spectral hp least squares finite element formulation for the navier stokes equations we consider the application of least squares finite element models combined with spectral hp methods for the numerical solution of viscous flow problems the paper presents the formulation validation and application of a spectral hp algorithm to the numerical solution of the navier stokes equations governing two and three dimensional stationary incompressible and low speed compressible flows the navier stokes equations are expressed as an equivalent set of first order equations by introducing vorticity or velocity gradients as additional independent variables and the least squares method is used to develop the finite element model high order element expansions are used to construct the discrete model the discrete model thus obtained is linearized by newton s method resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method spectral convergence of the l2 least squares functional and l2 error norms is verified using smooth solutions to the two dimensional stationary poisson and incompressible navier stokes equations numerical results for flow over a backward facing step steady flow past a circular cylinder three dimensional lid driven cavity flow and compressible buoyant flow inside a square enclosure are presented to demonstrate the predictive capability and robustness of the proposed formulation

adaptive least squares finite element approximations to stokes equations this paper is concerned with adaptive least squares methods for stokes equations based on velocity pressure stress and velocity vorticity pressure formulations to capture the stokes flow region an adaptive algorithm based on mesh redistribution is developed for a nonlinear weighted least squares functional a redistribution approach is considered to generate the optimal grids model problems considered are the flow past a planar channel and a 4 to 1 contraction problems numerical results of model problems illustrating the efficiency of the proposed scheme are presented adaptive least squares methods are used to solve the stokes equations velocity pressure stress and velocity vorticity pressure formulations are used mesh redistribution for planar and 4 to 1 contraction flow problems are developed lower order basis functions in all variables are used in least squares methods hybrid meshes of regular and graded grids are used to reduce the grid effects

numerical investigation on the stability of singular driven cavity flow by applying the singularity subtraction technique to the unsteady driven cavity problem the stability of the impulsively started flow is investigated without smoothing the corner singularity a second order spectral projection method allows localization of the critical reynolds number for the first hopf bifurcation in the interval 8017 6 8018 8

enhanced mass conservation in least squares methods for navier stokes equations there are many applications of the least squares finite element method for the numerical solution of partial differential equations because of a number of benefits that the least squares method has however one of the most well known drawbacks of the least squares finite element method is the lack of exact discrete mass conservation in some contexts due to the fact that the least squares method minimizes the continuity equation in l 2 norm in this paper we explore the reason for the mass loss and provide new approaches to retain the mass even in a severely underresolved grid

metastable patterns for the cahn hilliard equation part ii layer dynamics and slow invariant manifold abstract in this paper we study the dynamics of the 1 dimensional cahn hilliard equation u t 2 u xx w u xx in a finite interval in a neighborhood of an equilibrium with n 1 transition layers where is a small parameter and w is a double well energy density function with equal minima the lower bound of the layer motion speed is given explictly and the layer motion directions are described precisely if a solution of the cahn hilliard equation starts outside a neighborhood of the equilibrium of size o ln 1 it is proved that there is an n dimensional unstable invariant manifold which is a smooth graph over the approximate manifold constructed in j differential equations 111 1994 421 457 with its global lipschitz constant exponentially small and this unstable invariant manifold attracts solutions exponentially fast uniformly in

a least squares finite element method for incompressible flow in stress velocity pressure version abstract in this paper we are concerned with the incompressible flow in 2 d introducing additional variables of derivatives of velocity which are called stresses here the second order dynamic equations are reduced into a first order system with variables of stress velocity and pressure combining the compatability conditions and the divergence ice condition we have a system with six first order equations and six unknowns least squares method performed over this extended system the analysis shows that this method achieves optimal rates of convergence in the h1 norm as the h approaches to zero numerical experiences are also available

analysis of least squares finite element methods for the navier stokes equations in this paper we study finite element methods of least squares type for the stationary incompressible navier stokes equations in two and three dimensions we consider methods based on velocity vorticity pressure form of the navier stokes equations augmented with several nonstandard boundary conditions least squares minimization principles for these boundary value problems are developed with the aid of the agmon douglis nirenberg adn elliptic theory among the main results of this paper are optimal error estimates for conforming finite element approximations and analysis of some nonstandard boundary conditions results of several computational experiments with least squares methods which illustrate among other things the optimal convergence rates are also reported

