the universal r matrix for u_q rm sl 3 and beyond ther matrices for the quantised lie algebrasa n are constructed through the quantum double procedure given by drinfel d 6 the case ofu q sl 3 is thoroughly analysed initially to demonstrate the more subtle points of the calculation the ease of the calculation fora n is very dependent on a choice of generators for the borel subalgebrau q b and its dual and a certain ordering imposed on these generators which is related to the length of a certain word in the weyl group

new approach to the quantum nonlinear schrodinger equation a new technique is proposed which allows one to avoid the space and uv cutoffs when quantising field theoretic models integrable by means of the quantum spectral transform r matrix method the technique is based on a new definition of the monodromy matrix for the infinite interval for the nonlinear schrodinger equation repulsive case zero density the known excitation spectrum is reproduced

critical behavior of a kam surface i empirical results kolmogorov arnol d moser kam surfaces are studied in the context of a perturbed two dimensional twist map in particular we ask how a kam surface can disappear as the perturbation parameter is increased following greene we use cycles to numerically construct the kam curve and discover that at the critical coupling it shows structure at all length scales aspects of this structure are fitted by a scaling analysis critical indices and scaling functions are determined numerically some evidence is presented which suggests that the results are universal

finite element approximations for stokes darcy flow with beavers joseph interface conditions numerical solutions using finite element methods are considered for transient flow in a porous medium coupled to free flow in embedded conduits such situations arise for example for groundwater flows in karst aquifers the coupled flow is modeled by the darcy equation in a porous medium and the stokes equations in the conduit domain on the interface between the matrix and conduit beavers joseph interface conditions instead of the simplified beavers joseph saffman conditions are imposed convergence and error estimates for finite element approximations are obtained numerical experiments illustrate the validity of the theoretical results

coupling fluid flow with porous media flow the transport of substances back and forth between surface water and groundwater is a very serious problem we study herein the mathematical model of this setting consisting of the stokes equations in the fluid region coupled with the darcy equations in the porous medium coupled across the interface by the beavers joseph saffman conditions we prove existence of weak solutions and give a complete analysis of a finite element scheme which allows a simulation of the coupled problem to be uncoupled into steps involving porous media and fluid flow subproblems this is important because there are many legacy codes available which have been optimized for uncoupled porous media and fluid flow

iterative techniques for time dependent stokes problems abstract in this paper we consider solving the coupled systems of discrete equations which arise from implicit time stepping procedures for the time dependent stokes equations using a mixed finite element spatial discretization at each time step a two by two block system corresponding to a perturbed stokes problem must be solved although there are a number of techniques for iteratively solving this type of block system to be effective they require a good preconditioner for the resulting pressure operator schur complement in contrast to the time independent stokes equations where the pressure operator is well conditioned the pressure operator for the perturbed system becomes more ill conditioned as the time step is reduced and or the reynolds number is increased in this paper we shall describe and analyze preconditioners for the resulting pressure systems these preconditioners give rise to iterative rates of convergence which are independent of both the mesh size h as well as the time step and reynolds number parameter k

a domain decomposition method for the steady state navier stokes darcy model with beavers joseph interface condition this paper proposes and analyzes a robin type multiphysics domain decomposition method ddm for the steady state navier stokes darcy model with three interface conditions in addition to the two regular interface conditions for the mass conservation and the force balance the beavers joseph condition is used as the interface condition in the tangential direction the major mathematical difficulty in adopting the beavers joseph condition is that it creates an indefinite leading order contribution to the total energy budget of the system y cao et al comm math sci 8 2010 pp 1 25 y cao et al siam j numer anal 47 2010 pp 4239 4256 in this paper the well posedness of the navier stokes darcy model with beavers joseph condition is analyzed by using a branch of nonsingular solutions by following the idea in y cao et al numer math 117 2011 pp 601 629 the three physical interface conditions are utilized together to construct the robin type boundary conditions on th

effective preconditioning of uzawa type schemes for a generalized stokes problem the schur complement of a model problem is considered as a preconditioner for the uzawa type schemes for the generalized stokes problem the stokes problem with the additional term alpha boldmath u in the motion equation the implementation of the preconditioned method requires for each iteration only one extra solution of the poisson equation with neumann boundary conditions for a wide class of 2d and 3d domains a theorem on its convergence is proved in particular it is established that the method converges with a rate that is bounded by some constant independent of alpha some finite difference and finite element methods are discussed numerical results for finite difference mac scheme are provided

