a characterization of hybridized mixed methods for second order elliptic problems in this paper we give a new characterization of the approximate solution given by hybridized mixed methods for second order self adjoint elliptic problems we apply this characterization to obtain an explicit formula for the entries of the matrix equation for the lagrange multiplier unknowns resulting from hybridization we also obtain necessary and sufficient conditions under which the multipliers of the raviart thomas and the brezzi douglas marini methods of similar order are identical

a superconvergent ldg hybridizable galerkin method for second order elliptic problems we identify and study an ldg hybridizable galerkin method which is not an ldg method for second order elliptic problems in several space dimensions with remarkable convergence properties unlike all other known discontinuous galerkin methods using polynomials of degree k 0 for both the potential as well as the flux the order of convergence in l 2 of both unknowns is k 1 moreover both the approximate potential as well as its numerical trace superconverge in l 2 like norms to suitably chosen projections of the potential with order k 2 this allows the application of element by element postprocessing of the approximate solution which provides an approximation of the potential converging with order k 2 in l 2 the method can be thought to be in between the hybridized version of the raviart thomas and that of the brezzi douglas marini mixed methods

unified hybridization of discontinuous galerkin mixed and continuous galerkin methods for second order elliptic problems we introduce a unifying framework for hybridization of finite element methods for second order elliptic problems the methods fitting in the framework are a general class of mixed dual finite element methods including hybridized mixed continuous galerkin nonconforming and a new wide class of hybridizable discontinuous galerkin methods the distinctive feature of the methods in this framework is that the only globally coupled degrees of freedom are those of an approximation of the solution defined only on the boundaries of the elements since the associated matrix is sparse symmetric and positive definite these methods can be efficiently implemented moreover the framework allows in a single implementation the use of different methods in different elements or subdomains of the computational domain which are then automatically coupled finally the framework brings about a new point of view thanks to which it is possible to see how to devise novel methods displaying very localized and simple mortaring techniques as well as methods permitting an even further reduction of the number of globally coupled degrees of freedom

hopf bifurcation and turing instability in spatial homogeneous and inhomogeneous predator prey models abstract this paper is concerned with two species spatial homogeneous and inhomogeneous predator prey models with beddington deangelis functional response for the spatial homogeneous model the asymptotic behavior of the interior equilibrium and the existence of hopf bifurcation of nonconstant periodic solutions surrounding the interior equilibrium are considered furthermore the direction of hopf bifurcation and the stability of bifurcated periodic solutions are investigated for the model with no flux boundary conditions turing instability of the interior equilibrium solution is studied in particular turing instability region regarding the parameters is established finally to verify our theoretical results some numerical simulations are also included

boundary behavior of interior point algorithms in linear programming this paper studies the boundary behavior of some interior point algorithms for linear programming the algorithms considered are karmarkar s projective rescaling algorithm the linear rescaling algorithm which was proposed as a variation on karmarkar s algorithm and the logarithmic barrier technique the study includes both the continuous trajectories of the vector fields induced by these algorithms and also the discrete orbits it is shown that although the algorithms are defined on the interior of the feasible polyhedron they actually determine differentiable vector fields on the closed polyhedron conditions are given under which a vector field gives rise to trajectories that each visit the neighborhoods of all the vertices of the klee minty cube the linear rescaling algorithm satisfies these conditions thus limits of such trajectories obtained when a starting point is pushed to the boundary may have an exponential number of breakpoints it is shown that limits of projective rescaling trajectories may have a linear number of such breakpoints however projective rescaling trajectories may visit the neighborhoods of linearly many vertices the behavior of the linear rescaling algorithm near vertices is analyzed it is proved that all the trajectories have a unique asymptotic direction of convergence to the optimum

a hybrid mixed method for the compressible navier stokes equations we present a novel discretization method for nonlinear convection diffusion equations and in particular for the compressible navier stokes equations the method is based on a discontinuous galerkin dg discretization for convection terms and a mixed method using h div spaces for the diffusive terms furthermore hybridization is used to reduce the number of globally coupled degrees of freedom for the scalar case a local postprocessing procedure is used to enhance the quality of the approximate solution w h the method reduces to a dg scheme for pure convection and to a mixed method for pure diffusion while for the intermediate case the combined variational formulation requires no additional parameters we formulate and validate our scheme for nonlinear model problems as well as compressible flow problems

