global well posedness for the 3d primitive equations in anisotropic framework abstract this paper is concerned with the initial boundary value problem of the 3d primitive equations with only horizontal viscosities and horizontal diffusion we first establish the local well posedness result of this system with the initial data belongs to the anisotropic sobolev pace h 0 s s 1 2 then under the basic and high order energy estimates one can extend the local solution to the global one provided that s 1 the restrictions on the initial data is more weaker than the previous results which are obtained by cao et al 2016 8 and 2017 9 in which the initial data belongs to h 2 and h 1 l respectively

existence and regularity of invariant measures for the three dimensional stochastic primitive equations we establish the continuity of the markovian semigroup associated with strong solutions of the stochastic 3d primitive equations and prove the existence of an invariant measure the proof is based on new moment bounds for strong solutions the invariant measure is supported on strong solutions and is furthermore shown to have higher regularity properties

on the solutions of the 3d steady and unsteady primitive equations of the ocean abstract in this article the existence and uniqueness of the solution for the 3d steady viscous primitive equations pes for brevity of the ocean under the uniqueness condition are analyzed also the h 2 regularity results of the solution for the 3d steady viscous pes of the ocean under the stability condition is provided finally the convergence of the solution for the 3d unsteady viscous pes of the ocean to the solution for the 3d steady viscous pes of the ocean when t towards to infinity under the uniqueness condition is studied

mathematical analysis of geophysical balance models abstract an analysis of an approximation to the rotating shallow water equations is presented the approximation removes the fast waves without introducing secular terms and is valid for physical boundaries and prepared initial data in particular the shallow water equations are decomposed into two equations describing the slow and fast dynamics the basic idea is one of enslaving in which the fast part of the solution is expressed as a function of the slow part yielding an approximation to the slow dynamics existence and convergence theorems are given

global well posedness of the 3d primitive equations with partial vertical turbulence mixing heat diffusion the three dimensional incompressible viscous boussinesq equations under the assumption of hydrostatic balance govern the large scale dynamics of atmospheric and oceanic motion and are commonly called the primitive equations to overcome the turbulence mixing a partial vertical diffusion is usually added to the temperature advection or density stratification equation in this paper we prove the global regularity of strong solutions to this model in a three dimensional infinite horizontal channel subject to periodic boundary conditions in the horizontal directions and with no penetration and stress free boundary conditions on the solid top and bottom boundaries specifically we show that short time strong solutions to the above problem exist globally in time and that they depend continuously on the initial data

finite element methods of least squares type we consider the application of least squares variational principles to the numerical solution of partial differential equations our main focus is on the development of least squares finite element methods for elliptic boundary value problems arising in fields such as fluid flows linear elasticity and convection diffusion for many of these problems least squares principles offer numerous theoretical and computational advantages in the algorithmic design and implementation of corresponding finite element methods that are not present in standard galerkin discretizations most notably the use of least squares principles leads to symmetric and positive definite algebraic problems and allows us to circumvent stability conditions such as the inf sup condition arising in mixed methods for the stokes and navier stokes equations as a result application of least squares principles has led to the development of robust and efficient finite element methods for a large class of problems of practical importance

global strong lp well posedness of the 3d primitive equations with heat and salinity diffusion abstract consider the full primitive equations i e the three dimensional primitive equations coupled to the equation for temperature and salinity and subject to outer forces it is shown that this set of equations is globally strongly well posed for arbitrary large initial data lying in certain interpolation spaces which are explicitly characterized as subspaces of h 2 p p 1 p satisfying certain boundary conditions in particular global well posedness of the full primitive equations is obtained for initial data having less differentiability properties than h 1 hereby generalizing a result by cao and titi 2007 5 to the case of non smooth data in addition it is shown that the solutions are exponentially decaying provided the outer forces possess this property

on an efficient second order backward difference newton scheme for mhd system abstract for the 2d 3d time dependent magnetohydrodynamics mhd system we propose a new second order backward difference formula newton scheme sbdfn which is a combination of second order backward difference approximation of the time derivative terms and newton treatment of the nonlinear terms meanwhile the finite element method is applied as spatial discretization firstly the optimal convergence of spatially semi discrete form is deduced secondly the stability and well posedness of sbdfn scheme are provided under c where c is independent of h based on these results optimal error estimates of the scheme in time are proved by negative norm technique finally numerical experiments are carried out to validate the theoretical analysis