grad div stablilization for stokes equations in this paper a stabilizing augmented lagrangian technique for the stokes equations is studied the method is consistent and hence does not change the continuous solution we show that this stabilization improves the well posedness of the continuous problem for small values of the viscosity coefficient we analyze the influence of this stabilization on the accuracy of the finite element solution and on the convergence properties of the inexact uzawa method

on the role of the helmholtz decomposition in mixed methods for incompressible flows and a new variational crime abstract according to the helmholtz decomposition the irrotational parts of the momentum balance equations of the incompressible navier stokes equations are balanced by the pressure gradient unfortunately nearly all mixed methods for incompressible flows violate this fundamental property resulting in the well known numerical instability of poor mass conservation the origin of this problem is the lack of l 2 orthogonality between discretely divergence free velocities and irrotational vector fields therefore a new variational crime for the nonconforming crouzeix raviart element is proposed where divergence free lowest order raviart thomas velocity reconstructions reestablish l 2 orthogonality this approach allows to construct a cheap flow discretization for general 2d and 3d simplex meshes that possesses the same advantageous robustness properties like divergence free flow solvers in the stokes case optimal a priori error estimates for the velocity gradients and the pressure are derived moreover the discrete velocity is independent of the continuous pressure several detailed linear and nonlinear numerical examples illustrate the theoretical findings

on the accuracy of the rotation form in simulations of the navier stokes equations the rotation form of the navier stokes equations nonlinearity is commonly used in high reynolds number flow simulations it was pointed out by a few authors and not widely known apparently that it can also lead to a less accurate approximate solution than the usual email protected u form we give a different explanation of this effect related to i resolution of the bernoulli pressure and ii the scaling of the coupling between velocity and pressure error with respect to the reynolds number we show analytically that i the difference between the two nonlinearities is governed by the difference in the resolution of the bernoulli and kinematic pressures and ii a simple linear grad div stabilization ameliorates much of the bad scaling of the velocity error with respect to re the rotation form does have superior conservation properties to the alternatives and it appears to be amenable to more efficient preconditioners thus the rotational form with grad div stabilization is a promising method we also give experiments that show bad velocity approximation is tied to poor pressure resolution in either form

coupled generalized nonlinear stokes flow with flow through a porous medium in this article we analyze the flow of a fluid through a coupled stokes darcy domain the fluid in each domain is non newtonian modeled by the generalized nonlinear stokes equation in the free flow region and the generalized nonlinear darcy equation in the porous medium a flow rate is specified along the inflow portion of the free flow boundary we show existence and uniqueness of a variational solution to the problem we propose and analyze an approximation algorithm and establish a priori error estimates for the approximation

a unified stabilized method for stokes and darcy s equations we use the lowest possible approximation order piecewise linear continuous velocities and piecewise constant pressures to compute solutions to stokes equation and darcy s equation applying an edge stabilization term to avoid locking we prove that the formulation satisfies the discrete inf sup condition we prove an optimal a priori error estimate for both problems the formulation is then extended to the coupled case using a nitsche type weak formulation allowing for different meshes in the two subdomains finally we present some numerical examples verifying the theoretical predictions and showing the flexibility of the coupled approach

on quantum integrability of the landau lifshitz model we investigate the quantum integrability of the landau lifshitz ll model and solve the long standing problem of finding the local quantum hamiltonian for the arbitrary n particle sector the particular difficulty of the ll model quantization which arises due to the ill defined operator product is dealt with by simultaneously regularizing the operator product and constructing the self adjoint extensions of a very particular structure the diagonalizibility difficulties of the hamiltonian of the ll model due to the highly singular nature of the quantum mechanical hamiltonian are also resolved in our method for the arbitrary n particle sector we explicitly demonstrate the consistency of our construction with the quantum inverse scattering method due to sklyanin lett math phys 15 357 1988 and give a prescription to systematically construct the general solution which explains and generalizes the puzzling results of sklyanin for the particular two particle sector case moreover we demonstrate t

