a first order system least squares finite element method for the shallow water equations a least squares finite element method for the first order system of the shallow water equations is proposed and studied the method combines a characteristic based time discretization with a least squares finite element approach which approximates the water level and the velocity field in h1 and h div respectively the linearized least squares functional is shown to be elliptic uniformly as the time step size tau approaches zero with respect to a suitably weighted norm moreover lipschitz continuity of the frechet derivative is shown with respect to this norm this implies that the local evaluation of the nonlinear least squares functional constitutes an a posteriori error estimator on which an adaptive refinement technique may be based the efficiency of such an adaptive finite element approach is tested numerically for a test problem involving the surface flow in a widening channel leading to a recirculating velocity field

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

an alternative least squares formulation of the navier stokes equations with improved mass conservation the focus of this paper is on incompressible flows in three dimensions modeled by least squares finite element methods lsfem and using a novel reformulation of the navier stokes equations lsfem are attractive because the resulting discrete equations yield symmetric positive definite systems of algebraic equations and the functional provides both a local and global error measure on the other hand it has been documented for existing reformulations that certain types of boundary conditions and high aspect ratio domains can yield very poor mass conservation it has also been documented that improved mass conservation with lsfem can be achieved by strengthening the coupling between the pressure and velocity the new reformulation presented here is demonstrated to provide both improved multigrid convergence rates because it is differentially diagonally dominant and improved mass conservation over existing methods because it increases the pressure velocity coupling along the inflow and outflow boundaries

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

implicit time discretization schemes for mixed least squares finite element formulations abstract this work is an extension of the ideas in averweg et al 2019 with the focus on a detailed investigation of implicit time discretization schemes to model instationary fluid flow based on the incompressible navier stokes equations and linear elastodynamic structural behavior the variational approaches for fluid and solid mechanics are based on a mixed least squares finite element method the l 2 norm minimization of the residuals of the constructed first order systems of the governing differential equations is based on two field stress velocity sv functionals for the time discretization of the sv fluid formulation four different types of implicit integration schemes are investigated namely the houbolt method the crank nicolson method and two explicit singly diagonally implicit runge kutta methods esdirk the sv formulation for the solid is discretized applying the houbolt method the presented time integration schemes are validated investigating an unsteady fluid flow and an elastodynamic structural benchmark since both fluid and solid sv formulations are discretized using conforming finite element spaces in h div and h 1 respectively the inherent fulfillment of coupling conditions when modeling fluid structure interaction problems is given a priori therefore the applicability is also examined by two simplified fsi problems for small deformations in order to represent the main characteristics of the presented approach

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

a characteristic galerkin approximation to a system of shallow water equations summary characteristic methods are known to handle advective flow better than traditional galerkin methods and allow large time steps to be taken when compared to standard time stepping methods in this paper we investigate a characteristic galerkin approximation to the 2 dimensional system of shallow water equations we derive cal l infty left 0 t cal l 2 omega right bounds for elevation and velocity showing these to be optimal for velocity in cal l 2 left 0 t cal h 1 omega right

series parallel languages and the bounded width property abstract in order to model concurrency we extend automata theory from the usual languages of words to sets of labelled series parallel posets or sp languages we use the algebraic properties of the set of series parallel posets to investigate the notion of recognizability based on finite index congruences and a notion of regularity based on a new class of branching automata the main results of the paper concern the sp languages having bounded width we show that such languages are recognizable if and only if they are regular in addition the regular bounded width sp languages are exactly those which can be described by series rational expressions that is expressions using the letters the operations of sequential and parallel product and sequential iterations we also give a strictly algebraic characterization of the bounded width property in terms of depth nilpotent series parallel algebras

nested iteration and first order system least squares for incompressible resistive magnetohydrodynamics this paper develops a nested iteration algorithm to solve time dependent nonlinear systems of partial differential equations for each time step newton s method is used to form approximate solutions from a sequence of nested spaces where the resolution of the approximations increases as the algorithm progresses nested iteration results in most of the iterations being performed on coarser grids where minimal work is needed to reduce error to the level of discretization error the approximate solution on a given coarse grid is interpolated to a refined grid and is used as an initial guess for the problem posed there the approximation is then already close enough to the solution on the current grid that a minimal amount of work is needed to solve the refined problem due to the rapid convergence of newton s method near a solution the paper develops an algorithm that attempts to optimize accuracy per computational cost on each grid so that essentially no unnecessary work is done on any grid the nested iteration algorithm is then applied to a reduced two dimensional model of the incompressible resistive magnetohydrodynamic mhd equations using this algorithm on the mhd equations in the context of a first order system least squares finite element discretization and algebraic multigrid to solve the linearized systems instabilities in a model tokamak fusion reactor are simulated numerical results show that this highly complex nonlinear problem is solved in an equivalent of 30 80 fine grid relaxation sweeps per time step