analysis of a large eddy simulation model based on anisotropic filtering abstract we consider a new large eddy simulation model derived with the introduction of suitable horizontal anisotropic differential filters one main advantage of this filtering is that for channel flows there is no need for artificial boundary conditions hence we can deal with some realistic problems equipped with dirichlet boundary conditions in special bounded domains at least those bounded only in one direction recent numerical results for a similar model based on a derivation with wave number asymptotics are also recalled after a detailed analysis of the properties of the differential filter we prove that the resulting initial boundary value problem is well posed in suitable anisotropic sobolev spaces giving a strong mathematical support to the model we propose some remarks on higher accuracy approximate deconvolution models are also given in the last section

issues related to least squares finite element methods for the stokes equations least squares finite element methods have become increasingly popular for the approximate solution of first order systems of partial differential equations here after a brief review of some existing theories a number of issues connected with the use of such methods for the velocity vorticity pressure formulation of the stokes equations in two dimensions in realistic settings are studied through a series of computational experiments finite element spaces that are not covered by existing theories are considered included in these are piecewise linear approximations for the velocity mixed boundary conditions which are also not covered by existing theories are also considered as is enhancing mass conservation next problems in nonconvex polygonal regions and the resulting nonsmooth solutions are considered with a view toward seeing how accuracy can be improved a conclusion that can be drawn from this series of computational experiments is that the use of appropriate mesh dependent weights in the least squares functional almost always improves the accuracy of the approximations concluding remarks concerning three dimensional problems the nonlinear navier stokes equations and the conditioning of the discrete systems are provided

erratum to a bubble stabilized least squares finite element method for steady mhd duct flow problems at high hartmann numbers j comput phys 228 2009 8301 8320

further considerations on residual free bubbles for advective diffusive equations abstract we further consider the galerkin method for advective diffusive equations in two dimensions the finite dimensional space employed is of piecewise polynomials enriched with residual free bubbles rfb we show that in general this method does not coincide with the supg method unless the piecewise polynomials are spanned by linear functions furthermore a simple stability analysis argument displays the effect of the rfb on the reduced space of piecewise polynomials which in some situations is not equivalent to streamline diffusion for bilinears

a new finite element formulation for computational fluid dynamics viii the galerkin least squares method for advective diffusive equations abstract galerkin least squares finite element methods are presented for advective diffusive equations galerkin least squares represents a conceptual simplification of supg and is in fact applicable to a wide variety of other problem types a convergence analysis and error estimates are presented

f e m in steady mhd duct flow problems the governing equations and the boundary conditions of steady incompressible magnetohydrodynamic mhd problems are adapted to the case of steady mhd flow in simply connected cross sections with arbitrary wall conductance the problem is then solved numerically by means of the finite element method based on the weighted residual approach and solutions for industrially relevant hartmann numbers and non symmetric kinetic fields are presented

enhanced rfb method the residual free bubble method rfb is a parameter free stable finite element method that has been successfully applied to a wide range of boundary value problems exhibiting multiple scale behaviour if some local features of the solution are known a priori the approximation properties of the rfb finite element space can be improved by enriching it on selected edges of the partition by edge bubbles that are supported on pairs of neighbouring elements motivated by this idea we define and analyse an enhanced residual free bubble method for the solution of convection dominated convection diffusion problems in 2 d our a priori analysis highlights the limitations of the rfb method and the improved global approximation properties of the new method the theoretical results are supported by detailed numerical experiments

global well posedness of strong solutions to the 3d primitive equations with horizontal eddy diffusivity in this paper we consider the initial boundary value problem of the 3d primitive equations for oceanic and atmospheric dynamics with only horizontal diffusion in the temperature equation global well posedness of strong solutions are established with h2 initial data 2014 elsevier inc all rights reserved