fast iterative solution of stabilised stokes systems part ii using general block preconditioners mixed finite element approximation of the classical stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables iterative solution of such indefinite systems is feasible and is an attractive approach for large problems part i of this work described a conjugate gradient like method the method of preconditioned conjugate residuals which is applicable to symmetric indefinite problems a j wathen and d j silvester siam j numer anal 30 1993 pp 630 649 using simple arguments estimates for the eigenvalue distribution of the discrete stokes operator on which the convergence rate of the iteration depends are easily derived part i discussed the important case of diagonal preconditioning scaling this paper considers the more general class of block preconditioners where the partitioning into blocks corresponds to the natural partitioning into the velocity and pressure variables it is shown that provid

a class of iterative methods for solving saddle point problems we consider the numerical solution of indefinite systems of linear equations arising in the calculation of saddle points we are mainly concerned with sparse systems of this type resulting from certain discretizations of partial differential equations we present an iterative method involving two levels of iteration similar in some respects to the uzawa algorithm we relate the rates of convergence of the outer and inner iterations proving that under natural hypotheses the outer iteration achieves the rate of convergence of the inner iteration the technique is applied to finite element approximations of the stokes equations

segregated runge kutta time integration of convection stabilized mixed finite element schemes for wall unresolved les of incompressible flows in this work we develop a high performance numerical framework for the large eddy simulation les of incompressible flows the spatial discretization of the nonlinear system is carried out using mixed finite element fe schemes supplemented with symmetric projection stabilization of the convective term and a penalty term for the divergence constraint these additional terms introduced at the discrete level have been proved to act as implicit les models in order to perform meaningful wall unresolved simulations we consider a weak imposition of the boundary conditions using a nitsche s type scheme where the tangential component penalty term is designed to act as a wall law next segregated runge kutta srk schemes recently proposed by the authors for laminar flow problems are applied to the les simulation of turbulent flows by the introduction of a penalty term on the trace of the acceleration these methods exhibit excellent stability properties for both implicit and explicit treatment of the convective terms srk schemes are excellent for large scale simulations since they reduce the computational cost of the linear system solves by splitting velocity and pressure computations at the time integration level leading to two uncoupled systems the pressure system is a darcy type problem that can easily be preconditioned using a traditional block preconditioning scheme that only requires a poisson solver at the end only coercive systems have to be solved which can be effectively preconditioned by multilevel domain decomposition schemes which are both optimal and scalable the framework is applied to the taylor green and turbulent channel flow benchmarks in order to prove the accuracy of the convection stabilized mixed fes as les models and srk time integrators the scalability of the preconditioning techniques in space only has also been proven for one step of the srk scheme for the taylor green flow using uniform meshes moreover a turbulent flow around a naca profile is solved to show the applicability of the proposed algorithms for a realistic problem

weighted clement operator and application to the finite element discretization of the axisymmetric stokes problem we consider the stokes problem in an axisymmetric three dimensional domain with data which are axisymmetric and have angular component equal to zero we observe that the solution is also axisymmetric and the velocity has also zero angular component hence the solution satisfies a system of equations in the meridian domain the weak three dimensional problem reduces to a two dimensional one with weighted integrals the latter is discretized by taylor hood type finite elements a weighted clement operator is defined and approximation results are proved this operator is then used to derive the discrete inf sup condition and optimal a priori error estimates

a stabilized mixed finite element method for coupled stokes and darcy flows with transport abstract in this article we present a stabilized mixed finite element method for solving the coupled stokes and darcy flow equations with a solute transport the mathematical model includes the velocity and pressure equations and concentration equation where the viscosity depends on the concentration we propose a mixed weak formulation and use the nonconforming piecewise crouzeix raviart finite element piecewise constant and conforming piecewise linear finite element to approximate velocity pressure and concentration respectively the existence uniqueness of the approximate solution are obtained and optimal order a priori error estimates are derived no assumption on the boundness of the infinity norms of approximate velocity or concentration or the restriction about the time step and spatial meshsize is needed due to a new weak formulation introduced for the concentration equation numerical examples are presented to verify the theoretical results

finite element analysis for the axisymmetric laplace operator on polygonal domains abstract let l r 2 r r 2 z 2 we consider the equation l u f on a bounded polygonal domain with suitable boundary conditions derived from the three dimensional axisymmetric poisson s equation we establish the well posedness regularity and fredholm results in weighted sobolev spaces for possible singular solutions caused by the singular coefficient of the operator l as r 0 and by non smooth points on the boundary of the domain in particular our estimates show that there is no loss of regularity of the solution in these weighted sobolev spaces besides by analyzing the convergence property of the finite element solution we provide a construction of improved graded meshes such that the quasi optimal convergence rate can be recovered on piecewise linear functions for singular solutions the introduction of a new projection operator from the weighted space to the finite element subspace certain scaling arguments and a calculation of the index of the fredholm operator together with our regularity results are the ingredients of the finite element estimates