newton multigrid least squares fem for the v v p formulation of the navier stokes equations we solve the v v p vorticity velocity pressure formulation of the stationary incompressible navier stokes equations based on the least squares finite element method for the discrete systems we use a conjugate gradient cg solver accelerated with a geometric multigrid preconditioner for the complete system in addition we employ a krylov space smoother inside of the multigrid which allows a parameter free smoothing combining this linear solver with the newton linearization we construct a very robust and efficient solver we use biquadratic finite elements to enhance the mass conservation of the least squares method for the inflow outflow problems and to obtain highly accurate results we demonstrate the advantages of using the higher order finite elements and the grid independent solver behavior through the solution of three stationary laminar flow problems of benchmarking character the comparisons show excellent agreement between our results and those of the galerkin mixed finite element method as well as available reference solutions

a least squares finite element formulation for unsteady incompressible flows with improved velocity pressure coupling in the weak form galerkin formulation for incompressible flows the pressure has a well understood role at all times it may be interpreted as a lagrange multiplier that enforces the divergence free constraint on the velocity field this is not the case in least squares formulations for incompressible flows where the divergence free constraint is enforced in a least squares sense in a variational setting of residual minimization thus the role of the pressure in a least squares formulation is rather vague we find that this lack of velocity pressure coupling in least squares formulations may induce spurious temporal pressure oscillations when using the non stationary form of the equations we present a least squares formulation with improved velocity pressure coupling based on the use of a regularized divergence free constraint a first order system least squares fosls approach based on velocity pressure and vorticity is used to allow the use of practical c0 element expansions in the finite element model we use high order spectral element expansions in space and second and third order time stepping schemes excellent conservation of mass and accuracy of computed pressure metrics are demonstrated in the numerical results

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

an adaptive least squares mixed finite element method for elasto plasticity a least squares mixed finite element method for the incremental formulation of elasto plasticity using a plastic flow rule of von mises type with isotropic hardening is presented this approach is based on the use of the stress tensor in addition to the displacement field as independent process variables the nonlinear least squares functional is shown to constitute an a posteriori error estimator on which an adaptive refinement strategy may be based for the finite element implementation under plane strain conditions quadratic i e next to lowest order raviart thomas elements are used for the stress approximation while the displacement is represented by standard quadratic conforming elements computational results for a benchmark problem of elasto plasticity under plane strain conditions are presented in order to illustrate the effectiveness of the least squares approach

efficient stress velocity least squares finite element formulations for the incompressible navier stokes equations abstract in this contribution three different mixed least squares finite element methods lsfems are investigated with respect to accuracy and efficiency with simultaneous consideration of regular and adaptive meshing strategies the deliberations are made for the incompressible navier stokes equations the generally known first order div grad system in terms of the total stress velocity and pressure svp formulation is the basis for two further div grad least squares formulations in terms of total stress and velocity sv whereby both formulations are derived from different fountainheads the extended sv formulation is an enrichment of the known stress velocity formulation proposed by cai et al 2004 the second sv formulation is based on the substitution of the pressure in the svp formulation thus it is based on a discontinuous pressure interpolation advantage of the sv formulations is a smaller system matrix size due to the reduction of the relevant degrees of freedom nevertheless the approximation quality has to last the demand of the established least squares formulations the drawback of a poor mass conservation is well investigated for the lsfem and known to overcome by using high order interpolations for all solution variables therefore the main attention of this contribution is focused on accuracy while aiming for high efficiency which is intrinsic for the fundamental idea in here in the light of efficient lsfem solutions the advantage of the inherent a posteriori error estimator of the method is used for deliberations on different marking strategies in an h type adaptive mesh refinement

analysis of velocity flux least squares principles for the navier stokes equations part ii this paper continues the development of the least squares methodology for the solution of the incompressible navier stokes equations started in part i here we again use a velocity flux first order navier stokes system but our focus now is on a practical algorithm based on a discrete negative norm

