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

some potential theory for reflecting brownian motion in holder and lipschitz domains bounds are found on the transition densities and green functions for brownian motion with normal reflection in holder and lipschitz domains for lipschitz domains reflecting brownian motion and boundary local time are constructed a harnack inequality valid up to the boundary is proved a probabilistic solution to the neumann problem is given and the kuramochi boundary is identified

direct minimization of the least squares spectral element functional part i direct solver this paper describes an equivalent but improved least squares formulation for the numerical approximation of the solution of partial differential equations instead of using variational analysis to impose the conditions for minimizing the residual the residuals are minimized directly thus leading to a method we will denote by direct minimization dm dm circumvents setting up the normal equations which consists of matrix matrix multiplications matrix matrix multiplications are expensive may lead to loss of accuracy and destroy the sparsity pattern present in the original system the condition number of the dm formulation is the square root of the condition number which would be obtained if variational analysis was employed an element by element procedure will be presented which allows for parallelization of dm a computational comparison between dm and the conventional least squares formulation based on variational analysis will be presented

a new consistent splitting scheme for incompressible navier stokes flows a least squares spectral element implementation this paper presents a new consistent splitting scheme for the numerical solution of incompressible navier stokes flows allowing to consistently decouple the computation of velocity and pressure the scheme is not a pressure correction or velocity correction scheme and does not display the splitting error in pressure associated with these fractional step methods the linearized momentum equations are first solved based on an explicit treatment of the pressure resulting in an advection diffusion problem for each velocity component a least squares projection is used to numerically solve the advection diffusion problem next a div curl problem is solved to make the velocity field solenoidal this step is also handled by a least squares projection finally a pressure poisson problem is solved to obtain the pressure field induced by the solenoidal velocity field this is done by solving the weak poisson problem by a galerkin projection or alternatively by solving the strong poisson problem by a least squares projection at each stage we only see coefficient matrices with a symmetric positive definite structure and use matrix free preconditioned conjugate gradient methods to numerically solve for the velocity and pressure fields high order c 0 spectral basis are used to span the finite element spaces a verification benchmark shows optimal algebraic convergence rates in time for the velocity pressure and vorticity the scheme is further verified by simulating the two dimensional unsteady flow past a circular cylinder up to moderately high reynolds numbers

a div curl grad formulation for compressible buoyant flows solved by the least squares finite element method abstract the present paper reports the development of the least squares finite element method for simulating compressible buoyant flows at low mach numbers we propose a div curl grad formulation with unknowns including vorticity velocity heat fluxes temperature and pressure variation the formulation is proved to be elliptic such that permissible boundary conditions become self evident for a well posed flow problem in contrast to conventional approaches the present method evades the predicament of the singularity problem of low speed flows and no special treatment or artificial boundary condition is needed moreover the assembled coefficient matrix is symmetric and positive definite its inversion is implemented by an element by element jacobi conjugate gradient method as a numerical example we calculate two dimensional compressible buoyant flows inside a square enclosure at various rayleigh numbers for rayleigh number one million four secondary vortices were found embedded in the primary vortex due to significant temperature variations the fluid flows are highly compressible in the interior along the walls however the flows are incompressible the nusselt number rayleigh number correlation deduced from the numerical result compared favorably with previously reported data

stochastic differential equations with reflecting boundary conditions on resout des equations differentielles stochastiques a conditions aux limites reflechissantes par une approche directe basee sur le probleme de skorokhod

a parallel time space least squares spectral element solver for incompressible flow problems abstract the least squares spectral element method lsq sem is a very promising numerical method which combines the least squares formulation with the spectral element method in this article a parallel space time implementation of a least squares spectral element method for incompressible flows is presented in the space time formulation time is included as an extra dimension providing an efficient framework for handling time dependent independent multidimensional problems the algebraic solver is based on the shur complement method numerical experiments are performed for showing the capability and computational efficiency of the solver besides the scalability of the code is tested

fictitious domain approach with hp finite element approximation for incompressible fluid flow we consider the application of fictitious domain approach combined with least squares spectral elements for the numerical solution of fluid dynamic incompressible equations fictitious domain methods allow problems formulated on a complicated shaped domain to be solved on a simpler domain containing least squares spectral element method has been used to develop the discrete model as this scheme combines the generality of finite element methods with the accuracy of spectral methods moreover the least squares methods have theoretical and computational advantages in the algorithmic design and implementation this paper presents the formulation and validation of the fictitious domain least squares spectral element approach for the steady incompressible navier stokes equations the convergence of the approximated solution is verified solving two dimensional benchmark problems demonstrating the predictive capability of the proposed formulation