in memory of leo p kadanoff leo kadanoff has worked in many fields of statistical mechanics his contributions had an enormous impact this holds in particular for critical phenomena where he explained widom s homogeneity laws by means of block spin transformations and laid the basis for wilson s renormalization group equation i had the pleasure to work in his group for 1 year a short historical account is given

stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection abstract we consider the problem of poor mass conservation in mixed finite element algorithms for flow problems with large rotation free forcing in the momentum equation we provide analysis that suggests for such problems obtaining accurate solutions necessitates either the use of pointwise divergence free finite elements such as scott vogelius or heavy grad div stabilization of weakly divergence free elements the theory is demonstrated in numerical experiments for a benchmark natural convection problem where large irrotational forcing occurs with high rayleigh numbers

local projection stabilization of equal order interpolation applied to the stokes problem the local projection stabilization allows us to circumvent the babuska brezzi condition and to use equal order interpolation for discretizing the stokes problem the projection is usually done in a two level approach by projecting the pressure gradient onto a discontinuous finite element space living on a patch of elements we propose a new local projection stabilization method based on possibly enriched finite element spaces and discontinuous projection spaces defined on the same mesh optimal order of convergence is shown for pairs of approximation and projection spaces satisfying a certain inf sup condition examples are enriched simplicial finite elements and standard quadrilateral hexahedral elements the new approach overcomes the problem of an increasing discretization stencil and thus is simple to implement in existing computer codes numerical tests confirm the theoretical convergence results which are robust with respect to the user chosen stabilization parameter

grad div stabilization and subgrid pressure models for the incompressible navier stokes equations in this paper the grad div stabilization for the incompressible navier stokes finite element approximations is considered from two different viewpoints i as a variational multiscale approach for the pressure subgrid modeling and ii as a stabilization procedure of least square type some new error estimates for the linearized problem with the grad div stabilization are proved with the help of norms induced by the pressure schur complement operator we discuss the stabilization parameter choice arising in the frameworks of least square and multiscale methods and consider assumptions which allow to relate both approaches

analysis of the small viscosity and large reaction coefficient in the computation of the generalized stokes problem by a novel stabilized finite element method abstract in this paper we propose and analyze a novel stabilized finite element method fem for the system of generalized stokes equations arising from the time discretization of transient stokes problem the system involves a small viscosity which is proportional to the inverse of large reynolds number and a large reaction coefficient which is the inverse of small time step the proposed stabilized fem employs the c 0 piecewise linear elements for both velocity field and pressure on the same mesh and uses the residuals of the momentum equation and the divergence free equation to define the stabilization terms the stabilization parameters are fixed and element independent without a comparison of the viscosity the reaction coefficient and the mesh size using the finite element solution of an auxiliary boundary value problem as the interpolating function for velocity and the h 1 seminorm projection for pressure instead of the usual nodal interpolants we derive error estimates for the stabilized finite element approximations to velocity and pressure in the l 2 and h 1 norms and most importantly we explicitly establish the dependence of error bounds on the viscosity the reaction coefficient and the mesh size our analysis reveals that this stabilized fem is particularly suitable for the generalized stokes system with a small viscosity and a large reaction coefficient which has never been achieved before in the error analysis of other stabilization methods in the literature we then numerically confirm the effectiveness of the proposed stabilized fem comparisons made with other existing stabilization methods show that the newly proposed method can attain better accuracy and stability

a low order galerkin finite element method for the navier stokes equations of steady incompressible flow a stabilization issue and iterative methods a galerkin finite element method is considered to approximate the incompressible navier stokes equations together with iterative methods to solve a resulting system of algebraic equations this system couples velocity and pressure unknowns thus requiring a special technique for handling we consider the navier stokes equations in velocity kinematic pressure variables as well as in velocity bernoulli pressure variables the latter leads to the rotation form of nonlinear terms this form of the equations plays an important role in our studies a consistent stabilization method is considered from a new view point theory and numerical results in the paper address both the accuracy of the discrete solutions and the effectiveness of solvers and a mutual interplay between these issues when particular stabilization techniques are applied 2002 published by elsevier science b v

