some least squares galerkin procedures for first order time dependent convection diffusion system abstract four least squares galerkin finite element schemes are formulated to solve first order time dependent convection diffusion systems and the convergence of these schemes is analyzed

validation analysis of mirror descent stochastic approximation method the main goal of this paper is to develop accuracy estimates for stochastic programming problems by employing stochastic approximation sa type algorithms to this end we show that while running a mirror descent stochastic approximation procedure one can compute with a small additional effort lower and upper statistical bounds for the optimal objective value we demonstrate that for a certain class of convex stochastic programs these bounds are comparable in quality with similar bounds computed by the sample average approximation method while their computational cost is considerably smaller

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

a continuous interior penalty method for viscoelastic flows in this paper we consider a finite element discretization of the oldroyd b model of viscoelastic flows the method uses standard continuous polynomial finite element spaces for velocities pressures and stresses inf sup stability and stability for convection dominated flows are obtained by adding a term penalizing the jump of the solution gradient over element faces to increase robustness when the deborah number is high we add a nonlinear artificial viscosity of shock capturing type the method is analyzed on a linear model problem and optimal a priori error estimates are proven that are independent of the solvent viscosity eta s finally we demonstrate the performance of the method on some known benchmark cases

a variational multiscale method with bubble stabilization for the oseen problem based on two local gauss integrations abstract we present a variational multiscale method for the oseen equation with higher reynolds number based on the mini element discretization the stabilization term only depends on bubble functions and the best algorithmic feature of our method is using two local gauss integrations to replace projection operator without adding any extra storage a priori error estimate is proved and the effectiveness of the method is illustrated through several numerical experiments

two splitting positive definite mixed finite element methods for parabolic integro differential equations abstract two novel mixed finite element procedures are established for parabolic integro differential equations which can be split into two independent symmetric positive definite sub schemes and do not need to solve a coupled system of equations without requiring the lbb consistency condition the convergence analysis shows that both methods lead to the optimal order l 2 norm error estimate for u and optimal h div norm error estimate for a numerical example is presented to illustrate the theoretical analysis

adjustable robust solutions of uncertain linear programs we consider linear programs with uncertain parameters lying in some prescribed uncertainty set where part of the variables must be determined before the realization of the uncertain parameters non adjustable variables while the other part are variables that can be chosen after the realization adjustable variables we extend the robust optimization methodology 1 3 6 9 13 14 to this situation by introducing the adjustable robust counterpart arc associated with an lp of the above structure often the arc is significantly less conservative than the usual robust counterpart rc however in most cases the arc is computationally intractable np hard this difficulty is addressed by restricting the adjustable variables to be affine functions of the uncertain data the ensuing affinely adjustable robust counterpart aarc problem is then shown to be in certain important cases equivalent to a tractable optimization problem typically an lp or a semidefinite problem and in other cases having a tight approximation which is tractable the aarc approach is illustrated by applying it to a multi stage inventory management problem

relativistic gamow vectors the friedrichs model has often been used in order to obtain explicit formulas for eigenvectors associated to complex eigenvalues corresponding to lifetimes such eigenvectors are called gamow vectors and they acquire meaning in extensions of the conventional hilbert space of quantum theory to the so called rigged hilbert space in this paper gamow vectors are constructed for a solvable model of an unstable relativistic field as a result we obtain a time asymmetric relativistic extension of the fock space this extension leads to two distinct poincare semigroups the time reversal transformation maps one semigroup to the other as a result the usual pct invariance should be extended we show that irreversibility as expressed by dynamical semigroups is compatible with the requirements of relativity

a generalized streamline finite element approach for the analysis of incompressible flow problems including moving surfaces abstract in the present work a generalized streamline finite element formulation able to deal with incompressible flow problems is presented in the finite element framework this technique allows the use of equal order interpolation for the unknowns of the problem velocity and pressure in this context stable and convergent solutions can be obtained without requiring tuning parameters defined outside this model the tracking of moving surfaces is also included in the numerical model this formulation has been checked in 21 and 3d tests

robust stochastic approximation approach to stochastic programming in this paper we consider optimization problems where the objective function is given in a form of the expectation a basic difficulty of solving such stochastic optimization problems is that the involved multidimensional integrals expectations cannot be computed with high accuracy the aim of this paper is to compare two computational approaches based on monte carlo sampling techniques namely the stochastic approximation sa and the sample average approximation saa methods both approaches the sa and saa methods have a long history current opinion is that the saa method can efficiently use a specific say linear structure of the considered problem while the sa approach is a crude subgradient method which often performs poorly in practice we intend to demonstrate that a properly modified sa approach can be competitive and even significantly outperform the saa method for a certain class of convex stochastic problems we extend the analysis to the case of convex concave stochastic saddle point problems and present in our opinion highly encouraging results of numerical experiments