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

variable selection in heteroscedastic discriminant analysis abstract the likelihood ratio test statistic for the identity in means and covariance matrices of k normal populations has a well known step down decomposition measuring the contribution of each component of the vector observation this decomposition in turn gives rise to three components testing the residual homo scedasticity of each variable the parallelism of its regression on its predecessors and the identity of location a variety of uses of this decomposition in selecting variables is proposed

some methods of combining class information in multivariate normal discrimination for the classification of human chromosomes we consider the use of discriminant analysis based on an assumption of multivariate normality for allocating human chromosomes in an automated system in this context the assumptions which might be made about the covariance matrices for the different chromosome classes have important implications for the error rate of the system and the time required to allocate a chromosome linear discriminant functions based on the assumption of a common covariance matrix for all classes are fast but sometimes give bigger error rates than the assumption of a separate covariance matrix for each class the latter assumption requires many more calculations to evaluate the associated quadratic discriminant functions however it is possible to assume that the covariance matrices for the different classes are in various senses similar to one another in order to derive other methods of combining class information on variability these methods are here incorporated in the estimative maximum likelihood approach to discrimination the methods considered lead to machine classification times of human chromosomes intermediate between those for the assumptions of a common or unrelated covariance matrices they also require the simultaneous estimation of fewer parameters than the use of a separate covariance matrix for each chromosome class the methods are illustrated by three data sets of very different quality graphs of estimated error rate against classification time show that some of these ways of combining class information can be useful in the trade off of error rate against time

multilevel first order system least squares for nonlinear elliptic partial differential equations a fully variational approach is developed for solving nonlinear elliptic equations that enables accurate discretization and fast solution methods the equations are converted to a first order system that is then linearized via newton s method first order system least squares fosls is used to formulate and discretize the newton step and the resulting matrix equation is solved using algebraic multigrid amg the approach is coupled with nested iteration to provide an accurate initial guess for finer levels using coarse level computation a general theory is developed that confirms the usual full multigrid efficiency accuracy comparable to the finest level discretization is achieved at a cost proportional to the number of finest level degrees of freedom in a companion paper the theory is applied to elliptic grid generation egg and supported by numerical results

automatic context sensitive karyotyping of human chromosomes based on elliptically symmetric statistical distributions we introduce a statistical model of a metaphase cell consisting of independent chromosomes with elliptically symmetric feature vectors from this model we derive the ml classifier for classification in the 24 chromosomal classes taking into account the correct number of chromosomes in each class experimental results show that error rates of the best of these classifiers are less than 2 with respect to chromosomes if applied to the large copenhagen data set cpr simulation studies suggest that there should be even more information contained in the features of this data set

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

least squares finite element formulations for viscous incompressible and compressible fluid flows abstract we present least squares based finite element formulations for the numerical solution of viscous fluid flows governed by the navier stokes equations as an alternate approach to the well known weak form galerkin finite element formulations the use of least squares principles leads to a variational unconstrained minimization problem where compatibility conditions between approximation spaces never arise and the resulting linear algebraic problem will have a symmetric positive definite coefficient matrix we address the issue of norm equivalence of the least squares functional and its implications on the resulting finite element model in particular we develop an understanding of the compromise that must exist between the optimality and practicality of the finite element model the latter measured in terms of c k regularity across inter element boundaries we show through numerical examples that for the traditional c 0 basis such a compromise is possible when high p levels are used to span the finite element spaces when a low p level solution is desired guidelines are presented to obtain a reliable least squares collocation solution numerical examples are presented to demonstrate the high and low p level approaches these include incompressible flow past two circular cylinders in a side by side arrangement for gap sizes s d 2 0 and 0 85 incompressible flow past a square cylinder and subsonic transonic and supersonic compressible flow past a circular cylinder in addition we present a discontinuous least squares formulation where c k regularity across inter element boundaries is enforced in a weak sense through the least squares functional allowing for h and p type non conformities in the computational domain

space time least squares spectral elements for convection dominated unsteady flows legendre polynomials are employed in a space time least squares spectral element formulation applied to linear and nonlinear hyperbolic scalar equations no stabilization techniques are required to render a stable high order accurate scheme in parts of the domain where the underlying exact solution is smooth the scheme exhibits exponential convergence with polynomial enrichment whereas in parts of the domain where the underlying exact solution contains discontinuities the solution displays a gibbs like behavior numerical results will be given in which the capabilities of the space time formulation to capture discontinuities will be demonstrated