a spectral element least squares formulation for incompressible navier stokes flows using triangular nodal elements we present a least squares formulation for the numerical solution of incompressible flows using high order triangular nodal elements the fekete points of the triangle are used as nodes and numerical integration is performed using tensor product gauss legendre rules in a collapsed coordinate system for the standard triangle a first order system least squares fosls approach based on velocity pressure and vorticity is used to allow the use of practical c 0 element expansions in each triangle the numerical results demonstrate spectral convergence for smooth solutions excellent conservation of mass for steady and unsteady problems of the inflow outflow type and the flexibility of using triangles to partition domains where the use of quadrangles would be cumbersome or inefficient

basis functions for triangular and quadrilateral high order elements in this paper we present a unified description of new spectral bases suitable for high order hp finite element discretizations on hybrid two dimensional meshes consisting of triangles and quadrilaterals all bases presented are for c0 continuous discretizations and are described both as modal and as mixed modal nodal expansions general jacobi polynomials of mixed weights are employed that accommodate automatic exact numerical quadratures generalized tensor products and variable expansion order in each element the approximation properties of the bases are analyzed in the context of the projection linear advection and diffusion operators

penalty finite element analysis of 3 d navier stokes equations abstract the steady state navier stokes equations in three dimensions are solved by a penalty finite element method for the problems of wall driven cavity flow in a cubical box and natural convection in a cubical cavity subjected to differential side heating the present solutions are compared with recent finite difference solutions of the wall driven cavity problem for reynolds numbersre 100 and 400 the agreement is found to be very good

large scale computation of incompressible viscous flow by least squares finite element method abstract the least square finite element method lsfem based on the velocity pressure vorticity formulation is applied to large scale three dimensional steady incompressible navier stokes problems this method can accommodate equal order interpolations and results in a symmetric positive definite algebraic system which can be solved effectively by simple iterative methods the first order velocity bernoulli pressure vorticity formulation for incompressible viscous flows is also tested the first order velocity pressure stress formulation is not elliptic in the ordinary sense so we do not recommend its use for newtonian flows for three dimensional flows a compatibility equation i e zero divergence of vorticity vector is included to make the first order system elliptic as a by product of proving the ellipticity of first order systems a rigorous mathematical technique has been developed to justify the number of permissible boundary conditions for the navier stokes equations the simple substitution or newton s method is employed to linearize the partial differential equations the lsfem is used to obtain discretized equations and the system of algebraic equations is solved using the jacobi preconditioned conjugate gradient method which avoids formation of either element or global matrices matrix free to achieve high efficiency to show the validity of this method for large scale computation we give numerical results for the 2d driven cavity problem at re 10000 with 408 400 bilinear elements the flow in a 3d cavity is calculated at re 100 400 and 1000 with 50 52 50 trilinear elements the taylor gortler like vortices are observed for re 1000

discrete approximations to reflected brownian motion in this paper we investigate three discrete or semi discrete approximation schemes for reflected brownian motion on bounded euclidean domains for a class of bounded domains d in r n that includes all bounded lipschitz domains and the von koch snowflake domain we show that the laws of both discrete and continuous time simple random walks on d 2 k z n moving at the rate 2 2k with stationary initial distribution converge weakly in the space d 0 1 r n equipped with the skorokhod topology to the law of the stationary reflected brownian motion on d we further show that the following myopic conditioning algorithm generates in the limit a reflected brownian motion on any bounded domain d for every integer k 1 let x k j2 k j 0 1 2 be a discrete time markov chain with one step transition probabilities being the same as those for the brownian motion in d conditioned not to exit d before time 2 k we prove that the laws of x k converge to that of the reflected brownian motion on d these approximation schemes give not only new ways of constructing reflected brownian motion but also implementable algorithms to simulate reflected brownian motion

