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

least squares methods for stokes equations based on a discrete minus one inner product abstract the purpose of this paper is to develop and analyze least squares approximations for stokes and elasticity problems the major advantage of the least squares formulation is that it does not require that the classical ladyzhenskaya bab ska brezzi lbb condition be satisfied we provide two methods the first is posed in terms of the velocity pressure pair without the introduction of additional variables the second adds a vorticity variable in both cases we employ least squares functionals which involve a discrete inner product which is related to the inner product in h 1 the sobolev space of order minus one on the use of such inner products applied to second order problems was proposed in an earlier paper by bramble lazarov and pasciak 1994

error analysis for constrained first order system least squares finite element methods in this paper a general error analysis is provided for finite element discretizations of partial differential equations in a saddle point form with divergence constraint in particular this extends upon the work of j h adler and p s vassilevski springer proc math statist 45 springer new york 2013 pp 1 19 giving a general error estimate for finite element problems augmented with a divergence constraint and showing that these estimates are obtained for problems such as diffusion and stokes using the first order system least squares fosls finite element method the main result is that by enforcing the constraint on a mathbf h 1 equivalent fosls formulation one maintains optimal convergence of the fosls functional i e the energy norm of the error while guaranteeing the conservation of the divergence constraint i e mass conservation in some examples the error estimates and results depend on using finite elements for the constraint space that are inf sup stable when paired wi

first order system least squares for darcy stokes flow the subject of this paper is a first order system least squares formulation for the stokes equation which remains uniformly valid in the limit of vanishing viscosity for this so called darcy stokes flow problem we establish continuity and coercivity of the corresponding least squares functional in appropriate norms two types of finite element spaces for the approximation of the velocity field are investigated in detail the well known raviart thomas elements and an element recently introduced by mardal tai and winther specifically for mixed approaches to darcy stokes flow the computational results derived with next to lowest order raviart thomas elements as well as the mardal tai winther elements confirm our analysis

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

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

a least squares approach based on a discrete minus one inner product for first order systems the purpose of this paper is to develop and analyze a least squares approximation to a first order system the first order system represents a reformulation of a second order elliptic boundary value problem which may be indefinite and or nonsymmetric the approach taken here is novel in that the least squares functional employed involves a discrete inner product which is related to the inner product in h 1 the sobolev space of order minus one on the use of this inner product results in a method of approximation which is optimal with respect to the required regularity as well as the order of approximation even when applied to problems with low regularity solutions in addition the discrete system of equations which needs to be solved in order to compute the resulting approximation is easily preconditioned thus providing an efficient method for solving the algebraic equations the preconditioner for this discrete system only requires the construction of preconditioners for standard second order problems a task which is well understood

first order system least squares for incompressible resistive magnetohydrodynamics magnetohydrodynamics mhd is a fluid theory that describes plasma physics by treating the plasma as a fluid of charged particles hence the equations that describe the plasma form a nonlinear system that couples navier stokes equations with maxwell s equations this paper shows that the first order system least squares fosls finite element method is a viable discretization for these large mhd systems to solve this system a nested iteration newton fosls amg approach is taken most of the work is done on the coarse grid including most of the linearizations we show that at most one newton step and a few v cycles are all that are needed on the finest grid here we describe how the fosls method can be applied to incompressible resistive mhd and how it can be used to solve these mhd problems efficiently a 3d steady state and a reduced 2d time dependent test problem are studied the latter equations can simulate a large aspect ratio tokamak the goal is to resolve as much physics from the test problems with the least amount of computational work we show that this is achieved in a few dozen work units or fine grid residual evaluations

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

an adaptively refined least squares finite element method for generalized newtonian fluid flows using the carreau model we implemented an adaptively refined least squares finite element approach for the navier stokes equations that govern generalized newtonian fluid flows using the carreau model to capture the flow region we developed an adaptive mesh refinement approach based on the least squares method the generated refined grids agree well with the physical attributes of the flows we also proved that the least squares approximation converges to the linearized versions solutions of the carreau model at the best possible rate model problems considered in the study are the flow past a planar channel and 4 to 1 contraction problems we presented the numerical results of the model problems revealing the efficiency of the proposed scheme and investigated the physical parameter effects