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

on mass conserving least squares methods least squares variational methods have several practical and theoretical advantages for solving elliptic partial differential equations including symmetric positive definite discrete operators and a sharp error measure one of the potential drawbacks especially in three dimensions is that mass conservation is achieved only in a least squares sense and underresolved solutions are especially vulnerable to poor conservation for the stationary navier stokes equations which are typically rewritten as a larger system of first order equations the loss of mass in the approximate solution is strongly dependent upon the boundary conditions used a new first order system reformulation of the navier stokes equations is presented that admits a wider range of mass conserving boundary conditions this new formulation is shown to provide both excellent mass conservation and excellent algebraic multigrid performance for three different problems a square channel backward facing step and branching tubes with two generations of bifurcations

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

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

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

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

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

first order system ll fosll for general scalar elliptic problems in the plane this paper develops new first order system ll fosll formulations for scalar elliptic partial differential equations it extends the work of z cal et al siam j numer anal 39 2001 pp 1418 1445 where the fosll methodology was first introduced one focus of that paper was to develop fl formulations that allow the use of h1 conforming finite element spaces and optimal multigrid solution techniques to construct l2 approximations of the dependent variables in the presence of discontinuous coefficients the problems for which this goal was achieved were limited to those with no reaction term and with dirichlet and neumann boundaries that were individually connected that is each had at most one component here new fosll formulations are developed to achieve the same goals on a wider class of problems including problems with reaction terms dirichlet and neumann boundaries with multiple components reentrant corners and points at which dirichlet and neumann boundaries meet with an inner angle greater than pi 2 the efficiency of the improved fosll formulations is illustrated by a series of numerical examples

the recognition of series parallel digraphs we present a linear time algorithm to recognize the class of vertex series parallel vsp digraphs our method is based on the relationship between vsp digraphs and the class of edge series paralle

first order system least squares and the energetic variational approach for two phase flow this paper develops a first order system least squares fosls formulation for equations of two phase flow the main goal is to show that this discretization along with numerical techniques such as nested iteration algebraic multigrid and adaptive local refinement can be used to solve these types of complex fluid flow problems in addition from an energetic variational approach it can be shown that an important quantity to preserve in a given simulation is the energy law we discuss the energy law and inherent structure for two phase flow using the allen cahn interface model and indicate how it is related to other complex fluid models such as magnetohydrodynamics finally we show that using the fosls framework one can still satisfy the appropriate energy law globally while using well known numerical techniques

two equational theories of partial words abstract partial words are in our terminology isomorphism classes of labeled partial orders we consider two structures starting with the partial orders reduced to a unique element the first one is defined as the closure under the three operations of series product parallel product and power series product of the same partial word and the second one as the closure under the three operations of series product parallel product and product series product of possibly different partial words we show that the two equational theories can be axiomatized by an infinite collection of simple identities and that no finite axiomatization exists we characterize the free algebras in the corresponding varieties as algebras of certain partial words subject to order and graph theoretic conditions we also show that the first equational theory is decidable

least squares mixed finite element solution of variably saturated subsurface flow problems a least squares mixed finite element formulation is applied to the nonlinear elliptic problems arising in each time step of an implicit euler discretization for variably saturated flow problems this approach simultaneously constructs approximations to the flux in raviart thomas spaces and to the hydraulic potential by standard h1 conforming linear finite elements two important properties of the least squares approach are investigated in detail the local least squares functional provides an a posteriori error estimator and gauss newton methods are robust iterative solvers for the resulting nonlinear least squares problems computational experiments conducted for a realistic water table recharge problem illustrate the effectiveness of this approach

series parallel languages on scattered and countable posets in this paper we investigate the recognition by finite automata of languages of countable labelled posets we unify and generalize several previous results from two different directions the theory of finite or wn free posets and automata over countable and scattered linear orderings first we establish that the smallest class of posets obtained from the empty set and the singleton and closed under finite parallel operation and sequential concatenation indexed by all linear orderings corresponds precisely to the class of scattered and countable n free posets without infinite antichains next we prove a kleene like theorem we define automata and rational expressions for languages of countable scattered n free labelled posets without infinite antichains and show that both formalisms have the same expressive power