analysis of the inexact uzawa algorithm for saddle point problems in this paper we consider the so called inexact uzawa algorithm for iteratively solving linear block saddle point problems such saddle point problems arise for example in finite element and finite difference discretizations of stokes equations the equations of elasticity and mixed finite element discretization of second order problems we consider both the linear and nonlinear variants of the inexact uzawa iteration we show that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left hand block in the case of nonlinear iteration the inexact uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left hand block is of sufficient accuracy bounds for the nonlinear iteration are given in terms of this accuracy parameter and the rate of convergence of the preconditioned linear uzawa algorithm applications to the stokes equations and mixed finite element discretization of second order elliptic problems are discussed and finally the results of numerical experiments involving the algorithms are presented

numerical treatment of defective boundary conditions for the navier stokes equations we present a formulation for accommodating defective boundary conditions for the incompressible navier stokes equations where only averaged values are prescribed on measurable portions of the boundary in particular we consider the case where the flow rate is imposed on several domain sections this methodology has an interesting application in the numerical simulation of flow in blood vessels when only a reduced set of boundary data are generally available for the upstream and downstream sections

approximation of coupled stokes darcy flow in an axisymmetric domain abstract in this article we investigate the numerical approximation of coupled stokes and darcy fluid flow equations in an axisymmetric domain the fluid flow is assumed to be axisymmetric rewriting the problem in cylindrical coordinates reduces the 3 d problem to a problem in 2 d this reduction to 2 d requires the numerical analysis to be studied in suitably weighted hilbert spaces in this setting we show that the proposed approximation scheme has a unique solution and derive corresponding a priori error estimate computations for an example with a know solution are presented which support the a priori error estimate computations are also given for a model of fluid flow in the eye

adaptive partitioned methods for the time accurate approximation of the evolutionary stokes darcy system abstract this paper develops analyzes and tests a time accurate partitioned method for the stokes darcy equations the method combines a time filter and backward euler scheme is second order accurate and provides at no extra complexity an estimate of the temporal error this approach post processes the solutions of backward euler scheme by adding three lines to original codes to increase the time accuracy from first order to second order we prove long time stability and error estimates of backward euler plus time filter with constant time stepsize moreover we extend the approach to variable time stepsize and construct adaptive algorithms numerical tests show convergence of our method and support the theoretical analysis

an augmented cartesian grid method for stokes darcy fluid structure interactions summary a new finite difference method based on cartesian meshes is proposed for solving the fluid structure interaction between a fluid flow modeled by the stokes equations and a porous media modeled by the darcy s law the idea is to introduce several augmented variables along the interface between the fluid flow and the porous media so that the problem can be decoupled as several poisson equations the augmented variables should be chosen so that the beavers joseph saffman and other interface conditions are satisfied in the discretization the augmented variables have co dimension one compared with that of the primitive variables and are solved through the schur complement system a non trivial analytic solution with a circular interface is constructed to check second order convergency of the proposed method numerical examples with various interfaces and parameters are also presented some simulations show interesting behaviors of the fluid structure interaction between the fluid flow and the porous media the computational framework can be applied to other multi phase and multi physics problems copyright 2015 john wiley sons ltd

calculation of norms of bethe wave functions a class of two dimensional completely integrable models of statistical mechanics and quantum field theory is considered eigenfunctions of the hamiltonians are known for these models norms of these eigenfunctions in the finite box are calculated in the present paper these models include in particular the quantum nonlinear schrodinger equation and the heisenbergxxz model

