optimal bounds for a lagrange interpolation inequality for piecewise linear continuous finite elements in two space dimensions optimal bounds for a lagrange interpolation inequality for piecewise linear continuous finite elements in two space dimensions

generalized bundle methods we study a class of generalized bundle methods for which the stabilizing term can be any closed convex function satisfying certain properties this setting covers several algorithms from the literature that have been so far regarded as distinct under a different hypothesis on the stabilizing term and or the function to be minimized we prove finite termination asymptotic convergence and finite convergence to an optimal point with or without limits on the number of serious steps and or requiring the proximal parameter to go to infinity the convergence proofs leave a high degree of freedom in the crucial implementative features of the algorithm i e the management of the bundle of subgradients beta strategy and of the proximal parameter t strategy we extensively exploit a dual view of bundle methods which are shown to be a dual ascent approach to one nonlinear problem in an appropriate dual space where nonlinear subproblems are approximately solved at each step with an inner linearization approach this allows us to precisely characterize the changes in the subproblems during the serious steps since the dual problem is not tied to the local concept of varepsilon subdifferential for some of the proofs a generalization of inf compactness called compactness is required this concept is related to that of asymptotically well behaved functions

discontinuous galerkin finite element methods for second order hyperbolic problems abstract we prove a priori and a posteriori error estimates for a finite element method for linear second order hyperbolic equations based on a space time finite element discretization with the basis functions being continuous in space and discontinuous in time

on monotonicity preserving stabilized finite element approximations of transport problems the aim of this work is to design monotonicity preserving stabilized finite element techniques for transport problems as a blend of linear and nonlinear shock capturing stabilization as linear stabilization we consider and analyze a novel symmetric projection stabilization technique based on a local scott zhang projector next we design a weighting of the aforementioned linear stabilization such that when combined with a finite element discretization enjoying a discrete maximum principle usually attained via nonlinear stabilization it does not spoil these monotonicity properties then we propose novel nonlinear stabilization schemes in the form of an artificial viscosity method where the amount of viscosity is proportional to gradient jumps at either finite element boundaries or nodes for the nodal scheme we prove a discrete maximum principle for time dependent multidimensional transport problems numerical experiments support the numerical analysis and we show that the resulting methods prov

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

on the convergence of a finite element method for a nonlinear hyperbolic conservation law we consider a space time finite element discretization of a time dependent nonlinear hyperbolic conservation law in one space dimension burgers equation the finite element method is higher order accurate and is a petrov galerkin method based on the so called streamline diffusion modification of the test functions giving added stability we first prove that if a sequence of finite element solutions converges boundedly almost everywhere as the mesh size tends to zero to a function u then u is an entropy solution of the conservation law this result may be extended to systems of conservation laws with convex entropy in several dimensions we then prove using a compensated compactness result of murat tartar that if the finite element solutions are uniformly bounded then a subsequence will converge to an entropy solution of burgers equation we also consider a further modification of the test functions giving a method with improved shock capturing finally we present the results of some numerical experiments

decomposition for structured convex programs with smooth multiplier methods one drawback associated with the classical quadratic multiplier method augmented lagrangian is the fact that it is only differentiable once even when the problem s data process higher differentiability and therefore efficient newton type methods cannot be applied in fact such a lack of continuity in the second derivative can seriously slow down the rate of convergence of these methods and cause algorithmic failure one way of coping with this difficulty is to use the recently developed non quadratic multiplier methods based on entropy like proximal methods and leading to multiplier methods which as opposed to the classical quadratic multiplier are twice continuously differentiable if the original problem is also c 2 this is an important advantage since newton type methods can then be applied the second drawback associated with multiplier methods is the lack of its separability even when the original problem is separable however some careful reformulation of the problem e g by introducing additional variables may preserve some of the given separable structure thus giving to the augmented lagrangian framework still an important role to play in the development of efficient decomposition schemes examples of such methods include splitting alternating direction type algorithms we combine here these two ideas to develop decomposition schemes for solving structured convex programs with smooth lagrangians allowing the use of newton types methods we will concentrate our numerical analysis on an exponential and modified logarithmic barrier type of augmented lagrangians leading to an alternating direction type algorithm numerical examples illustrating the performance of this algorithm versus classical quadratic alternating direction methods will be presented

entropy like proximal methods in convex programming we study an extension of the proximal method for convex programming where the quadratic regularization kernel is substituted by a class of convex statistical distances called divergences which are typically entropy like in form after establishing several basic properties of these quasi distances we present a convergence analysis of the resulting entropy like proximal algorithm applying this algorithm to the dual of a convex program we recover a wide class of nonquadratic multiplier methods and prove their convergence