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

least squares finite element method with high continuity nurbs basis for incompressible navier stokes equations modern least squares finite element method lsfem has advantage over mixed finite element method for non self adjoint problem like navier stokes equations but has problem to be norm equivalent and mass conservative when using c 0 type basis in this paper lsfem with non uniform b splines nurbs is proposed for navier stokes equations high order continuity nurbs is used to construct the finite dimensional spaces for both velocity and pressure variational form is derived from the governing equations with primitive variables and the dofs due to additional variables are not necessary there is a novel k refinement which has spectral convergence of least squares functional the method also has same advantages as isogeometric analysis like automatic mesh generation and exact geometry representation several benchmark problems are solved using the proposed method the results agree well with the benchmark solutions available in literature the results also show good performance in mass conservation

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

an essentially non oscillatory semi lagrangian method for tidal flow simulations we develop an essentially non oscillatory semi lagrangian method for solving two dimensional tidal flows the governing equations are derived from the incompressible navier stokes equations with assumptions of shallow water flows including bed frictions eddy viscosity wind shear stresses and coriolis forces the method employs the modified method of characteristics to discretize the convective term in a finite element framework limiters are incorporated in the method to reconstruct an essentially non oscillatory algorithm at minor additional cost the central idea consists in combining linear and quadratic interpolation procedures using nodes of the finite element where departure points are localized the resulting semi discretized system is then solved by an explicit runge kutta chebyshev scheme with extended stages this scheme adds in a natural way a stabilizing stage to the conventional runge kutta method using the chebyshev polynomials the proposed method is verified for the recirculation tidal flow in a channel with forward facing step we also apply the method for simulation of tidal flows in the strait of gibraltar in both test problems the proposed method demonstrates its ability to handle the interaction between water free surface and bed frictions copyright 2009 john wiley sons ltd

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

the equational theory of pomsets abstract pomsets have been introduced as a model of concurrency since a pomset is a string in which the total order has been relaxed to be a partial order in this paper we view them as a generalization of strings and investigate their algebraic properties in particular we investigate the axiomatic properties of pomsets sets of pomsets and ideals of pomsets under such operations as concatenation parallel composition union and their associated closure operations we find that the equational theory of sets pomsets under concatenation parallel composition and union is finitely axiomatizable whereas the theory of languages under the analogous operations is not a similar result is obtained for ideals of pomsets which incorporate the notion of subsumption which is also known as augmentation finally we show that the addition of any closure operation parallel or serial leads to nonfinite axiomatizability of the resulting equational theory

proportionality of covariance matrices on considere l inference pour un modele statistique specifiant la proportionnalite entre deux distributions de wishart independantes

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

long words the theory of concatenation and o power it is shown that for any set a the algebra of ordinal words on the alphabet a equipped with the operations of concatenation and power is axiomatized by the equationsx y z x y z x y x y x xn x n 1 indeed the algebra freely generated by a in the variety determined by these equations is the algebra of tail finite ordinal words of length on the alphabet a it is further shown that this collection of identities cannot be replaced by any finite set last a polynomial algorithm is given for recognizing when two terms denote the same tail finite ordinal word

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

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

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

projection multilevel methods for quasilinear elliptic partial differential equations theoretical results the goal of this paper is to introduce a new multilevel solver for two dimensional elliptic systems of nonlinear partial differential equations pdes where the nonlinearity is of the type u partial v the incompressible navier stokes equations are an important representative of this class and are the target of this study using a first order system least squares fosls approach and introducing a new variable for partial v for this class of pdes we obtain a formulation in which the nonlinearity appears as a product of two different dependent variables the result is a system that is linear within each variable but nonlinear in the cross terms in this paper we introduce a new multilevel method that treats the nonlinearities directly this approach is based on a projection multilevel pml method s f mccormick multilevel projection methods for partial differential equations siam philadelphia 1992 applied to the fosls functional the implementation of the discretization process relaxation coarse grid correction and cycling strategies is discussed and optimal performance is established numerically a companion paper t a manteuffel s f mccormick and o rohrle siam j numer anal 44 2006 pp 139 152 establishes a two level convergence proof for this new multilevel method

the finite element method for parabolic equations in this first of two papers computable a posteriori estimates of the space discretization error in the finite element method of lines solution of parabolic equations are analyzed for time independent space meshes the effectiveness of the error estimator is related to conditions on the solution regularity mesh family type and asymptotic range for the mesh size for clarity the results are limited to a model problem in which piecewise linear elements in one space dimension are used the results extend straight forwardly to systems of equations and higher order elements in one space dimension while the higher dimensional case requires additional considerations the theory presented here provides the basis for the analysis and adaptive construction of time dependent space meshes which is the subject of the second paper computational results show that the approach is practically very effective and suggest that it can be used for solving more general problems

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