characteristic stabilized finite element method for the transient navier stokes equations abstract based on the lowest equal order conforming finite element subspace xh mh i e p1 p1 or q1 q1 elements a characteristic stabilized finite element method for transient navier stokes problem is proposed the proposed method has a number of attractive computational properties parameter free avoiding higher order derivatives or edge based data structures and averting the difficulties caused by trilinear terms existence uniqueness and error estimates of the approximate solution are proved by applying the technique of characteristic finite element method finally a serious of numerical experiments are given to show that this method is highly efficient for transient navier stokes problem

an exact approach to the oscillator radiation process in an arbitrarily large cavity starting from a solution of the problem of a mechanical oscillator coupled to a scalar field inside a reflecting sphere of radius r we study the behaviour of the system in free space as the limit of an arbitrarily large radius in the confined solution from a mathematical point of view we show that this way of addressing the problem is not equivalent to considering the system a priori embedded in infinite space in particular the matrix elements of the transformation turning the system to the principal axis do not tend to distributions in the limit of an arbitrarily large sphere as should be the case if the two procedures were mathematically equivalent also we introduce dressed coordinates which allow an exact description of the oscillator radiation process expanding in powers of the coupling constant we recover from our exact expressions the well known decay formulae from perturbation theory

mixed finite elements for second order elliptic problems in three variables two families of mixed finite elements one based on simplices and the other on cubes are introduced as alternatives to the usual raviart thomas nedelec spaces these spaces are analogues of those introduced by brezzi douglas and marini in two space variables error estimates inl2 andh s are derived

an anisotropic pressure stabilized finite element method for incompressible flow problems we consider a pressure stabilized finite element approximation of incompressible flow problems in primitive velocity pressure variables which is based on a projection of the gradient of the discrete pressure onto the space of discrete functions equal order interpolation for the velocity and the pressure can be employed with this formulation the method introduced here is specially developed to be used on anisotropic finite element meshes with large element aspect ratios

a velocity pressure streamline diffusion finite element method for navier stokes equations a streamline diffusion finite element method is introduced for the time dependent incompressible navier stokes equations in a bounded domain in r squared and r cubed in the case of a flow with a high reynolds number an error estimate is proved and numerical results are given the method is based on a mixed velocity pressure formulation using the same finite element discretization of space time for the velocity and the pressure spaces which consist of piecewise linear functions together with certain least squares modifications of the galerkin variational formulation giving added stability without sacrificing accuracy

a finite element variational multiscale method for incompressible flows based on two local gauss integrations in this article we present a finite element variational multiscale vms method for incompressible flows based on two local gauss integrations and compare it with common vms method which is defined by a low order finite element space l h on the same grid as x h for the velocity deformation tensor and a stabilization parameter a the best algorithmic feature of our method is using two local gauss integrations to replace projection operator we theoretically discuss the relationship between our method and common vms method for the taylor hood elements and show that the nonlinear system derived from our method by finite element discretization is much smaller than that of common vms method computationally additionally we present numerical simulations to demonstrate the effectiveness storage computational complexity of our method finally we give some numerical simulations of the nonlinear flow problems to show good stability and accuracy properties of the method

a stabilized finite element method for generalized stationary incompressible flows abstract in this paper we describe a finite element formulation for the numerical solution of the stationary incompressible navier stokes equations including coriolis forces and the permeability of the medium the stabilized method is based on the algebraic version of the sub grid scale approach we first describe this technique for general systems of convection diffusion reaction equations and then we apply it to the linearized flow equations the important point is the design of the matrix of stabilization parameters that the method has this design is based on the identification of the stability problems of the galerkin method and a scaling of variables argument to determine which coefficients must be included in the stabilization matrix this together with the convergence analysis of the linearized problem leads to a simple expression for the stabilization parameters in the general situation considered in the paper the numerical analysis of the linearized problem also shows that the method has optimal convergence properties

the least squares finite element method for low mach number compressible viscous flows the present paper reports the development of the least squares finite element method lsfem for simulating compressible viscous flows at low mach numbers in which the incompressible flows pose as an extreme conventional approach requires special treatments for low speed flows calculations finite difference and finite volume methods are based on the use of the staggered grid or the preconditioning technique and finite element methods rely on the mixed method and the operator splitting method in this paper however we show that such difficulty does not exist for the lsfem and no special treatment is needed the lsfem always leads to a symmetric positive definite matrix through which the compressible flow equations can be effectively solved two numerical examples are included to demonstrate the method first driven cavity flows at various reynolds numbers and buoyancy driven flows with significant density variation both examples are calculated by using full compressible flow equations