first order system least squares for coupled stokes darcy flow the coupled problem with stokes flow in one subdomain and a darcy flow model in a second subdomain is studied in this paper both flow problems are treated as first order systems involving pseudostress and velocity in the stokes case and using a flux pressure formulation in the darcy subdomain as process variables respectively the beavers joseph saffman interface conditions are treated by an appropriate interface functional which is added to the least squares functional associated with the subdomain problems a combination of h mathrm div conforming raviart thomas and standard h 1 conforming elements is used for the stokes as well as for the darcy subsystem the homogeneous least squares functional is shown to be equivalent to an appropriate norm allowing the use of standard finite element approximation estimates it also establishes the fact that the local evaluation of the least squares functional itself constitutes an a posteriori error estimator to be used for adaptive refinement strategies

numerical construction of optimal adaptive grids in two spatial dimensions abstract this article concerns a new procedure to generate a solution adaptive grid for convection dominated problems in two spatial dimensions based on finite element approximations the procedure extends a one dimensional equidistribution principle which minimizes the interpolation error in appropriate norms the idea in extending such a technique to two spatial dimensions is to select two directions which can reflect the physics of the problems and then the one dimensional equidistribution principle is applied to the chosen directions the final grids generated are connected through a sweep line based unstructured grid technique model problems considered are the two dimensional convection diffusion problems where boundary and interior layers occur numerical results of model problems illustrating the efficiency of the proposed scheme are presented comparisons of the solutions with an existing scheme will also be provided

a monolithic fem multigrid solver for non isothermal incompressible flow on general meshes we present special numerical simulation methods for non isothermal incompressible viscous fluids which are based on lbb stable fem discretization techniques together with monolithic multigrid solvers for time discretization we apply the fully implicit crank nicolson scheme of 2nd order accuracy while we utilize the high order q 2 p 1 finite element pair for discretization in space which can be applied on general meshes together with local grid refinement strategies including hanging nodes to treat the nonlinearities in each time step as well as for direct steady approaches the resulting discrete systems are solved via a newton method based on divided differences to calculate explicitly the jacobian matrices in each nonlinear step the coupled linear subproblems are solved simultaneously for all quantities by means of a monolithic multigrid method with local multilevel pressure schur complement smoothers of vanka type for validation and evaluation of the presented methodology we perform the mit benchmark 2001 m a christon p m gresho s b sutton computational predictability of natural convection flows in enclosures in first mit conference on computational fluid and solid mechanics vol 40 elsevier 2001 pp 1465 1468 of natural convection flow in enclosures to compare our results with respect to accuracy and efficiency additionally we simulate problems with temperature and shear dependent viscosity and analyze the effect of an additional dissipation term inside the energy equation moreover we discuss how these fem multigrid techniques can be extended to monolithic approaches for viscoelastic flow problems

optimal adaptive grids of least squares finite element methods in two spatial dimensions this article concerns a procedure to generate optimal adaptive grids for convection dominated problems in two spatial dimensions based on least squares finite element approximations the procedure extends a one dimensional equidistribution principle which minimizes the interpolation error in some norms the idea is to select two directions which can reflect the physics of the problems and then apply the one dimensional equidistribution principle to the chosen directions model problems considered are the two dimensional convection diffusion problems where boundary and interior layers occur numerical results of model problems illustrating the efficiency of the proposed scheme are presented in addition to avoid skewed mesh in the optimal grids generated by the algorithm an unstructured local mesh smoothing will be considered in the least squares approximations comparisons with the gakerkin finite element method will also be provided

first order system least squares for the stokes equations with application to linear elasticity following our earlier work on general second order scalar equations here we develop a least squares functional for the two and three dimensional stokes equations generalized slightly by allowing a pressure term in the continuity equation by introducing a velocity flux variable and associated curl and trace equations we are able to establish ellipticity in an h1 product norm appropriately weighted by the reynolds number this immediately yields optimal discretization error estimates for finite element spaces in this norm and optimal algebraic convergence estimates for multiplicative and additive multigrid methods applied to the resulting discrete systems both estimates are naturally uniform in the reynolds number moreover our pressure perturbed form of the generalized stokes equations allows us to develop an analogous result for the dirichlet problem for linear elasticity where we obtain the more substantive result that the estimates are uniform in the poisson ratio