mixed mimetic spectral element method for stokes flow a pointwise divergence free solution in this paper we apply the recently developed mimetic discretization method to the mixed formulation of the stokes problem in terms of vorticity velocity and pressure the mimetic discretization presented in this paper and in kreeft et al 51 is a higher order method for curvilinear quadrilaterals and hexahedrals fundamental is the underlying structure of oriented geometric objects the relation between these objects through the boundary operator and how this defines the exterior derivative representing the grad curl and div through the generalized stokes theorem the mimetic method presented here uses the language of differential k forms with k cochains as their discrete counterpart and the relations between them in terms of the mimetic operators reduction reconstruction and projection the reconstruction consists of the recently developed mimetic spectral interpolation functions the most important result of the mimetic framework is the commutation between differentiation at the continuous level with that on the finite dimensional and discrete level as a result operators like gradient curl and divergence are discretized exactly for stokes flow this implies a pointwise divergence free solution this is confirmed using a set of test cases on both cartesian and curvilinear meshes it will be shown that the method converges optimally for all admissible boundary conditions

a direct solver for the least squares spectral collocation system on rectangular elements for the incompressible navier stokes equations a least squares spectral collocation scheme for the stokes and incompressible navier stokes equations is proposed the original domain is decomposed into quadrilateral subelements and on the element interfaces continuity of the functions is enforced in the least squares sense the collocation conditions and the interface conditions lead to overdetermined systems these systems are directly solved by qr decomposition of the underlying matrices by numerical simulations it is shown that the direct method leads to better results than the approach with normal equations furthermore it is shown that the condition numbers can be reduced by introducing the clenshaw curtis quadrature rule for imposing the average pressure to be zero finally our scheme is successfully applied to the regularized and lid driven cavity flow problems

mass and momentum conservation of the least squares spectral collocation method for the navier stokes equations from the literature it is known that the least squares spectral element method lssem for the stationary stokes equations performs poorly with respect to mass conservation but compensates this lack by a superior conservation of momentum furthermore it is known that the least squares spectral collocation method lsscm leads to superior conservation of mass and momentum for the stationary stokes equations in the present paper we consider mass and momentum conservation of the lsscm for time dependent stokes and navier stokes equations we observe that the lsscm leads to improved conservation of mass and momentum for these problems furthermore the lsscm leads to the well known time dependent profiles for the velocity and the pressure profiles to obtain these results we use only a few elements each with high polynomial degree avoid normal equations for solving the overdetermined linear systems of equations and introduce the clenshaw curtis quadrature rule for imposing the average pressure to be zero furthermore we combined the transformation of gordon and hall transfinite mapping with the least squares spectral collocation scheme to discretize the internal flow problems

numerical calculation of the moments of the population balance equation the combined cfd pbm population balance models are computationally intensive so a possibility is to calculate only a few moments of the probability density function pdf of the pbm minimizing the computational costs however this formulation results in an excess of unknowns with respect to equations which is referred to as a closure problem one approach for dealing with this closure problem is to apply a numerical quadrature approximation on the other hand a different possibility is to compute the pdf and from this the moments of interest if required in this work the two mentioned approaches are discussed and numerical experiments are used to show the capability of the methods for predicting the moments of the pbe in particular the quadrature method of moments and a time space least squares spectral method will be discussed

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

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

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

stabilized mixed methods for the stokes problem the solution of the stokes problem is approximated by three stabilized mixed methods one due to hughes balestra and franca and the other two being variants of this procedure in each case the bilinear form associated with the saddle point problem of the standard mixed formulation is modified to become coercive over the finite element space error estimates are derived for each procedure

temporal spatial and thermal features of 3 d rayleigh benard convection by a least squares finite element method numerical solutions of 3 d time dependent rayleigh benard convection are presented in this work the temporal spatial and thermal features of convective patterns are studied for four different geometric aspect ratios 2 1 2 4 1 4 5 1 5 and 3 5 1 2 1 at supercritical rayleigh numbers ra 8 103 2 4 104 and prandtl numbers pr 0 71 2 5 several physical phenomena such as multicellular flow pattern oscillatory transient solution t shaped rolls at the ends of a rectangular box and roll alignment are observed in our simulations the numerical technique is based on an implicit fully coupled and time accurate method which consists of the crank nicolson scheme for time integration newton s method for the convective terms with extensive linearization steps and a least squares finite element method a matrix free algorithm of the jacobi conjugate gradient method is implemented to solve the symmetric positive definite linear system of equations