quantitative error control for finite element methods for one dimensional compressible flow we prove a posteriori l 2 l 2 and l infty h 1 residual based error estimates the estimates contain certain strong stability factors or weights related to the solution of a linearized dual problem associated with the euler equations we compute the stability factors and weights by solving the dual problem numerically and show that quantitative error control is possible for one dimensional 1 d compressible flow

multiphysics simulation of flow induced vibrations and aeroelasticity on parallel computing platforms abstract this article describes the application of multiphysics simulation on parallel computing platforms to model aeroelastic instabilities and flow induced vibrations multiphysics simulation is based on a single computational framework for the modeling of multiple interacting physical phenomena within the multiphysics framework the finite element treatment of fluids is based on the galerkin least squares gls method with discontinuity capturing operators the arbitrary lagrangian eulerian ale method is utilized to account for deformable fluid domains the finite element treatment of solids and structures is based on a three field variational principle fully coupled interaction constraints are enforced using the augmented lagrangian method the multiphysics architecture lends itself naturally to high performance parallel computing several applications are presented the importance of capturing the nonlinear effects and accounting for mesh movement is highlighted and the scalability of the software is illustrated

an analysis of some high accuracy finite element methods for hyperbolic problems some high accuracy finite element methods for hyperbolic problems are studied in this paper it is proven that over a finite element mesh the convergence order of linear finite element solutions for both linear and nonlinear equations can be higher than one even if the exact solutions are discontinuous the theoretical tools for the convergence analysis are some superclose error estimates that are also developed in this paper for nonsmooth solutions

subgrid stabilized projection method for 2d unsteady flows at high reynolds numbers a subgrid stabilization technique is developed for solving the two dimensional incompressible navier stokes equations at high reynolds numbers the time marching algorithm is based on a well established fractional step pressure correction projection method the advection diffusion step is enriched by an implicit subgrid stabilizing term and by an explicit dissipative shock capturing term the former is calculated by means of a hierarchical finite element setting the latter is included to avoid gibbs phenomenon in the boundary layer convergence tests on prototypical two dimensional examples are reported and the method is used to simulate the viscous incompressible flows around the airfoil naca0012 at zero incidence and reynolds numbers ranging from 105 to 106

convergence of a shock capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions convergence of a shock capturing streamline diffusion nite element method for a conservation law in two space dimensions

the highly accurate block grid method in solving laplace s equation for nonanalytic boundary condition with corner singularity the highly accurate block grid method for solving laplace s boundary value problems on polygons is developed for nonanalytic boundary conditions of the first kind the quadrature approximation of the integral representations of the exact solution around each reentrant corner singular part are combined with the 9 point finite difference equations on the nonsingular part in the integral representations and in the construction of the sixth order gluing operator the boundary conditions are taken into account with the help of integrals of poisson type for a half plane which are computed with e accuracy it is proved that the uniform error of the approximate solution is of order o h 6 e where h is the mesh step this estimation is true for the coefficients of singular terms also the errors of p order derivatives p 0 1 in the singular parts are o h 6 e r j 1 a j p r j is the distance from the current point to the vertex in question and a j p is the value of the interior angle of the jth vertex finally we give the numerical justifications of the obtained theoretical results

interior proximal methods and central paths for convex second order cone programming we make a unified analysis of interior proximal methods of solving convex second order cone programming problems these methods use a proximal distance with respect to second order cones which can be produced with an appropriate closed proper univariate function in three ways under some mild conditions the sequence generated is bounded with each limit point being a solution and global rates of convergence estimates are obtained in terms of objective values a class of regularized proximal distances is also constructed which can guarantee the global convergence of the sequence to an optimal solution these results are illustrated with some examples in addition we also study the central paths associated with these distance like functions and for the linear socp we discuss their relations with the sequence generated by the interior proximal methods from this we obtain improved convergence results for the sequence for the interior proximal methods using a proximal distance continuous at the boundary of second order cones

stabilised finite element methods for steady incompressible flow abstract we give a brief overview of frequently used stabilised finite element schemes for steady incompressible flow we derive taylor galerkin schemes for the flow equations and show how supg type and taylor galerkin methods can be interpreted within the same framework

on the symmetric form of systems of conservation laws with entropy abstract this paper reviews the symmetrizability of systems of conservation laws which possess entropy functions symmetric formulations in conservation form for the equations of gas dynamics are presented