least squares finite element method for the stokes problem with zero residual of mass conservation in this paper the simulation of incompressible flow in two dimensions by the least squares finite element method lsfem in the vorticity velocity pressure version is studied in the lsfem the equations for continuity of mass and momentum and a vorticity equation are minimized on a discretization of the domain of interest a problem is these equations are minimized in a global sense thus this method may not enforce that div underline u 0 at every point of the discretization in this paper a modified lsfem is developed which enforces near zero residual of mass conservation i e div underline u is nearly zero at every point of the discretization this is accomplished by adding an extra restriction in the divergence free equation through the lagrange multiplier strategy in this numerical method the inf sup or say lbb condition is not necessary and the matrix resulting from applying the method on a discretization is symmetric the uniqueness of the solution and the application of the conjugate gradient method are also valid numerical experience is given in simulating the flow of a cylinder with diameter 1 moving in a narrow channel of width 1 5 results obtained by the lsfem show that mass is created or destroyed at different points in the interior of discretization the results obtained by the modified lsfem show the mass is nearly conserved everywhere

convergence and optimality of adaptive least squares finite element methods the first order div least squares finite element methods lsfems allow for an immediate a posteriori error control by the computable residual of the least squares functional this paper establishes an adaptive refinement strategy based on some equivalent refinement indicators since the first order div lsfem measures the flux errors in h div the data resolution error measures the l 2 norm of the right hand side f minus the piecewise polynomial approximation pi f without a mesh size factor hence the data resolution term is neither an oscillation nor of higher order and consequently requires a particular treatment e g by the thresholding second algorithm due to binev and devore the resulting novel adaptive lsfem with separate marking converges with optimal rates relative to the notion of a nonlinear approximation class

least squares methods for incompressible newtonian fluid flow linear stationary problems this paper develops and analyzes two least squares methods for the numerical so lution of linear stationary incompressible newtonian fluid flow in two and three dimensions both approaches use the l 2 norm to define least squares functionals one is based on the stress velocity formulation see section 3 2 and it applies to general boundary conditions the other is based on an equivalent formulation for the pseudostress and velocity see section 4 2 and it applies to pure velocity dirichlet boundary conditions the velocity gradient and vorticity can be obtained alge braically from this new tensor variable it is shown that the homogeneous least squares functionals are elliptic and continuous in the h div d h 1 d norm this immediately implies optimal error estimates for conforming finite element approximations as well it admits optimal multigrid solution methods if raviart thomas finite element spaces are used to approximate the stress or the pseudostress tensor

a convergent adaptive algorithm for poisson s equation we construct a converging adaptive algorithm for linear elements applied to poisson s equation in two space dimensions starting from a macro triangulation we describe how to construct an initial triangulation from a priori information then we use a posteriors error estimators to get a sequence of refined triangulation and approximate solutions it is proved that the error measured in the energy norm decreases at a constant rate in each step until a prescribed error bound is reached extension to higher order elements in two space dimension and numerical results are included

analysis of velocity flux first order system least squares principles for the navier stokes equations part i this paper develops a least squares approach to the solution of the incompressible navier stokes equations in primitive variables as with our earlier work on stokes equations we recast the navier stokes equations as a first order system by introducing a velocity flux variable and associated curl and trace equations we show that a least squares principle based on l2 norms applied to this system yields optimal discretization error estimates in the h1 norm in each variable including the velocity flux an analogous principle based on the use of an h 1 norm for the reduced system with no curl or trace constraints is shown to yield similar estimates but now in the l2 norm for velocity flux and pressure although the h 1 least squares principle does not allow practical implementation these results are critical to the analysis of a practical least squares method for the reduced system based on a discrete equivalent of the negative norm a practical method of this type is the subject of a companion paper finally we establish optimal multigrid convergence estimates for the algebraic system resulting from the l2 norm approach