a stabilized mixed formulation using the proper generalized decomposition for fluid problems abstract the aim of this work is to derive a stabilized mixed formulation for flow problems while using a domain decomposition technique therefore we are generalizing the stabilized finite elements method using galerkin least squares gls into a stabilized mixed formulation valid for the separated representation as used in the proper generalized decomposition pgd formulation the resulting formulation is robust and can be used in a generalized multidimensional framework without undergoing any changes in either the pgd formulation or in the mesh conditioning we applied the stabilized pgd formulation for different benchmark problems including a 3d squeeze flow poiseuille flow and a lid driven cavity as well as solving high dimensionality parametric problems using the same approach

a relationship between stabilized finite element methods and the galerkin method with bubble functions abstract a relation between stabilized finite element methods and the galerkin method employing interpolations with bubble functions is established for the advective diffusive model and for the linearized compressible navier stokes equations the bubble functions are shown to help in stabilizing the advective operator without recourse to upwinding or any other numerical strategy in particular for the advective diffusive model the galerkin method employing piecewise linears with bubble functions is shown to be equivalent to the streamline upwind petrov galerkin supg method in the diffusive limit

convergence analyses of galerkin least squares methods for symmetric advective diffusive forms of the stokes and incompressible navier stokes equations abstract symmetric advective diffusive forms of the stokes and incompressible navier stokes equations are presented the galerkin least squares method for advective diffusive equations is used for both systems and is related to other stabilized methods previously studied the presentation reveals that the convergence analysis for advective diffusive equations as applied before to a linearized form of the compressible navier stokes equations carries over in a straightforward manner to the stokes problem and to a linearized form of the incompressible navier stokes equations

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

introduction to convex optimization in financial markets convexity arises naturally in financial risk management in risk preferences concerning random cash flows convexity corresponds to the fundamental diversification principle convexity is a basic property also of budget constraints both in classical linear models as well as in more realistic models with transaction costs and constraints moreover modern securities markets are based on trading protocols that result in convex trading costs the first part of this paper gives an introduction to certain basic concepts and principles of financial risk management in simple optimization terms the second part reviews some convex optimization techniques used in mathematical and numerical analysis of financial optimization problems

a remark on least squares mixed element methods for reaction diffusion problems in this paper we propose a least squares mixed element procedure for a reaction diffusion problem based on the first order system by selecting the least squares functional properly the resulting procedure can be split into two independent symmetric positive definite schemes one of which is for the unknown variable and the other of which is for the unknown flux variable which lead to the optimal order h 1 w and l 2 w norm error estimates for the primal unknown and optimal h div w norm error estimate for the unknown flux finally we give some numerical examples

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

duality and martingales a stochastic programming perspective on contingent claims the hedging of contingent claims in the discrete time discrete state case is analyzed from the perspective of modeling the hedging problem as a stochastic program application of conjugate duality leads to the arbitrage pricing theorems of financial mathematics namely the equivalence of absence of arbitrage and the existence of a probability measure that makes the price process into a martingale the model easily extends to the analysis of options pricing when modeling risk management concerns and the impact of spreads and margin requirements for writers of contingent claims however we find that arbitrage pricing in incomplete markets fails to model incentives to buy or sell options an extension of the model to incorporate pre existing liabilities and endowments reveals the reasons why buyers and sellers trade in options the model also indicates the importance of financial equilibrium analysis for the understanding of options prices in incomplete markets

an absolutely stabilized finite element method for the stokes problem on presente une formulation a elements finis absolument stabilisee pour le probleme de stokes on etablit des estimations d erreurs optimales dans la norme l 2 pour l approximation des champs de vitesses et de pressions

a space time formulation for multiscale phenomena abstract we develop subgrid scale models for a class of nonsymmetric linear evolution operators by applying the variational multiscale method in space time the results generalize those of hughes 14 which were confined to the steady case the subgrid scale models are shown to be a paradigm for bubble function finite element methods and provide a theoretical and practical framework for the development of so called stabilized methods

the lowest order characteristic mixed finite element scheme for convection dominated diffusion problem a new characteristic mixed finite element method mfem is proposed by combining the method of characteristics with the new mixed variational formulation for the convection dominated diffusion problem the error estimates for both the original variable u and auxiliary variable p are derived via the properties of the integral identity and mean value techniques at the same time some numerical experiments are provided to confirm validity of the theoretical analysis and excellent performance of the proposed method