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

least squares spectral method for solving advective population balance problems the combined cfd pbe population balance models are computationally intensive requiring efficient numerical methods for solving practical problems in this paper a high order method is presented based on the least squares method lsm for the solution of a spatial dependent population balance equation which includes advective processes numerical experiments are performed in order to study the behavior of the proposed method for one dimensional cases using model problems with analytical solutions

on reflecting diffusion processes and skorokhod decompositions letg be ad dimensional bounded euclidean domain h1 g the set off in l2 g such that f defined in the distribution sense is in l2 g reflecting diffusion processes associated with the dirichlet spaces h1 g on l2 g dx are considered in this paper where a aij is a symmetric bounded uniformly ellipticd d matrix valued function such thata ij h1 g for eachi j and h1 g is a positive bounded function ong which is bounded away from zero a skorokhod decomposition is derived for the continuous reflecting markov processes associated with h1 g having starting points ing under a mild condition which is satisfied when g has finite d 1 dimensional lower minkowski content

an augmented mixed finite element method for the vorticity velocity pressure formulation of the stokes equations this paper deals with the numerical approximation of the stationary two dimensional stokes equations formulated in terms of vorticity velocity and pressure with non standard boundary conditions here by introducing a galerkin least squares term we end up with a stabilized variational formulation that can be recast as a twofold saddle point problem we propose two families of mixed finite elements to solve the discrete problem in the first family the unknowns are approximated by piecewise continuous and quadratic elements brezzi douglas marini and piecewise constant finite elements respectively while in the second family the unknowns are approximated by piecewise linear and continuous raviart thomas and piecewise constant finite elements respectively the wellposedness of the resulting continuous and discrete variational problems are studied employing an extension of the babuska brezzi theory we establish a priori error estimates in the natural norms and we finally report some numerical experiments illustrating the behavior of the proposed schemes and confirming our theoretical findings on structured and unstructured meshes additional examples of cases not covered by our theory are also presented

least squares spectral elements applied to the stokes problem least squares spectral element methods are based on two important and successful numerical methods spectral hp element methods and least squares finite element methods in this respect least squares spectral element methods seem very powerful since they combine the generality of finite element methods with the accuracy of the spectral methods and also have the theoretical and computational advantages of the least squares methods these features make the proposed method a competitive candidate for the solution of large scale problems arising in scientific computing in order to demonstrate its competitiveness the method has been applied to an analytical problem and the theoretical optimal and suboptimal a priori estimates have been confirmed for various boundary conditions moreover the exponential convergence rates typical for a spectral element discretization have also been confirmed the comparison with the classical galerkin spectral element method revealed that the least squares spectral element method is as accurate as the galerkin method for the smooth model problem

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

hp adaptive least squares spectral element method for hyperbolic partial differential equations this paper describes a hp adaptive spectral element formulation which is used to discretize the weak formulation obtained by minimizing the residuals in the l 2 norm the least squares error indicator will be briefly discussed refinement of the numerical approximation is based on an estimate of the regularity of the underlying exact solution if the underlying exact solution is sufficiently smooth polynomial enrichment is employed in areas with limited regularity h refinement is used for this purpose the sobolev regularity is estimated functionally and geometrically non conforming neighbouring elements are patched together using so called mortar elements results of this approach are compared to uniform h and p refinement for a linear advection equation

least squares spectral element method for non linear hyperbolic differential equations the least squares spectral element method has been applied to the one dimensional inviscid burgers equation which allows for discontinuous solutions in order to achieve high order accuracy both in space and in time a space time formulation has been applied the burgers equation has been discretized in three different ways a non conservative formulation a conservative system with two variables and two equations one first order linear pde and one linearized algebraic equation and finally a variant on this conservative formulation applied to a direct minimization with a qr decomposition at elemental level for all three formulations an h p convergence study has been performed and the results are discussed in this paper

stochastic differential switching game in infinite horizon abstract we study a zero sum stochastic differential switching game in infinite horizon we prove the existence of the value of the game and characterize it as the unique viscosity solution of the associated system of quasi variational inequalities with bilateral obstacles we also obtain a verification theorem which provides an optimal strategy of the game finally some numerical examples with two regimes are given