finite element method solution of electrically driven magnetohydrodynamic flow the magnetohydrodynamic mhd flow in a rectangular duct is investigated for the case when the flow is driven by the current produced by electrodes placed one in each of the walls of the duct where the applied magnetic field is perpendicular the flow is steady laminar and the fluid is incompressible viscous and electrically conducting a stabilized finite element with the residual free bubble rfb functions is used for solving the governing equations the finite element method employing the rfb functions is capable of resolving high gradients near the layer regions without refining the mesh thus it is possible to obtain solutions consistent with the physical configuration of the problem even for high values of the hartmann number before employing the bubble functions in the global problem we have to find them inside each element by means of a local problem this is achieved by approximating the bubble functions by a nonstandard finite element method based on the local problem equivelocity and current lines are drawn to show the well known behaviours of the mhd flow those are the boundary layer formation close to the insulated walls for increasing values of the hartmann number and the layers emanating from the endpoints of the electrodes the changes in direction and intensity with respect to the values of wall inductance are also depicted in terms of level curves for both the velocity and the induced magnetic field

on the long time stability of a temporal discretization scheme for the three dimensional viscous primitive equations in this article a semi discretized euler scheme to solve the three dimensional viscous primitive equations is studied based on suitable assumptions on the initial data and forcing terms the long time stability of the proposed scheme is proven by showing that the h 1 norm in space variables of the solutions is bounded at each time step when the time step satisfies certain smallness condition

finite element approximation for grad div type systems in the plane this paper studies a 3 times 3 linear system of differential equations of grad div type with appropriate boundary conditions in constructing numerical approximate solutions to this boundary val

parallel computation of real solving bivariate polynomial systems by zero matching method we present a new algorithm for solving the real roots of a bivariate polynomial system s f x y g x y with a finite number of solutions by using a zero matching method the method is based on a lower bound for the bivariate polynomial system when the system is non zero moreover the multiplicities of the roots of s 0 can be obtained by the associated quotient ring technique and a given neighborhood from this approach the parallelization of the method arises naturally by using a multidimensional matching method this principle can be generalized to the multivariate equation systems

regularity and singularity in solutions of the three dimensional navier stokes equations higher moments of the vorticity field m t in the form of l 2 m norms graphic 1 are used to explore the regularity problem for solutions of the three dimensional incompressible navier stokes equations on the domain graphic 2 it is found that the set of quantities formula 3 provide a natural scaling in the problem resulting in a bounded set of time averages d m t on a finite interval of time 0 t the behaviour of d m 1 d m is studied on what are called good and bad intervals of 0 t which are interspersed with junction points neutral i for large but finite values of m with large initial data m 0 o gr 4 it is found that there is an upper bound formula 4 which is punctured by infinitesimal gaps or windows in the vertical walls between the good bad intervals through which solutions may escape while this result is consistent with that of leray leray 1934 acta math 63 193 248 doi 10 1007 bf02547354 5 and scheffer scheffer 1976 pacific j math 66 535 552 this estimate for m corresponds to a length scale well below the validity of the navier stokes equations 1 embed inline graphic 1 gif 2 embed inline graphic 2 gif 3 embed graphic 1 gif 4 embed graphic 2 gif 5 http dx doi org doi 10 1007 bf02547354

optimal control of the primitive equations of the ocean with state constraints abstract we investigate in this article pontryagin s maximum principle for a class of control problems associated with the primitive equations pes of the ocean these optimal problems involve a state constraint similar to that considered in wang 2002 7 for the three dimensional navier stokes ns equations the main difference between this work and wang 2002 7 is that the nonlinearity in the pes is stronger than in the three dimensional ns systems

a priori error analysis of residual free bubbles for advection diffusion problems we develop an a priori error analysis of a finite element approximation to the elliptic advection diffusion equation eps delta u conv cdot nabla u f subject to a homogeneous dirichlet boundary condition based on the use of residual free bubble functions an optimal order error bound is derived in the so called stability norm biggl eps nabla v 2 l 2 omega sum t h t conv cdot nabla v 2 l 2 t biggr 1 2 where ht denotes the diameter of element t in the subdivision of the computational domain

finite time blowup for the inviscid primitive equations of oceanic and atmospheric dynamics in an earlier work we have shown the global for all initial data and all time well posedness of strong solutions to the three dimensional viscous primitive equations of large scale oceanic and atmospheric dynamics in this paper we show that for certain class of initial data the corresponding smooth solutions of the inviscid non viscous primitive equations if they exist they blow up in finite time specifically we consider the three dimensional inviscid primitive equations in a three dimensional infinite horizontal channel subject to periodic boundary conditions in the horizontal directions and with no normal flow boundary conditions on the solid top and bottom boundaries for certain class of initial data we reduce this system into the two dimensional system of primitive equations in an infinite horizontal strip with the same type of boundary conditions and then we show that for specific sub class of initial data the corresponding smooth solutions of the reduced inviscid two dimensional system develop singularities in finite time