least squares mixed finite element methods for non selfadjoint elliptic problems ii performance of block ilu factorization methods the least squares mixed finite element technique developed in part i is applied to non selfadjoint second order elliptic problems this approach leads to a symmetric positive definite bilinear form which is coercive uniformly in the discretization parameter in this paper we consider an approximate block factorization technique recently proposed by chan and vassilevski in a framework for block ilu factorization using block size reduction math comp 64 1995 pp 129 156 and which is well defined for positive definite block tridiagonal matrices the method is analyzed and supported with extensive numerical experiments

a finite element method for first order elliptic systems in three dimensions abstract we consider finite element approximation schemes for boundary value problems for linear first order elliptic systems in three dimensions we also discuss div curl type systems in three independent variables both a least squares method and a new subdomain collocation least squares method are considered the analysis yield optimal asymptotic h 1 and l 2 error estimates

epi convergent discretizations of multistage stochastic programs in many dynamic stochastic optimization problems in practice the uncertain factors are best modeled as random variables with an infinite support this results in infinite dimensional optimization problems that can rarely be solved directly therefore the random variables stochastic processes are often approximated by finitely supported ones scenario trees which result in finite dimensional optimization problems that are more likely to be solvable by available optimization tools this paper presents conditions under which such finite dimensional optimization problems can be shown to epi converge to the original infinite dimensional problem epi convergence implies the convergence of optimal values and solutions as the discretizations are made finer our convergence result applies to a general class of convex problems where neither linearity nor complete recourse are assumed

a new finite element formulation for computational fluid dynamics viii the galerkin least squares method for advective diffusive equations abstract galerkin least squares finite element methods are presented for advective diffusive equations galerkin least squares represents a conceptual simplification of supg and is in fact applicable to a wide variety of other problem types a convergence analysis and error estimates are presented

an anisotropic gls stabilized finite element method for incompressible flow problems abstract we consider a stabilized finite element approximation of incompressible flow problems in primitive velocity pressure variables which is based on a gls formulation equal order interpolation for the velocity and the pressure can be employed with this formulation the method introduced here is specially developed to be used on anisotropic finite element meshes with large element aspect ratios rather than using a single stabilization parameter within each element the jacobian matrix of the affine transformation from the reference element is employed for stabilization purposes stability and anisotropic error estimates in a suitable mesh dependent norm are proved for the linear stokes problem numerical results confirm the accuracy and robustness of the proposed method compared to similar previous stabilized schemes

forchheimer law derived by homogenization of gas flow in turbomachines a general formulation of the homogenization problem of compressible fluid flow through a periodic porous material in turbomachines is presented here this formulation is able to derive a forchheimer law with a mean velocity dependent permeability as equivalent macroscopic behavior to specify this permeability additional flow problems are defined on the unit cell and solved by a mixed stabilized finite element discretization the application of the galerkin least square gls method requires the introduction of two stabilization terms with appropriate parameters the mixed finite element discretization of these unit cell problems is finally outlined

reproducing kernel hierarchical partition of unity part ii applications in this part of the work the meshless hierarchical partition of unity proposed in 1 referred here as part i is used as a multiple scale basis in numerical computations to solve practical problems the applications discussed in the present work fall into two categories 1 a wavelet adaptivity refinement procedure and 2 a so called wavelet petrov galerkin procedure in the applications of wavelet adaptivity the hierarchical reproducing kernels are used as a multiple scale basis to compute the numerical solutions of the helmholtz equation a model equation of wave propagation problems and to simulate shear band formation in an elasto viscoplastic material a problem dictated by the presence of the high gradient deformation in both numerical experiments numerical solutions with high resolution are obtained by inserting the wavelet like basis into the primary interpolation function basis a process that may be viewed as a spectral p type refinement by using the interpolant that has synchronized convergence property as a weighting function a wavelet petrov galerkin procedure is proposed to stabilize computations of some pathological problems in numerical computations such as advection diffusion problems and stokes flow problem it offers an alternative procedure in stablized methods and also provides some insight or new interpretation of the method detailed analysis has been carried out on the stability and convergence of the wavelet petrov galerkin method copyright 1999 john wiley sons ltd

approximation of the incompressible navier stokes equations using orthogonal subscale stabilization and pressure segregation on anisotropic finite element meshes abstract this paper describes a finite element model to solve the incompressible navier stokes equations based on the stabilization with orthogonal subscales and a pressure segregation the former consists of adding a least square form of the component orthogonal to the finite element space of the convective and pressure gradient terms this allows to deal with convection dominated flows and to use equal velocity pressure interpolation the pressure segregation is inspired in fractional step schemes although the converged solution corresponds to that of a monolithic time integration likewise we put special emphasis on the use of anisotropic grids in particular we describe some possible choices for the calculation of the element length that appears in the stabilization parameters and check their behavior in two classical numerical examples the preconditioning strategy used to solve the resulting algebraic system for very anisotropic meshes is also briefly described