the efficient implementation of a finite element multi resolution viscosity method for hyperbolic conservation laws it is well known that the classic galerkin finite element method is unstable when applied to hyperbolic conservation laws such as the euler equations for compressible flows adding a diffusion term to the equations stabilizes the method but sacrifices too much accuracy to be of any practical use an elegant solution developed in the context of spectral methods by eitan tadmor and coworkers is to add diffusion only to the high frequency modes of the solution and can lead to stabilization without sacrificing accuracy we incorporate this idea into the finite element framework by using hierarchical functions as a multi frequency basis the result is a new finite element method for solving hyperbolic conservation laws for this method convergence for a one dimensional scalar conservation law has previously been proved here the method is described in detail several issues connected with its efficient implementation are considered and numerical results for several examples involving one and two dimensional hyperbolic conservation laws are provided several advantageous features of the method are discussed including the ease for which discontinuities can be detected and artificial diffusion can be applied anisotropically and locally in physical as well as frequency space

modified barrier functions theory and methods the nonlinear rescaling principle employs monotone and sufficiently smooth functions to transform the constraints and or the objective function into an equivalent problem the classical lagrangian which has important properties on the primal and the dual spaces the application of the nonlinear rescaling principle to constrained optimization problems leads to a class of modified barrier functions mbf s and mbf methods mbfm s being classical lagrangians cl s for an equivalent problem the mbf s combine the best properties of the cl s and classical barrier functions cbf s but at the same time are free of their most essential deficiencies due to the excellent mbf properties new characteristics of the dual pair convex programming problems have been found and the duality theory for nonconvex constrained optimization has been developed the mbfm have up to a superlinear rate of convergence and are to the classical barrier functions cbf s method as the multipliers method for augmented lagrangians is to the classical penalty function method based on the dual theory associated with mbf the method for the simultaneous solution of the dual pair convex programming problems with up to quadratic rates of convergence have been developed the application of the mbf to linear lp and quadratic qp programming leads to a new type of multipliers methods which have a much better rate of convergence under lower computational complexity at each step as compared to the cbf methods the numerical realization of the mbfm leads to the newton modified barrier method nmbm the excellent mbf properties allow us to discover that for any nondegenerate constrained optimization problem there exists a hot start from which the nmbm has a better rate of convergence a better complexity bound and is more stable than the interior point methods which are based on the classical barrier functions

interior gradient and epsilon subgradient descent methods for constrained convex minimization we extend epsilon subgradient descent methods for unconstrained nonsmooth convex minimization to constrained problems over polyhedral sets in particular over r p this is done by replacing the usual squared quadratic regularization term used in subgradient schemes by the logarithmic quadratic distancelike function recently introduced by the authors we then obtain interior e subgradient descent methods which allow us to provide a natural extension of bundle methods and polyak s subgradient projection methods for nonsmooth convex minimization furthermore similar extensions are considered for smooth constrained minimization to produce interior gradient descent methods global convergence as well as an improved global efficiency estimate are established for these methods within a unifying principle and minimal assumptions on the problem s data

entropy stable shock capturing space time discontinuous galerkin schemes for systems of conservation laws we present a streamline diffusion shock capturing spacetime discontinuous galerkin dg method to approximate nonlinear systems of conservation laws in several space dimensions the degrees of freedom are in terms of the entropy variables and the numerical flux functions are the entropy stable finite volume fluxes we show entropy stability of the formally arbitrarily high order accurate method for a general system of conservation laws furthermore we prove that the approximate solutions converge to the entropy measure valued solutions for nonlinear systems of conservation laws convergence to entropy solutions for scalar conservation laws and for linear symmetrizable systems is also shown numerical experiments are presented to illustrate the robustness of the proposed schemes

a high resolution petrov galerkin method for the 1d convection diffusion reaction problem abstract we present the design of a high resolution petrov galerkin hrpg method using linear finite elements for the problem defined by the residual r t u x k 2 x 2 s f where k s 0 the structure of the method in 1d is identical to the consistent approximate upwind petrov galerkin cau pg method a c galeao e g dutra do carmo a consistent approximate upwind petrov galerkin method for the convection dominated problems comput methods appl mech engrg 68 1988 83 95 except for the definitions of the stabilization parameters such a structure may also be attained via the finite calculus fic procedure e onate derivation of stabilized equations for numerical solution of advective diffusive transport and fluid flow problems comput methods appl mech engrg 151 1998 233 265 e onate j miquel g hauke stabilized formulation for the advection diffusion absorption equation using finite calculus and linear finite elements comput methods appl mech engrg 195 2006 3926 3946 by an appropriate definition of the characteristic length the prefix high resolution is used here in the sense popularized by harten i e second order accuracy for smooth regular regimes and good shock capturing in nonregular regimes the design procedure embarks on the problem of circumventing the gibbs phenomenon observed in l 2 projections next we study the conditions on the stabilization parameters to circumvent the global oscillations due to the convective term a conjuncture of the two results is made to deal with the problem at hand that is usually plagued by gibbs global and dispersive oscillations in the numerical solution it is shown that the method indeed reproduces stabilized high resolution numerical solutions for a wide range of values of u k s and f finally some remarks are made on the extension of the hrpg method to multidimensions