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

stability of a σp unital continuous field algebra we study the question of whether stability is preserved under the operation of forming a continuous field algebra this is not necessarily true when the base space is infinite dimensional however it is always true when the base space is an n cube or an n torus and when the continuous field algebra is p unital specifically we prove the following theorem 0 1 let a be a p unital separable maximal full algebra of operator fields with base space either an n cube x 0 1 n or an n torus x tn and fibre algebras ax x x if ax is stable for all x x then a is a stable c algebra we also show that under the same hypotheses the corona factorization property is also preserved under the formation of continuous field algebras theorem 0 2 let a be a p unital separable maximal full algebra of operator fields with base space either an n cube x 0 1 n or an n torus x tn and fibre algebras ax x x if ax has the corona factorization property for all x x then a also has the corona factorization property

mathcal c _ 0 x algebras stability and strongly self absorbing mathcal c algebras we study permanence properties of the classes of stable and so called mathcal d stable mathcal c algebras respectively more precisely we show that a mathcal c 0 x algebra a is stable if all its fibres are provided that the underlying compact metrizable space x has finite covering dimension or that the cuntz semigroup of a is almost unperforated a condition which is automatically satisfied for mathcal c algebras absorbing the jiang su algebra mathcal z tensorially furthermore we prove that if mathcal d is a k 1 injective strongly self absorbing mathcal c algebra then a absorbs mathcal d tensorially if and only if all its fibres do again provided that x is finite dimensional this latter statement generalizes results of blanchard and kirchberg we also show that the condition on the dimension of x cannot be dropped along the way we obtain a useful characterization of when a mathcal c algebra with weakly unperforated cuntz semigroup is stable which allows us to show that stability passes to extensions of mathcal z absorbing mathcal c algebras

classification of extensions of classifiable c algebras abstract for a certain class of extensions e 0 b e a 0 of c algebras in which b and a belong to classifiable classes of c algebras we show that the functor which sends e to its associated six term exact sequence in k theory and the positive cones of k 0 b and k 0 a is a classification functor we give two independent applications addressing the classification of a class of c algebras arising from substitutional shift spaces on one hand and of graph algebras on the other the former application leads to the answer of a question of carlsen and the first named author concerning the completeness of stabilized matsumoto algebras as an invariant of flow equivalence the latter leads to the first classification result for nonsimple graph c algebras

first order system least squares methods for an optimal control problem by the stokes flow the least squares approximations of an optimal control problem governed by the stokes equations are considered which leads to an unconstrained coupled optimization problem by the lagrange multiplier method the least squares functionals for the two and three dimensional first order coupled optimality systems are employed by modifying those functionals in z cai t a manteuffel and s f mccormick siam j numer anal 34 1997 pp 1727 1741 the established ellipticity and continuity in a product h 1 norm yield the optimal discretization error estimates in the finite element spaces for numerical tests we apply v cycle multigrid methods to the whole discrete algebraic system

quasi least squares finite element method for steady flow and heat transfer with system rotation two quasi least squares finite element schemes based on l2 inner product are proposed to solve a steady navier stokes equations coupled to the energy equation by the boussinesq approximation and augmented by a coriolis forcing term to account for system rotation the resulting nonlinear systems are linearized around a characteristic state resulting in linearized least squares models that yield algebraic systems with symmetric positive definite coefficient matrices existence of solutions are investigated and a priori error estimates are obtained the performance of the formulation is illustrated by using a direct iteration procedure to treat the nonlinearities and shown theoretical convergent rate for general initial guess

on the number of simple modules in a block of a finite group abstract we prove that if b is a p block with non trivial defect group d of a finite p solvable group g then l b p r where r is the sectional rank of d we remark that there are infinitely many p blocks b with non abelian defect groups and l b p r 1 we conjecture that the inequality l b p r holds for an arbitrary p block with defect group of sectional rank r we show this to hold for a large class of p blocks of various families of quasi simple and nearly simple groups

a least squares penalty method for distributed optimal control problems for stokes equations the purpose of this paper is to construct an unconstrained optimal control problem by using a least squares approach for the constrained distributed optimal control problem associated with incompressible stokes equations the constrained equations are reformulated to the equivalent first order system by introducing vorticity and then the least squares functional corresponding to the system is enforced via a penalty term to the objective functional the existence of a solution of the unconstrained optimal control problem is proved and the convergence of this solution to that of unpenalized one is demonstrated as the penalty parameter tends to zero finite element approximations with error estimates are studied and the relevant computational experiments are presented

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

analysis of the l2 least squares finite element method for the velocity vorticity pressure stokes equations with velocity boundary conditions a theoretical analysis of the l 2 least squares finite element method lsfem for solving the stokes equations in the velocity vorticity pressure vvp first order system formulation with the dirichlet velocity boundary condition is given the least squares energy functional is defined to be the sum of the squared l 2 norms of the residuals in the partial differential equations weighted appropriately by the viscosity constant n it is shown that with many advantages the method is stable and convergent without requiring extra smoothness of the exact solution and the piecewise linear finite elements can be used to approximate all the unknowns furthermore with respect to the order of approximation for smooth exact solutions the method exhibits an optimal rate of convergence in the h 1 norm for velocity and a suboptimal rate of convergence in the l 2 norm for vorticity and pressure some numerical experiments in two and three dimensions are given which confirm the a priori error estimates since the boundary of the bounded domain under consideration is polygonal in r 2 or polyhedral in r 3 instead of c 1 smooth the authors adopt the more direct technique of bramble pasciak and cai manteuffel mccormick which departs from the agmon douglis nirenberg theory in showing the coercivity bound of the least squares functional