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

a subdomain galerkin least squares method for first order elliptic systems in the plane a finite element method for the approximation of solutions of linear first order elliptic systems in two dimensional domains is considered the method differs from previous techniques in that the discretization is effected prior to the application of a least squares method optimal error estimates in text bf h 1 omega and text bf l 2 omega norms are derived for piecewise linear finite element trial spaces numerical examples are provided that illustrate the theoretical convergence rates and that indicate that the method is easy to implement and is computationally efficient

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

a tutorial in elementary finite element error analysis a systematic presentation of a priori and a posteriori error estimates abstract this paper provides a tutorial on the derivation of a priori and explicit residual based a posteriori error estimates for galerkin finite element discretizations of general linear elliptic operators for simplicity the presentation is in one dimension although extensions to multidimensions should be straightforward the structures of explicit residual based a posteriori error estimators and a priori error estimators are very similar they are each in the form of an upper bound on the error measured in some norm the structures of their derivations also follow similar lines it is therefore appropriate to present and discuss a priori and a posteriori error estimators in parallel in this paper error bounds are obtained for the error measured in the energy norm and the l2 norm in addition symmetric nonsymmetric positive definite and indefinite operators are discussed a priori estimates for nonsymmetric operators are treated with a novel approach involving the introduction of the skew norm special attention is given to the advection diffusion equation a nonsymmetric operator and the helmholtz equation an indefinite operator we systematically describe the necessary steps in deriving a priori and a posteriori error estimates and provide a simplified understanding of their differences as well as their similarities

epi convergent discretizations of multistage stochastic programs via integration quadratures this paper presents procedures for constructing numerically solvable discretizations of multistage stochastic programs that epi converge to the original problem as the discretizations are made finer epi convergence implies in particular that the cluster points of the first stage solutions of the discretized problems are optimal first stage solutions of the original problem the discretization procedures apply to a general class of nonlinear stochastic programs where the uncertain factors are driven by time series models using existing routines for numerical integration allows for an easy and efficient implementation of the procedures

pressure stabilization of finite element approximations of time dependent incompressible flow problems in this paper we develop and analyze pressure stabilized finite element methods for the solution of the transient stokes problem which is a linear model problem of transient incompressible flow a model for the bubble enrichment is proposed to stabilize the numerical solution and an implicit backward euler method is considered for the time approximation different methods are obtained depending on whether bubbles are allowed to evolve with time or they are considered quasi static stability estimates are provided for the two methods considered both for the velocity and the pressure solutions numerical results obtained on test cases are also given which confirm the theoretical stability results

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

two level variational multiscale finite element methods for navier stokes type variational inequality problem in this paper we present two level variational multiscale finite element method based on two local gauss integrations for navier stokes equations with friction boundary conditions which are of the form of navier stokes type variational inequality of the second kind we solve navier stokes type variational inequality problem on the coarse mesh and solve linearized navier stokes type variational inequality problem corresponding to newton iteration on the fine mesh the error estimates in h 1 norm for velocity and l 2 norm for pressure are derived meanwhile uzawa iteration schemes are constructed to solve the subproblems in this two level method finally the numerical results are displayed to support the theoretical analysis

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

a domain decomposition method for two phase transport model in the cathode of a polymer electrolyte fuel cell using kirchhoff transformation we develop a dirichlet neumann alternating iterative domain decomposition method for a 2d steady state two phase model for the cathode of a polymer electrolyte fuel cell pefc which contains a channel and a gas diffusion layer gdl this two phase pefc model is represented by a nonlinear coupled system which typically includes a modified navier stokes equation with darcy s drag as an additional source term of the momentum equation and a convection diffusion equation for the water concentration with discontinuous and degenerate diffusivity for both cases of dry and wet gas channel we employ kirchhoff transformation and dirichlet neumann alternating iteration with appropriate interfacial conditions on the gdl channel interface to treat the jump nonlinearities in the water equation numerical experiments demonstrate that fast convergence as well as accurate numerical solutions are obtained simultaneously owing to the implementation of the above described numerical techniques along with a combined finite element upwind finite volume discretization to automatically control the dominant convection terms arising in the gas channel