lagrangian duality and related multiplier methods for variational inequality problems we consider a new class of multiplier interior point methods for solving variational inequality problems with maximal monotone operators and explicit convex constraint inequalities developing a simple lagrangian duality scheme which is combined with the recent logarithmic quadratic proximal lqp theory introduced by the authors we derive three algorithms for solving the variational inequality vi problem this provides a natural extension of the methods of multipliers used in convex optimization and leads to smooth interior point multiplier algorithms we prove primal dual and primal dual convergence under very mild assumptions eliminating all the usual assumptions used until now in the literature for related algorithms applications to complementarity problems are also discussed

convergence of proximal like algorithms we analyze proximal methods based on entropy like distances for the minimization of convex functions subject to nonnegativity constraints we prove global convergence results for the methods with approximate minimization steps and an ergodic convergence result for the case of finding a zero of a maximal monotone operator we also consider linearly constrained convex problems and establish a quadratic convergence rate result for linear programs our analysis allows us to simplify and extend the available convergence results for these methods

an adaptive petrov galerkin formulation for the compressible euler and navier stokes equations in this paper we combine a stable petrov galerkin formulation for the compressible euler and navier stokes equations with an h adaptive remeshing refinement including directional stretching and stretching ratio in the mesh regeneration procedure the result is an accurate and efficient scheme appropriated to solve problems presenting shocks and boundary layers

adaptive streamline diffusion finite element methods for stationary convection diffusion problems adaptive finite element methods for stationary convectiondiffusion problems are designed and analyzed the underlying discretization scheme is the shock capturing streamline diffusion method the adaptive algorithms proposed are based on a posteriori error estimates for this method leading to reliable methods in the sense that the desired error control is guaranteed a priori error estimates are used to show that the algorithms are efficient in a certain sense

stability and convergence of a class of finite element schemes for hyperbolic systems of conservation laws we propose a class of finite element schemes for systems of hyperbolic conservation laws that are based on finite element discretizations of appropriate relaxation models we consider both semidiscrete and fully discrete finite element schemes and show that the schemes are stable and when the compensated compactness theory is applicable do converge to a weak solution of the hyperbolic system the schemes use piecewise polynomials of arbitrary degree and their consistency error is of high order we also prove that the rate of convergence of the relaxation system to a smooth solution of the conservation laws is of order o eps

augmented lagrangian applied to convex quadratic problems an algorithm based on the augmented lagrangian method is proposed to solve convex quadratic programming problem the quadratic penalty is considered here hence the augmented lagrangian function is quadratic when applied to quadratic programming problem for this penalty we show that if the lagrangian function associated with the original problem is strict convex or convex then the hessian matrix of augmented lagrangian function is positive definite or positive semi definite numerical experiments are presented illustrating the performance of the algorithm for the cute test set

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

classification of the four dimensional power commutative real division algebras a classification of all four dimensional power commutative real division algebras is given it is shown that every four dimensional power commutative real division algebra is an isotope of a particular kind of a quadratic division algebra the description of such isotopes in dimensions four and eight is reduced to the description of quadratic division algebras in dimension four this leads to a complete and irredundant classification as a special case the finite dimensional power commutative real division algebras that have a unique non zero idempotent are characterized

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

real quadratic flexible division algebras abstract in this paper we give a new process called vectorial isotopy in order to classify the eight dimensional real quadratic flexible division algebras and we solve the isomorphism problem

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

logarithmic quasi distance proximal point scalarization method for multi objective programming recently gregorio and oliveira developed a proximal point scalarization method applied to multi objective optimization problems for an abstract strict scalar representation with a variant of the logarithmic quadratic function of auslender et al as regularization in this study a variation of this method is proposed using the regularization with logarithm and quasi distance by restricting it to a certain class of quasi distances that are lipschitz continuous and coercive in any of their arguments we show that any sequence x k z k r n r m generated by the method satisfies zk is convergent and xk is bounded and its accumulation points are weak pareto solutions of the unconstrained multi objective optimization problem

multiplicative iterative algorithms for convex programming abstract we study multiplicative iterative algorithms for the minimization of a differentiable convex function defined on the positive orthant of r n if the objective function has compact level sets and has a locally lipschitz continuous gradient then these algorithms converge to a solution of the minimization problem moreover the convergence is nearly monotone in the sense of kullback leibler divergence special cases of these algorithms have been applied in position emission tomography and are formally related to the em algorithm for position emission tomography