error analysis of some galerkin least squares methods for the elasticity equations the recent technique of stabilizing mixed finite element methods by augmenting the galerkin formulation with least squares terms calculated separately on each element is considered the error analysis is performed in a unified manner yielding improved results for some methods introduced earlier in addition a new formulation is introduced and analyzed method is used within two different contexts the first is in connection with the plain strain or three dimensional problems near the incompressible limit for which the poisson ratio equals one half it is well known that for these problems a standard lower order displacement finite element method will break down by now it is also well known how a working method can be obtained a new unknown the pressure is introduced and discretized with a finite element space different from that used for the displacements the finite element method obtained falls into the class of saddlepoint problems for which there is a general theory developed by brezzi and babuska 5 7 the application of this theory shows that the method converges optimally if the discrete spaces for the displacement and pressure satisfy a stability condition for a particular choice of displacement and pressure approximations it can be a difficult task to verify this babuska brezzi or inf sup condition during the last decade this problem has been intensively studied and the performances of many combinations are now known cf 18 and 23 26 for some recent extensions in addition there exists a general technique utilizing bubble functions by which a stable method can

space time coupled spectral hp least squares finite element formulation for the incompressible navier stokes equations we consider least squares finite element models for the numerical solution of the non stationary navier stokes equations governing viscous incompressible fluid flows the paper presents a formulation where the effects of space and time are coupled resulting in a true space time least squares minimization procedure as opposed to a space time decoupled formulation where a least squares minimization procedure is performed in space at each time step the formulation is first presented for the linear advection diffusion equation and then extended to the navier stokes equations the formulation has no time step stability restrictions and is spectrally accurate in both space and time to allow the use of practical c0 element expansions in the resulting finite element model the navier stokes equations are expressed as an equivalent set of first order equations by introducing vorticity as an additional independent variable and the least squares method is used to develop the finite element model of the governing equations 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 in matrix free form spectral convergence of the l2 least squares functional and l2 error norms in space time is verified using a smooth solution to the two dimensional non stationary incompressible navier stokes equations numerical results are presented for impulsively started lid driven cavity flow oscillatory lid driven cavity flow transient flow over a backward facing step and flow around a circular cylinder the results demonstrate the predictive capability and robustness of the proposed formulation even though the space time coupled formulation is emphasized we also present the formulation and numerical results for least squares space time decoupled finite element models the numerical results show that the space time coupled formulation has superior predictive capabilities for flows demanding high space time resolution exemplified here by the transient flow over a backward facing step

reaction diffusion on metric graphs from 3d to 1d abstract we provide stochastic foundations for the analysis of a class of reaction diffusion systems using as an example the known temporal analysis of products tap experiments showing how to effectively obtain explicit solutions to the associated equations by approximating the 3 dimensional domain of diffusion u the reactor by 1 dimensional network models in a typical tap experiment a pulse of reactant gas of species a is injected into u which is filled with chemically inert material permeable to gas diffusion particles of catalyst are placed amid this inert medium forming active sites where the reaction a b may occur on part of the boundary of u designated as the exit a mixture of a and b can escape we study the problem of determining the molar fraction of product gas in the mixture after u is fully evacuated this fraction is identified with the reaction probability that is the probability of a single diffusing molecule reacting before leaving u specifically we are interested in how this probability depends on the reaction rate constant k after giving a stochastic formulation of the problem and the boundary value problem whose solution is this probability we study a class of domains called fat graphs comprising a network of thin tubes with active sites at junctures the main result of the paper is that in the limit as the thin tubes approach curves in 3 dimensional space reaction probability converges to functions of the point of gas injection that can be computed explicitly in terms of a rescaled parameter k by this 3d to 1d reduction the simpler processes on metric graphs can be used as model systems for more realistic 3 dimensional configurations this is illustrated with analytic and numerical examples one example in particular illustrates an important application of our method finding the catalyst configuration that maximizes reaction probability at a given reaction rate constant

diffusion processes on graphs stochastic differential equations large deviation principle ito s rule is established for the diffusion processes on the graphs we also consider a family of diffusions processes with small noise on a graph large deviation principle is proved for these diffusion processes and their local times at the vertices

a priori and a posteriori error analysis of a mixed scheme for the brinkman problem this paper deals with the analysis of new mixed finite element methods for the brinkman equations formulated in terms of velocity vorticity and pressure employing the babuska brezzi theory it is proved that the resulting continuous and discrete variational formulations are well posed in particular we show that raviart thomas elements of order k ge 0 k 0 for the approximation of the velocity field piecewise continuous polynomials of degree k 1 k 1 for the vorticity and piecewise polynomials of degree k for the pressure yield unique solvability of the discrete problem on the other hand we also show that families of finite elements based on brezzi douglas marini elements of order k 1 k 1 for the approximation of velocity piecewise continuous polynomials of degree k 2 k 2 for the vorticity and piecewise polynomials of degree k for the pressure ensure the well posedness of the associated galerkin scheme we note that these families provide exactly divergence free approximations of the velocity field we establish a priori error estimates in the natural norms with constants independent of the viscosity and we carry out the reliability and efficiency analysis of a residual based a posteriori error estimator finally we report several numerical experiments illustrating the behaviour of the proposed schemes and confirming our theoretical results on unstructured meshes additional examples of cases not covered by our theory are also presented