dressed coordinates the path integral approach the recently introduced dressed coordinates are studied in the path integral approach these coordinates are defined in the context of a harmonic oscillator linearly coupled to massless scalar field and it is shown that in this model the dressed coordinates appear as a coordinate transformation preserving the path integral functional measure the analysis also generalizes the sum rules established in a previous work

log optimal and rapid paths in von neumann gale dynamical systems abstract von neumann gale dynamical systems are defined in terms of multivalued operators in spaces of random vectors possessing certain properties of convexity and homogeneity a central role in the theory of such systems is played by a special class of paths trajectories called rapid they grow over each time period t 1 t in a sense faster than others the paper establishes existence and characterization theorems for such paths showing in particular that any trajectory maximizing a logarithmic functional over a finite time horizon is rapid the proof of this result is based on the methods of convex analysis in spaces of measurable functions the study is motivated by the applications of the theory of von neumann gale dynamical systems to the modeling of capital growth in financial markets with friction

a finite element formulation for incompressible flow problems using a generalized streamline operator abstract a finite element formulation for solving incompressible flow problems is presented in this paper the generalized streamline operator presented by hughes et al comput methods appl mech engrg 1986 58 305 328 for compressible flows is adapted to the incompressible navier stokes equations this new methodology allows the use of equal order interpolation for the unknowns of the problem velocity and pressure in this context the definition of the upwinding tensor does not require parameters defined outside this model this formulation has been checked in classical tests with satisfactory results finally a moving surface problem cruchaga et al comput numer methods engrg 1986 59 85 99 is also presented

virtual bubbles and galerkin least squares type methods ga l s abstract the equivalence between stabilized finite element methods or galerkin least squares tyoe methods ga l s and the standard galerkin method with bubble functions is established in an abstract framework the results are applicable to various finite element spaces including high order elements and applications to the advective diffusive model and to the stokes problem are presented illustrating the potential of the abstract theory introduced here

analysis of least squares mixed finite element methods for nonlinear nonstationary convection diffusion problems some least squares mixed finite element methods for convection diffusion problems steady or nonstationary are formulated and convergence of these schemes is analyzed the main results are that a new optimal a priori l 2 error estimate of a least squares mixed finite element method for a steady convection diffusion problem is developed and that four fully discrete least squares mixed finite element schemes for an initial boundary value problem of a nonlinear nonstationary convection diffusion equation are formulated also some systematic theories on convergence of these schemes are established

gls and evss methods for a three field stokes problem arising from viscoelastic flows in this paper order one finite elements together with galerkin least squares gls methods are used for solving a three held stokes problem arising from the numerical study of viscoelastic hows stability and convergence results are established even when the solvent viscosity is small compared to the viscosity due to the polymer chains an iterative algorithm decoupling velocity pressure and stress calculations is proposed the link with the modified elastic viscous split stress evss method studied in m fortin r guenette r pierre comput methods appl mech engrg 143 1997 79 95 r guenette m fortin j non newtonian fluid mech 60 1995 27 52 is presented numerical results are in agreement with theoretical predictions and with those presented in m fortin r guenette r pierre comput methods appl mech engrg 143 1997 79 95 c 2001 elsevier science b v all rights reserved

positivity preservation and adaptive solution for the k model of turbulence a simple change of dependent variables that guarantees positivity of turbulence variables in numerical simulation codes is presented the approach consists of solving for the natural logarithm of the turbulence variables which are known to be strictly positive the approach is valid for any numerical scheme be it finite difference a finite volume or a finite element method the work focuses on the advantages of the proposed change of dependent variables within the framework of an adaptive finite element method the turbulence equations in logarithmic variables are presented for the standard e model error estimation and mesh adaptation procedures are described the formulation is validated on a shear layer case for which an analytical solution is available this provides a framework for rigorous comparison of the proposed approach with the standard solution technique which makes use of k and e as dependent variables the approach is then applied to solve turbulent flow over a naca0012 airfoil for which experimental measurements are available the proposed procedure results in a robust adaptive algorithm improved predictions of turbulence variables are obtained using the proposed formulation

regular article numerical simulation of free surface flows a numerical model is presented for the simulation of complex fluid flows with free surfaces the unknowns are the velocity and pressure fields in the liquid region together with a function defining the volume fraction of liquid although the mathematical formulation of the model is similar to the volume of fluid vof method the numerical schemes used to solve the problem are different a splitting method is used for the time discretization at each time step two advection problems and a generalized stokes problem are to be solved two different grids are used for the space discretization the two advection problems are solved on a fixed structured grid made out of small rectangular cells using a forward characteristic method the generalized stokes problem is solved using a finite element method on a fixed unstructured mesh numerical results are presented for several test cases the filling of an s shaped channel the filling of a disk with core the broken dam in a confined domain

convex duality in stochastic optimization and mathematical finance this paper proposes a general duality framework for the problem of minimizing a convex integral functional over a space of stochastic processes adapted to a given filtration the framework unifies many well known duality frameworks from operations research and mathematical finance the unification allows the extension of some useful techniques from these two fields to a much wider class of problems in particular combining certain finite dimensional techniques from convex analysis with measure theoretic techniques from mathematical finance we are able to close the duality gap in some situations where traditional topological arguments fail