an analysis of the em algorithm and entropy like proximal point methods the em algorithm is a popular method for maximum likelihood estimation from incomplete data this method may be viewed as a proximal point method for maximizing the log likelihood function using an integral form of the kullback leibler distance function motivated by this interpretation we consider a proximal point method using an integral form of entropy like distance function we give a convergence analysis of the resulting proximal point method in the case where the cluster points lie in the interior of the objective function domain this result is applied to a normal independent example and a gaussian mixture example to establish convergence of the em algorithm on these examples further convergence analysis of the method for maximization over an orthant is given in low dimensions sublinear convergence and schemes for accelerating convergence are also discussed

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

entropic proximal mappings with applications to nonlinear programming we introduce a family of new transforms based on imitating the proximal mapping of moreau and the associated moreau yosida proximal approximation of a function the transforms are constructed in terms of the a divergence functional a generalization of the relative entropy and of bregman s measure of distance an analogue of moreau s theorem associated with these entropy like distances is proved we show that the resulting entropic proximal maps share properties similar to the proximal mapping and provide a fairly general framework for constructing approximation and smoothing schemes for optimization problems applications of the results to the construction of generalized augmented lagrangians for nonlinear programs and the minimax problem are presented

convergence rate analysis of nonquadratic proximal methods for convex and linear programming the divergence proximal method is an extension of the proximal minimization algorithm where the usual quadratic proximal term is substituted by a class of convex statistical distances called divergences in this paper we study the convergence rate of this nonquadratic proximal method for convex and particularly linear programming we identify a class of divergences for which superlinear convergence is attained both for optimization problems with strongly convex objectives at the optimum and linear programming problems when the regularization parameters tend to zero we prove also that with regularization parameters bounded away from zero convergence is at least linear for a wider class of divergences when the method is applied to the same kinds of problems we further analyze the associated class of augmented lagrangian methods for convex programming with nonquadratic penalty terms and prove convergence of the dual generated by these methods for linear programming problems under a weak nondegeneracy assumption

stabilized shock hydrodynamics i a lagrangian method abstract a new supg stabilized formulation for lagrangian hydrodynamics of materials satisfying mie gruneisen equation of state is proposed it allows the use of simplex type triangular tetrahedral meshes as well as the more commonly used brick type quadrilateral hexahedral meshes the proposed method yields a globally conservative formulation in which equal order interpolation p1 or q1 isoparametric finite elements is applied to velocities displacements and pressure as a direct consequence and in contrast to traditional cell centered multidimensional hydrocode implementations the proposed formulation allows a natural representation of the pressure gradient on element interiors the supg stabilization involves additional design requirements specific to the lagrangian formulation a discontinuity capturing operator in the form of a noh type viscosity with artificial heat flux is used to preserve stability and smoothness of the solution in shock regions a set of challenging shock hydrodynamics benchmark tests for the euler equations of gas dynamics in one and two space dimensions is presented in the two dimensional case computations performed on quadrilateral and triangular grids are analyzed and compared these results indicate that the new formulation is a promising technology for hydrocode applications

an interior proximal algorithm and the exponential multiplier method for semidefinite programming we introduce an interior proximal algorithm for semidefinite optimization problems and establish its convergence properties we also study the corresponding dual algorithm leading to an exponential multiplier method for semidefinite programs potential applications and extensions are also discussed

on a cell entropy inequality for discontinuous galerkin methods we prove a cell entropy inequality for a class of high order dis continuous galerkin finite element methods approximating conservation laws which implies convergence for the one dimensional scalar convex case

entropy viscosity method for nonlinear conservation laws a new class of high order numerical methods for approximating nonlinear conservation laws is described entropy viscosity method the novelty is that a nonlinear viscosity based on the local size of an entropy production is added to the numerical discretization at hand this new approach does not use any flux or slope limiters applies to equations or systems supplemented with one or more entropy inequalities and does not depend on the mesh type and polynomial approximation various benchmark problems are solved with finite elements spectral elements and fourier series to illustrate the capability of the proposed method

construction of stabilization operators for galerkin least squares discretizations of compressible and incompressible flows the design and analysis of a class of stabilization operators suitable for space time galerkin least squares finite element discretizations of the symmetrized compressible navier stokes equations is discussed the obtained stabilization matrix is well defined in the incompressible limit and reduces to the matrix described in m polner j j w van der vegt and r m j van damme analysis of stabilization operators for galerkin least squares discretizations of the incompressible navier stokes equations comput meth appl mech eng 195 2006 982 1006 the stabilization matrix is investigated to give necessary and sufficient conditions for its positive definiteness under these conditions on the stabilization matrix the galerkin least squares method for the symmetrized compressible navier stokes equations satisfies the entropy condition which ensures global nonlinear stability of the discretization