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

a posteriori dual mixed adaptive finite element error control for lame and stokes equations a unified and robust mathematical model for compressible and incompressible linear elasticity can be obtained by rephrasing the herrmann formulation within the hellinger reissner principle this quasi optimally converging extension of peers plane elasticity element with reduced symmetry is called dual mixed hybrid formulation dmh explicit residual based a posteriori error estimates for dmh are introduced and are mathematically shown to be locking free reliable and efficient the estimator serves as a refinement indicator in an adaptive algorithm for effective automatic mesh generation numerical evidence supports that the adaptive scheme leads to optimal convergence for lame and stokes benchmark problems with singularities

a mixed method for the biharmonic problem based on a system of first order equations we introduce a new mixed method for the biharmonic problem the method is based on a formulation where the biharmonic problem is rewritten as a system of four first order equations a hybrid form of the method is introduced which allows us to reduce the globally coupled degrees of freedom to only those associated with lagrange multipliers which approximate the solution and its derivative at the faces of the triangulation for k ge 1 a projection of the primal variable error superconverges with order k 3 while the error itself converges with order k 1 only this fact is exploited by using local postprocessing techniques that produce new approximations to the primal variable converging with order k 3 we provide numerical experiments that validate our theoretical results

analysis of hdg methods for stokes flow in this paper we analyze a hybridizable discontinuous galerkin method for numerically solving the stokes equations the method uses polynomials of degree k for all the components of the approximate solution of the gradient velocity pressure formulation the novelty of the analysis is the use of a new projection tailored to the very structure of the numerical traces of the method it renders the analysis of the projection of the errors very concise and allows us to see that the projection of the error in the velocity superconverges as a consequence we prove that the approximations of the velocity gradient the velocity and the pressure converge with the optimal order of convergence of k 1 in l 2 for any k 0 moreover taking advantage of the superconvergence properties of the velocity we introduce a new element by element postprocessing to obtain a new velocity approximation which is exactly divergence free h div conforming and converges with order k 2 for k 1 and with order 1 for k 0 numerical experiments are presented which validate the theoretical results

analysis of nonlinear poro elastic and poro visco elastic models we consider the initial and boundary value problem for a system of partial differential equations describing the motion of a fluid solid mixture under the assumption of full saturation the ability of the fluid phase to flow within the solid skeleton is described by the permeability tensor which is assumed here to be a multiple of the identity and to depend nonlinearly on the volumetric solid strain in particular we study the problem of the existence of weak solutions in bounded domains accounting for non zero volumetric and boundary forcing terms we investigate the influence of viscoelasticity on the solution functional setting and on the regularity requirements for the forcing terms the theoretical analysis shows that different time regularity requirements are needed for the volumetric source of linear momentum and the boundary source of traction depending on whether or not viscoelasticity is present the theoretical results are further investigated via numerical simulations based on a novel dual mixed hybridized finite element discretization when the data are sufficiently regular the simulations show that the solutions satisfy the energy estimates predicted by the theoretical analysis interestingly the simulations also show that in the purely elastic case the darcy velocity and the related fluid energy might become unbounded if indeed the data do not enjoy the time regularity required by the theory

error analysis of variable degree mixed methods for elliptic problems via hybridization a new approach to error analysis of hybridized mixed methods is proposed and applied to study a new hybridized variable degree raviart thomas method for second order elliptic problems the approach gives error estimates for the lagrange multipliers without using error estimates for the other variables error estimates for the primal and flux variables then follow from those for the lagrange multipliers in contrast traditional error analyses obtain error estimates for the flux and primal variables first and then use it to get error estimates for the lagrange multipliers the new approach not only gives new error estimates for the new variable degree raviart thomas method but also new error estimates for the classical uniform degree method with less stringent regularity requirements than previously known estimates the error analysis is achieved by using a variational characterization of the lagrange multipliers wherein the other unknowns do not appear this approach can be applied to other hybridized mixed methods as well

a new elasticity element made for enforcing weak stress symmetry we introduce a new mixed method for linear elasticity the novelty is a simplicial element for the approximate stress for every positive integer k the row wise divergence of the element space spans the set of polynomials of total degree k the degrees of freedom are suited to achieve continuity of the normal stresses what makes the element distinctive is that its dimension is the smallest required for enforcing a weak symmetry condition on the approximate stress this is achieved using certain bubble matrices which are special divergence free matrix valued polynomials we prove that the approximation error is of order k 1 in both the displacement and the stress and that a postprocessed displacement approximation converging at order k 2 can be computed element by element we also show that the globally coupled degrees of freedom can be reduced by hybridization to those of a displacement approximation on the element boundaries