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

stabilized finite element approximation of transient incompressible flows using orthogonal subscales abstract in this paper we present a stabilized finite element method to solve the transient navier stokes equations based on the decomposition of the unknowns into resolvable and subgrid scales the latter are approximately accounted for so as to end up with a stable finite element problem which in particular allows to deal with convection dominated flows and the use of equal velocity pressure interpolations three main issues are addressed the first is a method to estimate the behavior of the stabilization parameters based on a fourier analysis of the problem for the subscales secondly the way to deal with transient problems discretized using a finite difference scheme is discussed finally the treatment of the nonlinear term is also analyzed a very important feature of this work is that the subgrid scales are taken as orthogonal to the finite element space in the transient case this simplifies considerably the numerical scheme

a consistent and stabilized continuous discontinuous galerkin method for fourth order incompressible flow problems this paper presents a new consistent and stabilized finite element formulation for fourth order incompressible flow problems the formulation is based on the c 0 interior penalty method the galerkin least square gls scheme which assures that the formulation is weakly coercive for spaces that fail to satisfy the inf sup condition and considers discontinuous pressure interpolations a stability analysis through a lemma establishes that the proposed formulation satisfies the inf sup condition thus confirming the robustness of the method this lemma indicates that at the element level there exists an optimal or quasi optimal gls stability parameter that depends on the polynomial degree used to interpolate the velocity and pressure fields the geometry of the finite element and the fluid viscosity term numerical experiments are carried out to illustrate the ability of the formulation to deal with arbitrary interpolations for velocity and pressure and to stabilize large pressure gradients

the role of bulk viscosity in stabilized finite element formulations for incompressible flow a review abstract in stabilized finite element methods for incompressible flows like e g galerkin least squares gls formulations a bulk viscosity or penalty like term is sometimes present this paper reviews the appearance of this term and its role in stability accuracy and convergence speed issues

stochastic programs without duality gaps this paper studies dynamic stochastic optimization problems parameterized by a random variable such problems arise in many applications in operations research and mathematical finance we give sufficient conditions for the existence of solutions and the absence of a duality gap our proof uses extended dynamic programming equations whose validity is established under new relaxed conditions that generalize certain no arbitrage conditions from mathematical finance

least squares mixed finite elements for second order elliptic problems a theoretical analysis of a least squares mixed finite element method for second order elliptic problems in two and three dimensional domains is presented it is proved that the method is not subj

an error estimate of the least squares finite element method for the stokes problem in three dimensions in this paper we are concerned with the stokes problem in three dimensions see recent works of the author and b n jiang for the two dimensional case it is a linear system of four pdes with velocity 11 and pressure p as unknowns with the additional variable a curl the second order problem is reduced to a first order system considering the compatibility condition d i v g 0 we have a system with eight first order equations and seven unknowns a least squares method is applied to this extended system and also to the corresponding boundary conditions the analysis based on works of agmon douglis and nirenberg wendland zienkiewicz owen and niles etc shows that this method is stable in the h version for instance if we choose continuous piecewise polynomials to approximate 11 a and p this method achieves optimal rates of convergence in the hi norms let sz be an open bounded and connected subset of ir3 with a smooth boundary t let be a given function representing the body f e l r force the stokes can be posed as where g p with p 1 0 and v are respectively velocity pressure and kinematic viscosity constant all of which are assumed to be nondimensionalized over the past two decades many engineers and mathematicians have studied the above problem the mixed galerkin method solves this problem successfully in most cases the elements are required to satisfy a saddle point condition 4 5 8 9 221 which is not necessary for our method received by the editor march 16 1992 and in revised form october 12 1992 1991 mathematics subject classification primary 65n30 35f15 this work was performed when the author was an oaijcwru summer faculty fellow participant at nasa lewis research center cleveland oh 1991 and was revised at the university of texas at arlington 1992 and at wright patterson air force base dayton oh 1993 1994 american mathematical society 0025 5718194 1 00 25 per page

