwashita math ann 285 265 288 1989 kobayashi and shibata math ann 310 1 45 1998 maremonti and solonnikov ann sc norm super pisa 24 395 449 1997 and shibata in amann arendt hieber neubrander nicaise von below eds functional analysis and evolution equations the gunter lumer volume birkhauser basel pp 595 611 2008

abstract lp estimates for the cauchy problem with applications to the navier stokes equations in exterior domains abstract we apply an abstract perturbation theorem to derive global in time lq estimates for the cauchy problem and lq ls estimates for the nonstationary stokes equations in exterior domains this will be applied to obtain various new global in time estimates for weak solutions of the navier stokes equations

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

decay rate of solutions of navier stokes equations in 3 dimensional half space abstract in this article we derive the l 2 spatial decay rate of a weak solution of the incompressible flow in the half space when the half space or some other fluid region with boundary is concerned pressure estimate is main obstacle since we do not have enough information of the pressure on the boundary in this paper we give an idea which does not require any pressure information on the boundary and we hope that our idea could be applied to other problems such as exterior domain problem

a general approach to time periodic incompressible viscous fluid flow problems this article develops a general approach to time periodic incompressible fluid flow problems and semilinear evolution equations it yields on the one hand a unified approach to various classical problems in incompressible fluid flow and on the other hand gives new results for periodic solutions to the navier stokes oseen flow the navier stokes flow past rotating obstacles and in the geophysical setting for ornstein uhlenbeck and various diffusion equations with rough coefficients the method is based on a combination of interpolation and topological arguments as well as on the smoothing properties of the linearized equation

l q l r estimates for solutions of the nonstationary stokes equations in an exterior domain and the navier stokes initial value problems inl q spaces

energy decay for a weak solution of the navier stokes equation with slowly varying external forces we consider the navier stokes system with slowly decaying external forces formula we show that the energy norm of a weak solution has non uniform decay u t 2 0 t under suitable conditions on the data f and a which make the energy of solution bounded in time also we show the exact rate of the decay uniform decay of the energy u t 2 c 1 t e for external forces with a given explicit decay rate

optimal low order finite element methods for incompressible flow abstract in this paper we consider the characterization of optimal stabilisation parameters associated with mixed finite element approximations of stokes flow we show how the optimal parameter in terms of the approximating power of the method is related to the parameter value which minimises the condition number of the associated schur complement system for the pressure field we illustrate the tightness of our abstract theoretical results in the cases of the q 1 p 0 quadrilateral and p 1 p 1 triangular elements

semi discrete stabilized finite element methods for navier stokes equations with nonlinear slip boundary conditions based on regularization procedure based on the pressure projection stabilized methods the semi discrete finite element approximation to the time dependent navier stokes equations with nonlinear slip boundary conditions is considered in this paper because this class of boundary condition includes the subdifferential property then the variational formulation is the navier stokes type variational inequality problem using the regularization procedure we obtain a regularized problem and give the error estimate between the solutions of the variational inequality problem and the regularized problem with respect to the regularized parameter varepsilon which means that the solution of the regularized problem converges to the solution of the navier stokes type variational inequality problem as the parameter varepsilon longrightarrow 0 moreover some regularized estimates about the solution of the regularized problem are also derived under some assumptions about the physical data the pressure projection stabilized finite element methods are used to the regularized problem and some optimal error estimates of the finite element approximation solutions are derived

domains of fractional powers of the stokes operator in lr spaces soit d un domaine borne de r n n 2 a frontiere lisse s soit 1 r et soit x r la fermeture dans l r d n de tous les champs vectoriels solenoidaux lisses a support compact dans d soit w r m d l espace de sobolev d ordre m et soit le laplacien on considere l operateur de stokes a r p r de domaine d a r w w r 2 d n w 5 0 mx 2 soit a un nombre reel positif pour chaque positif il y a une constante ce a telle que a r z ce a exp im z pour tout z satisfaisant a rez 0 ou a r z est la norme operateur de a r z

a green s function approach to analysing the effects of random synaptic background activity in a model neural network the effect of randomly distributed synaptic background activity on the states of self sustained firing in a model neural network with shunting is investigated using mean field theory the steady state of the network is expressed in terms of an ensemble averaged single neuron green s function this green s function is shown to satisfy a matrix equation identical in form to that found in the tight binding alloy model of excitations on a one dimensional disordered lattice the ensemble averaging is then performed using a coherent potential approximation thus allowing the steady state firing rate of the network to be determined the firing rate is found to decrease as the mean level of background activity across the network is increased a uniform background zero variance leads to a greater reduction than a randomly distributed one non zero variance

parallel iterative stabilized finite element algorithms based on the lowest equal order elements for the stationary navier stokes equations abstract based on a fully overlapping domain decomposition technique and the lowest equal order finite elements three parallel iterative stabilized finite element algorithms for the stationary navier stokes equations are proposed and studied where the stabilization term is based on two local gauss integrations at element level in these parallel algorithms each processor independently computes a local stabilized solution in its own subdomain making the algorithms have low communication cost and easy to implement based on a sequential solver the algorithms can yield an approximate solution with an accuracy comparable to that of the standard stabilized finite element solution with a substantial reduction in computational time theoretical and numerical results demonstrated the effectiveness and efficiency of the algorithms

regularity criteria in terms of pressure for the 3 d navier stokes equations in a generic domain we consider the 3 d navier stokes equations in the half space 3 or a bounded domain with smooth boundary or else an exterior domain with smooth boundary some new sufficient conditions on pressure or the gradient of pressure for the regularity of weak solutions to the navier stokes equations are obtained

the navier stokes equations in the weak l n space with time dependent external force abstract we consider the navier stokes equations with time dependent external force either in the whole time or in positive time with initial data with domain either the whole space the half space or an exterior domain of dimension n ge 3 we give conditions on the external force sufficient for the unique existence of small solutions in the weak l n space bounded for all time in particular this result gives sufficient conditions for the unique existence and the stability of small time periodic solutions or almost periodic solutions this result generalizes the previous result on the unique existence and the stability of small stationary solutions in the weak l n space with time independent external force