on spurious oscillations at layers diminishing sold methods for convection diffusion equations part i a review abstract an unwelcome feature of the popular streamline upwind petrov galerkin supg stabilization of convection dominated convection diffusion equations is the presence of spurious oscillations at layers since the mid of the 1980s a number of methods have been proposed to remove or at least to diminish these oscillations without leading to excessive smearing of the layers the paper gives a review and state of the art of these methods discusses their derivation proposes some alternative choices of parameters in the methods and categorizes them some numerical studies which supplement this review provide a first insight into the advantages and drawbacks of the methods

convergence of a residual based artificial viscosity finite element method we present a residual based artificial viscosity finite element method to solve conservation laws the galerkin approximation is stabilized by only residual based artificial viscosity without any least squares supg or streamline diffusion terms we prove convergence of the method applied to a scalar conservation law in two space dimensions toward an unique entropy solution for implicit time stepping schemes

accurate fic fem formulation for the multidimensional steady state advection diffusion absorption equation abstract in this paper we present an accurate stabilized fic fem formulation for the multidimensional steady state advection diffusion absorption equation the stabilized formulation is based on the galerkin fem solution of the governing differential equations derived via the finite increment calculus fic method using two stabilization parameters the value of the two stabilization parameters ensuring an accurate nodal fem solution using uniform meshes of linear elements is obtained from the optimal values for the 1d problem the accuracy of the new fic fem formulation is demonstrated in the solution of 2d steady state advection diffusion absorption problems for a range of physical parameters and boundary conditions

inexact proximal point methods for variational inequality problems we present a new family of proximal point methods for solving monotone variational inequalities our algorithm has a relative error tolerance criterion in solving the proximal subproblems our convergence analysis covers a wide family of regularization functions including double regularizations recently introduced by silva eckstein and humes jr siam j optim 12 2001 pp 238 261 and the bregman distance induced by h x sum i 1 n x i log x i we do not use in our analysis the assumption of paramonotonicity which is standard in proving convergence of bregman based proximal methods

an extension of proximal methods for quasiconvex minimization on the nonnegative orthant abstract in this paper we propose an extension of proximal methods to solve minimization problems with quasiconvex objective functions on the nonnegative orthant assuming that the function is bounded from below and lower semicontinuous and using a general proximal distance it is proved that the iterations given by our algorithm are well defined and stay in the positive orthant if the objective function is quasiconvex we obtain the convergence of the iterates to a certain set which contains the set of optimal solutions and convergence to a kkt point if the function is continuously differentiable and the proximal parameters are bounded furthermore we introduce a sufficient condition on the proximal distance such that the sequence converges to an optimal solution of the problem

new error bounds and their applications to convergence analysis of iterative algorithms we present two new error bounds for optimization problems over a convex set whose objective function f is either semianalytic or strictly convex with 1 we then apply these error bounds to analyze the rate of convergence of a wide class of iterative descent algorithms for the aforementioned optimization problem our analysis shows that the function sequence f xk converges at least at the sublinear rate of k for some positive constant where k is the iteration index moreover the distances from the iterate sequence xk to the set of stationary points of the optimization problem converge to zero at least sublinearly

skew selfadjoint form for systems of conservation laws abstract hyperbolic systems of conservation laws augumented with an entropy inequality are studied it is shown that such systems can be written in a quasilinear skew selfadjoint form centered differencing of such a form under the smooth regime ends up with a systematic recipe for constructing quasiconservative schemes where the global entropy conservation is recovered employing the above formulation in bounded regions under the nonsmooth regime as well a local entropy decay estimate is also concluded examples of the shallow water and the full gasdynamics equations are explicitly treated

the simulation of texture evolution with finite elements over orientation space i development abstract a new class of finite element schemes is presented for the modeling of arbitrary polycrystalline microstructure a framework is constructed by viewing texture as arising out of maps that associate crystals with orientations drawn from a fundamental region of orientation space the state of the microstructure is characterized by an orientation distribution function and a hardness field over a fundamental region differential equations are derived to describe the evolution of the two fields finite element schemes cast over discretized fundamental regions are considered for the solution of the differential set properties of the schemes considered are illustrated by application to the texturing of a planar polycrystalline microstructure