expected utility penalty functions and duality in stochastic nonlinear programming we consider nonlinear programming problem p with stochastic constraints the lagrangean corresponding to such problems has a stochastic part which in this work is replaced by its certainty equivalent in the sense of expected utility theory it is shown that the deterministic surrogate problem ce p thus obtained contains a penalty function which penalizes violation of the constraints in the mean the approach is related to several known methods in stochastic programming such as chance constraints stochastic goal programming reliability programming and mean variance analysis the dual problem of ce p is studied for problems with stochastic righthand sides in the constraints and a comprehensive duality theory is developed by introducing a new certainty equivalent nce concept motivation for the nce and its potential role in decision theory are discussed as well as mean variance approximations

least squares methods for the velocity pressure stress formulation of the stokes equations we formulate and study finite element methods for the solution of the incompressible stokes equations based on the application of least squares minimization principle to an equivalent first order velocity pressure stress system our least squares functional involves the l norms of the residuals of each equation multiplied by a mesh dependent weight each weight is determined according to the agmon douglis nirenberg index of the corresponding equation as a result the approximating spaces are not subject to the lbb condition and conforming discretizations are possible with merely continuous finite element spaces moreover the resulting discrete problems involve only symmetric positive definite systems of linear equations i e assembly of the discretization matrix is not required even at the element level we prove that the least squares approximadons converge to the solutions of the stokes problem at the best possible rate and then present some numerical examples illustrating our theoretical results among other things these numerical examples indicate that the method is not optimal without the weights in the least squares functional

stabilization of incompressibility and convection through orthogonal sub scales in finite element methods two apparently different forms of dealing with the numerical instability due to the incompressibility constraint of the stokes problem are analyzed in this paper the first of them is the stabilization through the pressure gradient projection which consists of adding a certain least squares form of the difference between the pressure gradient and its l2 projection onto the discrete velocity space in the variational equations of the problem the second is a sub grid scale method whose stabilization effect is very similar to that of the galerkin least squares gls method for the stokes problem it is shown here that the first method can also be recast in the framework of sub grid scale methods with a particular choice for the space of sub scales this leads to a new stabilization procedure whose applicability to stabilize convection is also studied in this paper

object oriented finite elements iv symbolic derivations and automatic programming of nonlinear formulations reference epfl article 102770view record in web of science record created on 2007 04 23 modified on 2016 08 08

numerical verification of a non residual orthogonal term by term stabilized finite element formulation for incompressible convective flow problems abstract the modeling of turbulent flows is relevant in many engineering applications and is an active field of research on numerical methods convergence and stability of proposed formulations are crucial to predict transitional flows from laminar to turbulent flows in this work a recently developed stabilized finite element formulation is used as a powerful tool to describe such kind of problems the essential point of the formulation is the time dependent nature of the subscales and contrary to residual based formulations the introduction of two velocity subscale components the theoretical rate of convergence of the method is verified numerically using linear and quadratic equal order finite element discretizations to this end a standard convergence test of l 2 norm is presented where the computed solutions are compared when manufactured solutions are imposed at gauss point level moreover the hopf bifurcation is studied for two well known benchmark problems flow past a cylinder and the three dimensional lid driven cavity flow for the flow past a cylinder case the hopf bifurcation is verified using dynamic subscales and is assessed so that they do not disturb the solution in particular a dominant convective problem re 4000 is solved using both the quasistatic and dynamic versions of the method evaluating the performance of each one in the quality of the solution and in the cpu time needed to obtain a converged solution for the 3d lid driven cavity flow problem the hopf bifurcation is determined using two different boundary conditions analyzing their effect on the dynamics of the problem and on the thickness and shape of the boundary layer the final test case is the turbulent 3d lid driven cavity problem re 12000 where velocity profiles are compared with experimental les and dns reference solutions additionally pressure and velocity spectra are shown at certain representative points of the domain as well as phase diagrams correlation function graphs the poincare map and the lyapunov exponent typical mathematical tools used in dynamical system analysis from the results the method is robust and accurate for all the numerical tests both in viscous dominant problems as in dominant convective ones

portfolio selection with transaction costs in this paper optimal consumption and investment decisions are studied for an investor who has available a bank account paying a fixed rate of interest and a stock whose price is a log normal diffusion this problem was solved by merton and others when transactions between bank and stock are costless here we suppose that there are charges on all transactions equal to a fixed percentage of the amount transacted it is shown that the optimal buying and selling policies are the local times of the two dimensional process of bank and stock holdings at the boundaries of a wedge shaped region which is determined by the solution of a nonlinear free boundary problem an algorithm for solving the free boundary problem is given

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100