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

existence of strong solutions and decay of turbulent solutions of navier stokes flow with nonzero dirichlet boundary data abstract recently leray s problem of the l 2 decay of a special weak solution to the navier stokes equations with nonhomogeneous boundary values was studied by the authors exploiting properties of the approximate solutions converging to this solution in this paper this result is generalized to the case of an arbitrary weak solution satisfying the strong energy inequality

on a local energy decay estimate of solutions to the hyperbolic type stokes equations abstract in this paper we discuss a local energy decay estimate of solutions to the initial boundary value problem for the hyperbolic type stokes equations of incompressible fluid flow in an exterior domain and a perturbed half space the equations are linearized version of the hyperbolic navier stokes equations introduced by racke and saal 15 which are obtained as a delayed case for the deformation tensor in the incompressible navier stokes equations our proof of the local energy decay estimate is based on dan and shibata 2 in 2 they treated the dissipative wave equations in an exterior domain and discussed the local energy decay estimate our approach uses the fact that applying the helmholtz projection to the hyperbolic type stokes equations we obtain equations similar to the dissipative wave ones

an absolutely stable pressure poisson stabilized finite element method for the stokes equations the pressure poisson stabilized galerkin method for the stokes equation requires the choice of a positive parameter existing theoretical predictions for the range of parameter values that yield stable discretizations seem to be very pessimistic when compared to the computational evidence motivated by this wide gap we first examine a continuous prototype for this class of schemes we show that the prototype is absolutely stable i e it is stable for all parameter values and is optimally accurate we then define a new practical variant of the well known pressure poisson stabilized scheme we prove that the new method is absolutely stable just like its continuous prototype and that it achieves optimal convergence rates with respect to the same mesh independent norms the new method differs from the standard pressure poisson stabilized method in several important aspects first its definition does not degrade to a penalty formulation for the lowest order nodal spaces second the method is absolutely stable with respect to the natural norm for the problem while the standard pressure poisson stabilized method is stable with respect to a mesh dependent norm

stabilization of low order mixed finite elements for the stokes equations we present a new family of stabilized methods for the stokes problem the focus of the paper is on the lowest order velocity pressure pairs while not lbb compliant their simplicity and attractive computational properties make these pairs a popular choice in engineering practice our stabilization approach is motivated by terms that characterize the lbb deficiency of the unstable spaces the stabilized methods are defined by using these terms to modify the saddle point lagrangian associated with the stokes equations the new stabilized methods offer a number of attractive computational properties in contrast to other stabilization procedures they are parameter free do not require calculation of higher order derivatives or edge based data structures and always lead to symmetric linear systems furthermore the new methods are unconditionally stable achieve optimal accuracy with respect to solution regularity and have simple and straightforward implementations we present numerical results in two and three dimensions that showcase the excellent stability and accuracy of the new methods

two level picard and modified picard methods for the navier stokes equations abstract iterative methods of picard type for the navier stokes equations are known to converge only for quite small reynolds numbers however we study methods involving just one such iteration at general reynolds numbers for the initial approximation a coarse mesh of width h0 is used the corrected approximation is computed by just one picard or modified picard step on a fine mesh of width h1 for example h1 may be of order o h 0 2 when linear velocity elements are used the resulting method requires the solution of a small system of nonlinear equations on the coarse mesh and only one larger linear system on the fine mesh this two level picard method is proven to converge for fixed reynolds number as h 0 further the fine mesh solution satisfies a quasi optimal error bound the error constants grow as re as for the usual finite element method one very heuristic explanation why one step of the divergent picard method might work when beginning with a coarse mesh approximation is that the terms neglected involve lower order derivatives thus they are approximated with higher accuracy on the coarse mesh this is linked with a smoothing property of the fine mesh step

an existence theorem for the navier stokes flow in the exterior of a rotating obstacle we consider the three dimensional navier stokes initial value problem in the exterior of a rotating obstacle it is proved that a unique solution exists locally in time if the initial velocity possesses the regularity l 1 2 this regularity assumption is the same as that in the famous paper of fujita kato an essential step for the proof is the deduction of a certain smoothing property together with estimates near t 0 of the semigroup which is not an analytic one generated by the operator in the space l 2 where stands for the angular velocity of the rotating obstacle and p denotes the projection associated with the helmholtz decomposition

liouville theorems for the stokes equations with applications to large time estimates abstract we study liouville theorems for the non stationary stokes equations in exterior domains in r n under decay conditions for spatial variables as applications we prove that the stokes semigroup is a bounded analytic semigroup on l of angle 2 for n 3 we also prove large time estimates for n 2 with zero net force