the finite element method for mhd flow at high hartmann numbers abstract a stabilized finite element method using the residual free bubble functions rfb is proposed for solving the governing equations of steady magnetohydrodynamic duct flow a distinguished feature of the rfb method is the resolving capability of high gradients near the layer regions without refining mesh we show that the rfb method is stable by proving that the numerical method is coercive even not only at low values but also at moderate and high values of the hartmann number numerical results confirming theoretical findings are presented for several configurations of interest the approximate solution obtained by the rfb method is also compared with the analytical solution of shercliff s problem

a least squares finite element method using elsasser variables for magnetohydrodynamic equations abstract there are various different forms of magnetohydrodynamic mhd equations and they have been studied for years due to its complicated coupling between variables this paper proposes to use an equivalently transformed mhd equations with elsasser variables and the least squares finite element method to find the approximation to them introducing new variables by combining fluid velocity and magnetic field yields a navier stokes like system then the first order system least squares method using displacement recasts the transformed mhd equations into a system of first order partial differential equations and the newton s algorithm linearizes the problem an l 2 residual functional is defined to minimize and the unique existence of corresponding weak solution is shown finally the convergence of proposed approximation is analyzed and several numerical examples are presented to verify the theory

first order decoupled finite element method of the three dimensional primitive equations of the ocean this paper is concerned with a first order fully discrete decoupled method for solving the three dimensional 3d primitive equations of the ocean with the dirichlet boundary conditions on the side where a decoupled semi implicit scheme is used for the time discretization and the p 1 p 1 p 1 p 1 p 1 finite element for velocity pressure and density is used for the spatial discretization of these equations the h 1 l 2 h 1 optimal error estimates for the numerical solution u h n p h n theta h n and the l 2 optimal error estimate for u n h theta h n are established under the restriction of 0 h le beta 1 and 0 tau le beta 2 for some positive constants beta 1 and beta 2 moreover numerical investigations are provided to show that the first order decoupled method is of almost unconditional convergence with accuracy mathcal o h tau in the h 1 norm and mathcal o h 2 tau in the l 2 norm for solving the 3d primitive equations of the ocean numerical results are gi

global well posedness and finite dimensional global attractor for a 3 d planetary geostrophic viscous model in this paper we consider a three dimensional planetary geostrophic viscous model of the gyre scale mid latitude ocean we show the global existence and uniqueness of the weak and strong solutions to this model moreover we establish the existence of a nite dimensional global attractor to this dissipative evolution system c 2003 wiley periodicals inc

on the stability of residual free bubbles for convection diffusion problemsand their approximation by a two level finite element method we consider the galerkin finite element method for partial diffferential equations in two dimensions where the finite dimensional space used consists of piecewise isoparametric polynomials enriched with bubble functions writing l for the differential operator we show that for elliptic convection diffusion problems the component of the bubble enrichment that stabilizes the method is essentially equivalent to a petrov galerkin method with an l spline exponentially fitted trial space and piecewise polynomial test space the remaining component of the bubble influences the accuracy of the method a stability inequality recently obtained by brezzi franca and russo for a limiting case of bubbles applied to convection diffusion problems is shown to be slightly weaker than the standard stability inequality that is obtained for the sdfem supg method thereby demonstrating that the bubble approach is in general slightly less stable than the streamline diffusion method when the trial functions are piecewise linear we show that residual free bubbles are as stable as sdfem supg and we extend this stability inequality to include positive mesh peclet numbers in the convection dominated regime approximate computations of the residual free bubbles are performed using a two level finite element method

revisiting stabilized finite element methods for the advective diffusive equation we give a brief overview of stabilized finite element methods and illustrate the developments applied to the advection diffusion equation this article presents a concise perspective of the developments emanated from the works started in the 1980s through today 2005 elsevier b v all rights reserved pacs 47 11 j