conservation of mass and momentum of the least squares spectral collocation scheme for the stokes problem from the literature it is known that spectral least squares schemes perform poorly with respect to mass conservation and compensate this lack by a superior conservation of momentum this should be revised since the here presented new least squares spectral collocation scheme leads to an outstanding performance with respect to conservation of momentum and mass the reasons can be found in using only a few elements each with high polynomial degree avoiding normal equations for solving the overdetermined linear systems of equations and by introducing the clenshaw curtis quadrature rule for imposing the average pressure to be zero furthermore we combined the transformation of gordon and hall transfinite mapping with our least squares spectral collocation scheme to discretize the internal flow problems

spectral element method for three dimensional elliptic problems with smooth interfaces abstract in this paper we propose a least squares spectral element method for three dimensional elliptic interface problems the differentiability estimates and the main stability theorem using non conforming spectral element functions are proven the proposed method is free from any kind of first order reformulation a suitable preconditioner is constructed with help of the regularity estimate and proposed stability estimates which is used to control the condition number we show that these preconditioners are spectrally equivalent to the quadratic forms by which we approximate them we obtain the error estimates which show the exponential accuracy of the method numerical results are obtained for both straight and curved interfaces to show the efficiency of the proposed method

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

a collocation penalty least squares finite element formulation for incompressible flows abstract we present a penalty least squares finite element formulation for incompressible flows the pressure degree of freedom is eliminated using the classical penalty approach and the least squares model is formed in terms of velocity vorticity and dilatation an iterative penalization is implemented which allows the use of low penalty parameters and retains a manageable conditioning number of the global coefficient matrix the h convergence is verified using the exact solution of kovasznay s flow numerical results are presented for a number of benchmark problems e g steady flow past a large circular cylinder in a channel transient flow over a backward facing step unsteady flow past a circular cylinder and buoyancy driven natural convection in a square cavity for all numerical examples the effect of penalty parameter on the accuracy is investigated thoroughly and it is concluded that the present model produces accurate results for low penalty parameters 10 100 these problems are also solved on coarse meshes to show that good mass conservation is achieved

spectral hp penalty least squares finite element formulation for the steady incompressible navier stokes equations in this paper we present spectral hp penalty least squares finite element formulation for the numerical solution of steady incompressible navier stokes equations the continuity equation is treated as a constraint on velocity field and the constraint is enforced using the penalty method which is equivalent to replacing the pressure variable p with p n p n email protected u where u is the velocity field c is the penalty parameter and superscript n is non linear iteration number the modified navier stokes equations are expressed as an equivalent set of first order equations by introducing vorticity dilatation or stresses as additional independent variables and the least squares method is used to develop the finite element model high order basis functions are used to construct the discrete model unlike other penalty finite element formulations equal order integration is used for all terms of the coefficient matrix schur complement method is used and all interior element degrees of freedom are condensed out and interface degrees of freedom are solved either by direct solver banded cholesky factorization or iterative solver preconditioned conjugate gradient present formulation produces very accurate results for even very low penalty parameters 10 40 in this way present formulation and implementation fixes perennial problem of bad conditioning of coefficient matrix associated with penalty formulations proposed formulation is verified using the exact solution of kovasznay flow numerical results are also presented for a number of two dimensional benchmark problems e g 2d lid driven cavity flow flow over a backward facing step steady flow past a circular cylinder and 2d flow past a large circular cylinder in a channel finally the proposed formulation is extended to coupled velocity temperature problems and buoyant flow inside a square enclosure is solved to test the penalty least squares finite element model developed herein