measure valued solutions of scalar conservation laws with boundary conditions we define a solution concept for measure valued solutions to scalar conservation laws with initial conditions and boundary conditions and prove a uniqueness theorem for such solutions this result may be used to prove convergence towards the unique solution for approximate solutions which are uniformly bounded in l weakly consistent with certain entropy inequalities and strongly consistent with the initial condition i e without using derivative estimates as an example convergence of a finite element method is demonstrated

continuous interior penalty finite element method for the time dependent navier stokes equations space discretization and convergence this paper focuses on the numerical analysis of a finite element method with stabilization for the unsteady incompressible navier stokes equations incompressibility and convective effects are both stabilized adding an interior penalty term giving l 2 control of the jump of the gradient of the approximate solution over the internal faces using continuous equal order finite elements for both velocities and pressures in a space semi discretized formulation we prove convergence of the approximate solution the error estimates hold irrespective of the reynolds number and hence also for the incompressible euler equations provided the exact solution is smooth

adaptive finite element methods for conservation laws based on a posteriori error estimates we prove a posteriori error estimates for a finite element method for systems of strictly hyperbolic conservation laws in one space dimension and design corresponding adaptive methods the proof of the a posteriori error estimates is based on a strong stability estimate for an associated dual problem together with the galerkin orthogonality of the finite element method the strong stability estimate uses the entropy condition for the system in an essential way 1995 john wiley sons inc

numerical crosswind smear in the streamline diffusion method abstract the streamline diffusion method is a popular element method for approximating convection dominated transport equations the solution to such equations may develop boundary or internal layers in the crosswind direction orthogonal to the flow direction and this results in numerical crosswind layers it is known that the size of these layers is o h 1 2 or perhaps o h 3 4 h being the size of the mesh used in the numerical computation but in practice o h is observed if the gridlines of the method are aligned with the streamlines of the problem this connection is studied and an explanation is given

revisiting stabilized finite element methods for the advective diffusive equation we give a brief overview of stabilized finite element methods and illustrate the developments applied to the advection diffusion equation this article presents a concise perspective of the developments emanated from the works started in the 1980s through today 2005 elsevier b v all rights reserved pacs 47 11 j

interior proximal and multiplier methods based on second order homogeneous kernels we study a class of interior proximal algorithms and nonquadratic multiplier methods for solving convex programs where the usual proximal quadratic term is replaced by an homogeneous functional of order two defined in terms of a convex function we prove under mild assumptions several new convergence results in both the primal and dual framework allowing also for approximate minimization in particular we introduce a new class of interior proximal methods which is globally convergent and allows for generating c multiplier methods with bounded hessians which exhibit strong convergence properties we also consider linearly constrained convex problems and establish global quadratic convergence rates results for linear programs we then study in detail a particular realization of these algorithms leading to a new class of logarithmic quadratic interior point algorithms which are shown to enjoy several attractive properties for solving constrained convex optimization problems

nonlinear rescaling and proximal like methods in convex optimization the nonlinear rescaling principle nrp consists of transforming the objective function and or the constraints of a given constrained optimization problem into another problem which is equivalent to the original one in the sense that their optimal set of solutions coincides a nonlinear transformation parameterized by a positive scalar parameter and based on a smooth sealing function is used to transform the constraints the methods based on nrp consist of sequential unconstrained minimization of the classical lagrangian for the equivalent problem followed by an explicit formula updating the lagrange multipliers we first show that the nrp leads naturally to proximal methods with an entropy like kernel which is defined by the conjugate of the scaling function and establish that the two methods are dually equivalent for convex constrained minimization problems we then study the convergence properties of the nonlinear rescaling algorithm and the corresponding entropy like proximal methods for convex constrained optimization problems special cases of the nonlinear rescaling algorithm are presented in particular a new class of exponential penalty modified barrier functions methods is introduced

a conservative nodal variational multiscale method for lagrangian shock hydrodynamics abstract a new method based on a continuous piece wise linear approximation of the equations for lagrangian shock hydrodynamics is presented numerical instabilities are controlled by a stabilizing operator derived using the paradigm of the variational multiscale analysis encouraging numerical comparisons with existing methods in the case of quadrilateral and hexahedral elements indicate that the proposed method is capable of preventing hourglass patterns in the solution while maintaining accuracy in regions of smooth flow the proposed approach satisfies galilean invariance properties and hinges upon the interpretation of the lagrangian shock hydrodynamics equations as a system of nonlinear wave equations a specific implementation in terms of a predictor multi corrector version of the mid point time integrator guarantees global conservation of mass momentum and total energy for each iterate stability and formal order of accuracy are investigated applying the von neumann analysis to the linearized shock hydrodynamics equations in one dimension this approach yields tight bounds for stable time step estimation formal second order accuracy of the method in time and space and valuable indications on the choice of the most appropriate values for the stabilization parameters present in the formulation