a bubble stabilized least squares finite element method for steady mhd duct flow problems at high hartmann numbers in this paper we devise a stabilized least squares finite element method using the residual free bubbles for solving the governing equations of steady magnetohydrodynamic duct flow we convert the original system of second order partial differential equations into a first order system formulation by introducing two additional variables then the least squares finite element method using c 0 linear elements enriched with the residual free bubble functions for all unknowns is applied to obtain approximations to the first order system the most advantageous features of this approach are that the resulting linear system is symmetric and positive definite and it is capable of resolving high gradients near the layer regions without refining the mesh thus this approach is possible to obtain approximations consistent with the physical configuration of the problem even for high values of the hartmann number before incoorperating the bubble functions into the global problem we apply the galerkin least squares method to approximate the bubble functions that are exact solutions of the corresponding local problems on elements therefore we indeed introduce a two level finite element method consisting of a mesh for discretization and a submesh for approximating the computations of the residual free bubble functions numerical results confirming theoretical findings are presented for several examples including the shercliff problem

global well posedness of the three dimensional viscous primitive equations of large scale ocean and atmosphere dynamics in this paper we prove the global existence and uniqueness regularity of strong solutions to the three dimensional viscous primitive equations which model large scale ocean and atmosphere dynamics 1 introduction large scale dynamics of oceans and atmosphere is governed by the primitive equations which are derived from the navier stokes equations with rotation coupled to thermodynamics and salinity diffusion transport equations which account for the buoyancy forces and stratification effects under the boussinesq approximation moreover and due to the shallowness of the oceans and the atmosphere i e the depth of the fluid layer is very small in comparison to the radius of the earth the vertical large scale motion in the oceans and the atmosphere is much smaller than the horizontal one which in turn leads to modeling the vertical motion by the hydrostatic balance as a result

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

least squares mixed finite elements for second order elliptic problems a theoretical analysis of a least squares mixed finite element method for second order elliptic problems in two and three dimensional domains is presented it is proved that the method is not subj

an exponential compact difference scheme for solving 2d steady magnetohydrodynamic mhd duct flow problems in this article an exponential high order compact ehoc difference scheme on the nine point stencil is developed for the solution of the coupled equations representing the steady incompressible viscous magnetohydrodynamic mhd flow through a straight channel of rectangular section a key property of the ehoc scheme is that it has excellent stability and higher accuracy so that the high gradients near the boundary layer areas can be effectively resolved without refining the mesh numerical experiments are carried out to validate the performance of the currently proposed scheme computation results of the mhd flow in the 2d square channel problems with different wall conductivities are presented for hartmann numbers ranging from 10 to 10 6 the numerical solutions obtained with the newly developed ehoc scheme are also compared with analytic solutions and numerical results by other available methods in the literature

formulation of the equations of the humid atmosphere in the context of variational inequalities abstract the equations of the humid atmosphere with saturation are considered in the case where the saturation concentration q s is not assumed to be constant and obeys the laws of thermodynamics 22 23 32 this article starts from the observation that these equations are not correct in the extreme cases where the atmosphere is totally dry or totally wet that is when the concentration q of water vapor is equal to 0 or 1 a new formulation of the equations is proposed to remedy this difficulty in the context of variational inequalities results of existence maximum principle and regularity of solutions are derived for these new variational inequalities

two new upwind difference schemes for a coupled system of convection diffusion equations arising from the steady mhd duct flow problems in this paper we develop two new upwind difference schemes for solving a coupled system of convection diffusion equations arising from the steady incompressible mhd duct flow problem with a transverse magnetic field at high hartmann numbers such an mhd duct flow is convection dominated and its solution may exhibit localized phenomena such as boundary layers namely narrow boundary regions where the solution changes rapidly most conventional numerical schemes cannot efficiently solve the layer problems because they are lacking in either stability or accuracy in contrast the newly proposed upwind difference schemes can achieve a reasonable accuracy with a high stability and they are capable of resolving high gradients near the layer regions without refining the grid the accuracy of the first new upwind scheme is o h k and the second one improves the accuracy to o e 2 h k email protected h 2 k 2 h 3 k 3 where 0 email protected 1 email protected 1 and m is the high hartmann number numerical examples are provided to illustrate the performance of the newly proposed upwind difference schemes