on the roles of minimization and linearization in least squares finite element models of nonlinear boundary value problems in this paper we examine the roles of minimization and linearization in the least squares finite element formulations of nonlinear boundary values problems the least squares principle is based upon the minimization of the least squares functional constructed via the sum of the squares of appropriate norms of the residuals of the partial differential equations in the present case we consider l 2 norms since the least squares method is independent of the discretization procedure and the solution scheme the least squares principle suggests that minimization should be performed prior to linearization where linearization is employed in the context of either the picard or newton iterative solution procedures however in the least squares finite element analysis of nonlinear boundary value problems it has become common practice in the literature to exchange the sequence of application of the minimization and linearization operations the main purpose of this study is to provide a detailed assessment on how the finite element solution is affected when the order of application of these operators is interchanged the assessment is performed mathematically through an examination of the variational setting for the least squares formulation of an abstract nonlinear boundary value problem and also computationally through the numerical simulation of the least squares finite element solutions of both a nonlinear form of the poisson equation and also the incompressible navier stokes equations the assessment suggests that although the least squares principle indicates that minimization should be performed prior to linearization such an approach is often impractical and not necessary

one parameter continuous fields of kirchberg algebras we prove that all unital separable continuous fields of c algebras over 0 1 with fibers isomorphic to the cuntz algebra mathcal o n 2 leq n leq infty are trivial more generally we show that if a is a separable unital or stable continuous field over 0 1 of kirchberg c algebras satisfying the uct and having finitely generated k theory groups then a is isomorphic to a trivial field if and only if the associated k theory presheaf is trivial for fixed d in 0 1 we also show that under the additional assumption that the fibers have torsion free k d group and trivial k d 1 group the k d sheaf is a complete invariant for separable stable continuous fields of kirchberg algebras

decomposition rank and absorbing extensions of type i algebras abstract we prove the following let a and b be separable c algebras suppose that b is a type i c algebra such that i b has only infinite dimensional irreducible representations and ii b has finite decomposition rank if 0 b c a 0 is a unital homogeneous exact sequence with busby invariant then the extension is absorbing in the case of infinite decomposition rank we provide a counterexample specifically we construct a unital homogeneous split exact sequence of the form 0 c z k c c 0 which is not absorbing in this example z is an infinite dimensional compact second countable topological space this gives a counterexample to the natural infinite dimensional generalization of the result of pimsner popa and voiculescu

the least squares pseudo spectral method for navier stokes equations a spectral collocation approximation of first order system least squares for incompressible stokes equations was analyzed in kim et al 2004 12 and finite element approximations for incompressible navier stokes equations were developed in bochev et al 1998 1999 9 10 the aim of this paper is to analyze the first order system least squares pseudo spectral method for incompressible navier stokes equations the paper will be an extension of the result in kim et al 2004 12 to the navier stokes equations our least squares functional is defined by the sum of discrete spectral norms of a first order system of equations corresponding to the navier stokes equations based on legendre gauss lobatto points we show its ellipticity and continuity over an appropriate product space and spectral convergences of discretization errors are derived in the h 1 norm and the l 2 norm in each variable finally we present some numerical examples

on stability ofc algebras letabe a unitalc algebra i e aadmits a countable approximate unit it is proved thatais stable i e ais isomorphic toa k where k is the algebra of compact operators on a separable hilbert space if and only if for each positive elementa aand eache 0 there exists a positive elementb asuch that ab eandx x a xx bfor somexina using this characterization it is proved among other things that the inductive limit of any sequence of unital stablec algebras is stable and that the crossed product of a unital stablec algebra by a discrete group is again stable

enhancing least squares finite element methods through a quantity of interest in this paper we introduce an approach that augments least squares finite element formulations with user specified quantities of interest the method incorporates the quantity of interest into the least squares functional and inherits the global approximation properties of the standard formulation as well as increased resolution of the quantity of interest we establish theoretical properties such as optimality and enhanced convergence under a set of general assumptions central to the approach is that it offers an element level estimate of the error in the quantity of interest as a result we introduce an adaptive approach that yields efficient adaptively refined approximations several numerical experiments for a range of situations are presented to support the theory and highlight the effectiveness of our methodology notably the results show that the new approach is effective at improving the accuracy per total computational cost

newton s method for the navier stokes equations with finite element initial guess of stokes equations abstract it is shown that finite element solutions of stokes equations may be chosen as the initial guess for the quadratic convergence of newton s algorithm applied to navier stokes equations provided there are sufficiently small mesh size h and the moderate reynold s number we provide also a mixed convergence analysis in terms of iterations and finite error estimates of the initial guess with a regularity estimate and error analysis for each newton s step

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