analysis of fully mixed finite element methods for the stokes darcy coupled problem in this paper we analyze fully mixed finite element methods for the coupling of fluid flow with porous media flow flows are governed by the stokes and darcy equations respectively and the corresponding transmission conditions are given by mass conservation balance of normal forces and the beavers joseph saffman law the fully mixed concept employed here refers to the fact that we consider dual mixed formulations in both the stokes domain and the darcy region which means that the main unknowns are given by the pseudostress and the velocity in the fluid together with the velocity and the pressure in the porous medium in addition the transmission conditions become essential which leads to the introduction of the traces of the porous media pressure and the fluid velocity as the associated lagrange multipliers we apply the fredholm and babuska brezzi theories to derive sufficient conditions for the unique solvability of the resulting continuous formulation since the equations and unknowns can be ordered in several different ways we choose the one yielding a doubly mixed structure for which the inf sup conditions of the off diagonal bilinear forms follow straightforwardly next adapting to the discrete case the arguments of the continuous analysis we are able to establish suitable hypotheses on the finite element subspaces ensuring that the associated galerkin scheme becomes well posed in addition we show that the existence of uniformly bounded discrete liftings of the normal traces simplifies the derivation of the corresponding stability estimates a feasible choice of subspaces is given by raviart thomas elements of lowest order and piecewise constants for the velocities and pressures respectively in both domains together with continuous piecewise linear elements for the lagrange multipliers this example confirms that with the present approach the stokes and darcy flows can be approximated with the same family of finite element subspaces without adding any stabilization term finally several numerical results illustrating the good performance of the method with these discrete spaces and confirming the theoretical rate of convergence are provided

scaling and universality in statistical physics abstract the twin concepts of scaling and universality have played an important role in the description of statistical systems hydrodynamics contains many applications of scaling including descriptions of the behavior of boundary layers prandtl blasius and of the fluctuating velocity in turbulent flow kolmogorov heisenberg onsager phenomenological theories of behavior near critical points of phase transitions made extensive use of both scaling to define the size of various fluctuations and universality to say that changes in the model would not change the answers these two ideas were combined via the statement that elimination of degrees of freedom and a concomitant scale transformation left the answers quite unchanged in wilson s hands this mode of thinking led to the renormalization group approach to critical phenomena subsequently feigenbaum showed how scaling universality and renormalization group ideas could be applied to dynamical systems specifically this approach enabled us to see how chaos first arises in those systems in which but a few degrees of freedom are excited in parallel libchaber developed experiments aimed at understanding the onset of chaos the results of which were subsequently used to show that feigenbaum s universal behavior was in fact realized in honest to goodness hydrodynamical systems more recently gemunu gunaratne mogens jensen and itamar procaccia have indicated that they believe that a different and weaker universality might hold for the fully chaotic behavior of low dimensional dynamical systems dynamically generated situations often seem to show kinds of scaling and universality quite different from that seen in critical phenomena a technical difference which seems to arise in these intrinsically dynamical processes is that instead of having a denumerable list of different critical quantities each with their critical index instead there is continuum of critical indices this so called multifractal behavior may nonetheless show some kinds of universality and indeed this might be the kind of scaling and universality shown by those hydrodynamical systems in which many degrees of freedom are excited

low order raviart thomas approximations of axisymmetric darcy flow abstract in this paper we study the lowest order mixed finite element method for the axisymmetric darcy problem using raviart thomas elements in contrast to the cartesian setting the method is non conforming in the sense that the discretely divergence free functions are not solenoidal we derive several estimates that measure the inconsistency of the method and derive error estimates of the discrete pressure and velocity solutions we show that if the domain is convex then the errors converge with optimal order modulo logarithmic terms numerical experiments are presented and they indicate that the estimates are sharp

analysis of iterative methods for saddle point problems a unified approach in this paper two classes of iterative methods for saddle point problems are considered inexact uzawa algorithms and a class of methods with symmetric preconditioners in both cases the iteration matrix can be transformed to a symmetric matrix by block diagonal matrices a simple but essential observation which allows one to estimate the convergence rate of both classes by studying associated eigenvalue problems the obtained estimates apply for a wider range of situations and are partially sharper than the known estimates in literature a few numerical tests are given which confirm the sharpness of the estimates

a connection between scott vogelius and grad div stabilized taylor hood fe approximations of the navier stokes equations this article studies two methods for obtaining excellent mass conservation in finite element computations of the navier stokes equations using continuous velocity fields with a particular mesh construction the scott vogelius element pair has recently been shown to be inf sup stable and have optimal approximation properties while also providing pointwise mass conservation we present herein the first numerical tests of this element pair for the time dependent navier stokes equations we also prove that the limit of the grad div stabilized taylor hood solutions to the navier stokes problem converges to the scott vogelius solution as the stabilization parameter tends to infinity that is we provide theoretical justification that choosing the grad div parameter large does not destroy the solution numerical tests are provided which verify the theory and show how both scott vogelius and grad div stabilized taylor hood with large stabilization parameter elements can provide accurate results with excellent mass conservation for navier stokes approximations