finite element methods for first order hyperbolic systems with particular emphasis on the compressible euler equations a petrov galerkin finite element formulation is presented for first order hyperbolic systems of conservation laws with particular emphasis on the compressible euler equations applications of the methodology are made to one and two dimensional steady and unsteady flows with shocks results obtained suggest the potential of the type of methods developed

normal forms for the g2 action on the real symmetric 7 7 matrices by conjugation the exceptional lie group g2 o7 r acts on the set of real symmetric 7 7 matrices by conjugation we solve the normal form problem for this group action in view of the earlier results g m b

message passing for graph structured linear programs proximal methods and rounding schemes the problem of computing a maximum a posteriori map configuration is a central computational challenge associated with markov random fields there has been some focus on tree based linear programming lp relaxations for the map problem this paper develops a family of super linearly convergent algorithms for solving these lps based on proximal minimization schemes using bregman divergences as with standard message passing on graphs the algorithms are distributed and exploit the underlying graphical structure and so scale well to large problems our algorithms have a double loop character with the outer loop corresponding to the proximal sequence and an inner loop of cyclic bregman projections used to compute each proximal update we establish convergence guarantees for our algorithms and illustrate their performance via some simulations we also develop two classes of rounding schemes deterministic and randomized for obtaining integral configurations from the lp solutions our deterministic rounding schemes use a re parameterization property of our algorithms so that when the lp solution is integral the map solution can be obtained even before the lp solver converges to the optimum we also propose graph structured randomized rounding schemes applicable to iterative lp solving algorithms in general we analyze the performance of and report simulations comparing these rounding schemes

a space time galerkin least squares finite element formulation of the navier stokes equations for moving domain problems a space time galerkin least squares finite element formulation of the navier stokes equations is presented for the analysis of free surface flows moving spatial configurations and deforming fluid structure interfaces the variational equation is based on the time discontinuous galerkin method employing the physical entropy variables the space time elements are oriented in time to accommodate the spatial deformations if the elements are oriented along the particle paths the formulation is lagrangian and if they are fixed in time it is eulerian consequently this formulation is analogous to the arbitrary lagrangian eulerian ale technique a novel mesh rezoning strategy is presented to orient the elements in time and adapt the fluid mesh to the changing spatial configuration numerical results are presented to show the performance of the method

on the convergence of shock capturing streamline diffusion finite element methods for hyperbolic conservation laws we extend our previous analysis of streamline diffusion finite element methods for hyperbolic systems of conservation laws to include a shock capturing term adding artificial viscosity depending on the local absolute value of the residual of the finite element solution and the meh size with this term present we prove a maximum norm bound for finite element solutionsof burgers equation an thus complete an earlier convergence proof for this equation we further prove using entropy variables that a strong limit of finite element solutions is a weak solution of the system of conservation laws and satisfies the entropy inequality asociated with the entropy variables results of some numerical experiments for the time dependent compressible euler equations in two dimensions are also reported

an adaptive finite element method for linear elliptic problems on propose une methode adaptative d elements finis basee sur une estimation d erreur optimale de norme maxima pour les problemes elliptiques lineaires

time finite element methods for structural dynamics summary time finite element methods are developed for the equations of structural dynamics the approach employs the time discontinuous galerkin method and incorporates stabilizing terms having least squares form these enable a general convergence theorem to be proved in a norm stronger than the energy norm results are presented from finite difference analyses of the time discontinuous galerkin and least squares methods with various temporal interpolations and commonly used finite difference methods for structural dynamics these results show that for particular interpolations the time finite element method exhibits improved accuracy and stability

on the convergence of the exponential multiplier method for convex programming in this paper we analyze the exponential method of multipliers for convex constrained minimization problems which operates like the usual augmented lagrangian method except that it uses an exponential penalty function in place of the usual quadratic we also analyze a dual counterpart the entropy minimization algorithm which operates like the proximal minimization algorithm except that it uses a logarithmic entropy proximal term in place of a quadratic we strengthen substantially the available convergence results for these methods and we derive the convergence rate of these methods when applied to linear programs

an entropy satisfying muscl scheme for systems of conservation laws for the high order numerical approximation of hyperbolic systems of conservation laws we propose to use as a building principle an entropy diminishing criterion instead of the familiar total variation diminishing criterion introduced by harten for scalar equations based on this new criterion we derive entropy diminishing projections that ensure both the second order of accuracy and all of the classical discrete entropy inequalities the resulting scheme is a nonlinear version of the classical van leer s muscl scheme strong convergence of this second order entropy satisfying scheme is proved for systems of two equations numerical tests demonstrate the interest of our theory