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

least squares spectral element preconditioners for fourth order elliptic problems abstract in this paper we propose preconditioners for the system of linear equations that arise from a discretization of fourth order elliptic problems in two and three dimensions d 2 3 using spectral element methods these preconditioners are constructed using separation of variables and can be diagonalized and hence easy to invert for second order elliptic problems this technique has proven to be successful and performs better than other preconditioners in the framework of least squares methods we show that these preconditioners are spectrally equivalent to the quadratic forms by which we approximate them numerical results for the condition number reflects the effectiveness of the preconditioners

a multiscale finite element method for the incompressible navier stokes equations this paper presents a new multiscale finite element method for the incompressible navier stokes equations the proposed method arises from a decomposition of the velocity field into coarse resolved scales and fine unresolved scales modeling of the unresolved scales corrects the lack of stability of the standard galerkin formulation and yields a method that possesses superior properties like that of the streamline upwind petrov galerkin supg method and the galerkin least squares gls method the multiscale method allows arbitrary combinations of interpolation functions for the velocity and the pressure fields specifically the equal order interpolations that are easy to implement but violate the celebrated babuska brezzi condition a significant feature of the present method is that the structure of the stabilization tensor s appears naturally via the solution of the fine scale problem a family of 2 d elements comprising 3 and 6 node triangles and 4 and 9 node quadrilaterals has been developed convergence studies for the method on uniform skewed as well as composite meshes are presented numerical simulations of the nonlinear steady and transient flow problems are shown that exhibit the good stability and accuracy properties of the method 2005 elsevier b v all rights reserved

exponentially accurate nonconforming least squares spectral element method for elliptic problems on unbounded domain abstract non conforming approximation methods are becoming increasingly popular because of the potential to apply to multi material and multi model analysis for both bounded and unbounded domains in this paper we present a least square approximation based method to solve the one or two dimensional elliptic problems on an unbounded domain the method gives exponential accuracy and shows superior performance when compared to other numerical methods differentiability estimates and the main stability estimate theorem using a non conforming spectral element method are also discussed the exponential convergence rate of the proposed method is also shown through rigorous error estimate and specific numerical examples

least squares finite element formulation for shear deformable shells abstract we present a least squares based finite element formulation for the numerical analysis of shear deformable shell structures the variational problem is obtained by minimizing the least squares functional defined as the sum of the squares of the shell equilibrium equations residuals measured in suitable norms of hilbert spaces the use of least squares principles leads to a variational unconstrained minimization problem where compatibility conditions between approximation spaces never arise i e stability requirements such as inf sup conditions never arise the proposed formulation retains the generalized displacements and stress resultants as independent variables and in view of the nature of the variational setting upon which the finite element model is built allows for equal order interpolation a p type hierarchical basis is used to construct the discrete finite element model based on the least squares formulation exponentially fast decay of the least squares functional is verified for increasing order of the modal expansions several well established benchmark problems are solved to demonstrate the predictive capability of the least squares based shell elements shell elements based on this formulation are shown to be effective in both membrane and bending dominated states

a discontinuous petrov galerkin methodology for adaptive solutions to the incompressible navier stokes equations the discontinuous petrov galerkin methodology with optimal test functions dpg of demkowicz and gopalakrishnan 18 20 guarantees the optimality of the solution in an energy norm and provides several features facilitating adaptive schemes whereas bubnov galerkin methods use identical trial and test spaces petrov galerkin methods allow these function spaces to differ in dpg test functions are computed on the fly and are chosen to realize the supremum in the inf sup condition the method is equivalent to a minimum residual method for well posed problems with sufficiently regular solutions dpg can be shown to converge at optimal rates the inf sup constants governing the convergence are mesh independent and of the same order as those governing the continuous problem 48 dpg also provides an accurate mechanism for measuring the error and this can be used to drive adaptive mesh refinements we employ dpg to solve the steady incompressible navier stokes equations in two dimensions building on previous work on the stokes equations and focusing particularly on the usefulness of the approach for automatic adaptivity starting from a coarse mesh we apply our approach to a manufactured solution due to kovasznay as well as the lid driven cavity flow backward facing step and flow past a cylinder problems

diffusion processes in thin tubes and their limits on graphs the present paper is concerned with diffusion processes running on tubular domains with conditions on nonreaching the boundary respectively reflecting at the boundary and corresponding processes in the limit where the thin tubular domains are shrinking to graphs the methods we use are probabilistic ones for shrinking we use big potentials respectively reflection on the boundary of tubes we show that there exists a unique limit process and we characterize the limit process by a second order differential generator acting on functions defined on the limit graph with kirchhoff boundary conditions at the vertices

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