a two grid method of a mixed stokes darcy model for coupling fluid flow with porous media flow we study numerical methods for solving a coupled stokes darcy problem in porous media flow applications a two grid method is proposed for decoupling the mixed model by a coarse grid approximation to the interface coupling conditions error estimates are derived for the proposed method both theoretical analysis and numerical experiments show the efficiency and effectiveness of the two grid approach for solving multimodeling problems potential extensions and future directions are discussed

characterizing the inf sup condition on product spaces in this paper we establish characterization results for the continuous and discrete inf sup conditions on product spaces the inf sup condition for each component of the bilinear form involved and suitable decompositions of the pivot space in terms of the associated null spaces are the key ingredients of our theorems we illustrate the theory through its application to bilinear forms arising from the variational formulations of several boundary value problems

on stability accuracy and fast solvers for finite element approximations of the axisymmetric stokes problem by hood taylor elements we provide a proof of both the stability and the approximation property for the finite element approximations of the axisymmetric stokes problem by continuous piecewise polynomials of degree kappa 1 for the velocity and continuous piecewise polynomials of degree kappa for the pressure with any kappa geq1 new techniques are designed so that in this perspective by a simple transformation the existing theory developed in three dimensional cartesian coordinates can be effectively exploited in fact this perspective provides a new way of developing theories for the axisymmetric stokes problems and it can be applied potentially to other problems as well a simple illustration is provided for the application in the development and analysis of fast solvers for the resulting discrete saddle point problems sample numerical experiments have been presented as well to confirm the theoretical results

augmented lagrangian preconditioner for large scale hydrodynamic stability analysis abstract hydrodynamic linear stability analysis of large scale three dimensional configurations is usually performed with a time stepping approach based on the adaptation of existing solvers for the unsteady incompressible navier stokes equations we propose instead to solve the nonlinear steady equations with the newton method and to determine the largest growth rate eigenmodes of the linearized equations using a shift and invert spectral transformation and a krylov schur algorithm the solution of the shifted linearized navier stokes problem which is the bottleneck of this approach is computed via an iterative krylov subspace solver preconditioned by the modified augmented lagrangian mal preconditioner benzi et al 2011 the well known efficiency of this preconditioned iterative strategy for solving the real linearized steady state equations is assessed here for the complex shifted linearized equations the effect of various numerical and physical parameters is investigated numerically on a two dimensional flow configuration confirming the reduced number of iterations over state of the art steady state and time stepping based preconditioners a parallel implementation of the steady navier stokes and eigenvalue solvers developed in the freefem language suitably interfaced with the petsc slepc libraries is described and made openly available to tackle three dimensional flow configurations its application on a small scale three dimensional problem shows the good performance of this iterative approach over a direct l u factorization strategy in regards of memory and computational time on a large scale three dimensional problem with 75 million unknowns a 80 parallel efficiency on 256 up to 2048 processes is obtained

analysis of a streamline diffusion finite element method for the stokes and navier stokes equations for the stokes equations with convection and the incompressible navier stokes equations the authors analyze a streamline diffusion finite element method that is capable of balancing both the convection and the pressure thus allowing the use of arbitrary pairs of velocity pressure spaces for the linear problem the authors obtain for all mesh peclet numbers optimal error estimates in natural norms including in particular the l 2 norm of the pressure the same holds for the nonlinear problem which close to a regular branch of solutions i e the linearized operator is an isomorphism of the norm of the inverse of which still depends on the reynolds number consequently the dependence of the error constants on the reynolds number is not completely resolved in this case

locally conservative coupling of stokes and darcy flows a locally conservative numerical method for solving the coupled stokes and darcy flows problem is formulated and analyzed the approach employs the mixed finite element method for the darcy region and the discontinuous galerkin method for the stokes region a discrete inf sup condition and optimal error estimates are derived

multigrid methods for a parameter dependent problem in primal variables in this paper we consider multigrid methods for the parameter dependent problem of nearly incompressible materials we construct and analyze multilevel projection algorithms which can be applied to the mixed as well as to the equivalent non conforming finite element scheme in primal variables for proper norms we prove that the smoothing property and the approximation property hold with constants that are independent of the small parameter thus we obtain robust and optimal convergence rates for the w cycle and the variable v cycle multigrid methods the numerical results pretty well conform the robustness and optimality of the multigrid methods proposed