convergence analysis of hybrid expanded mixed finite element method for elliptic equations abstract in this article we utilize the approach of the variational characterization of the lagrange multipliers in cockburn and gopalakrishnan 2005 and present the convergence analysis of hybrid expanded mixed finite element method for elliptic equations the unknowns in the expanded mixed formulation can be expressed in terms of lifting operators the analysis provides a novel perspective for error estimates of the hybrid expanded mixed finite element method numerical examples are presented to confirm the convergence analysis

quasi optimal convergence rate for an adaptive hybridizable c 0 discontinuous galerkin method for kirchhoff plates in this paper we present an adaptive hybridizable c 0 discontinuous galerkin hcdg method for kirchhoff plates a reliable and efficient a posteriori error estimator is produced for this hcdg method quasi orthogonality and discrete reliability are established with the help of a postprocessed bending moment and the discrete helmholtz decomposition based on these the contraction property between two consecutive loops and complexity of the adaptive hcdg method are studied thoroughly the key points in our analysis are a postprocessed normal normal continuous bending moment from the hcdg method solution and a lifting of jump residuals from inter element boundaries to element interiors

polyhedral extensions of some theorems of linear programming three theorems of linear programming form our starting point tucker s theorem 1956 concerning the existence of optimal solutions satisfying the complementary slackness conditions strictly and williams two theorems 1970 concerning the coordinatewise complementary behavior of feasible and optimal solutions here we establish that the same phenomena hold in another more versatile framework involving general polyhedral convexity as one main application the results are transferred into the context of the monotone complementarity problem several other theoretical applications are indicated

new locally conservative finite element methods on a rectangular mesh a new family of locally conservative finite element methods for a rectangular mesh is introduced to solve second order elliptic equations our approach is composed of generating pde adapted local basis and solving a global matrix system arising from a flux continuity equation quadratic and cubic elements are analyzed and optimal order error estimates measured in the energy norm are provided for elliptic equations next this approach is exploited to approximate stokes equations numerical results are presented for various examples including the lid driven cavity problem

the embedded discontinuous galerkin method application to linear shell problems a new discontinuous galerkin method for elliptic problems which is capable of rendering the same set of unknowns in the final system of equations as for the continuous displacement based galerkin method is presented those equations are obtained by the assembly of element matrices whose structure in particular cases is also identical to that of the continuous displacement approach this makes the present formulation easily implementable within the existing commercial computer codes the proposed approach is named the embedded discontinuous galerkin method it is applicable to any system of linear partial differential equations but it is presented here in the context of linear elasticity an application of the method to linear shell problems is then outlined and numerical results are presented

hierarchical mixed hybridized methods for elliptic problems abstract in this article we propose a computational procedure for the efficient implementation of dual mixed hybridized methods of arbitrary degree the procedure relies on the decomposition of the finite element spaces into a vertical p type hierarchy consisting of a lower order part and a defect correction coupled with an additional horizontal decomposition of the defect correction space for the vector variable based on the helmholtz principle an appropriate definition of the basis function set allows us to obtain a systematic substructuring of the block matrix system this property in turn naturally gives rise to an efficient implementation of the procedure through an approximate fixed point block iteration exploiting the equivalence between the principle of defect correction and the variational multiscale modeling theory we also devise and numerically validate a hierarchical a posteriori error estimator for dual mixed methods in hybridized form

an implicit high order hybridizable discontinuous galerkin method for nonlinear convection diffusion equations in this paper we present hybridizable discontinuous galerkin methods for the numerical solution of steady and time dependent nonlinear convection diffusion equations the methods are devised by expressing the approximate scalar variable and corresponding flux in terms of an approximate trace of the scalar variable and then explicitly enforcing the jump condition of the numerical fluxes across the element boundary applying the newton raphson procedure and the hybridization technique we obtain a global equation system solely in terms of the approximate trace of the scalar variable at every newton iteration the high number of globally coupled degrees of freedom in the discontinuous galerkin approximation is therefore significantly reduced we then extend the method to time dependent problems by approximating the time derivative by means of backward difference formulae when the time marching method is p 1 th order accurate and when polynomials of degree p 0 are used to represent the scalar variable each component of the flux and the approximate trace we observe that the approximations for the scalar variable and the flux converge with the optimal order of p 1 in the l 2 norm finally we apply element by element postprocessing schemes to obtain new approximations of the flux and the scalar variable the new approximate flux which has a continuous interelement normal component is shown to converge with order p 1 in the l 2 norm the new approximate scalar variable is shown to converge with order p 2 in the l 2 norm the postprocessing is performed at the element level and is thus much less expensive than the solution procedure for the time dependent case the postprocessing does not need to be applied at each time step but only at the times for which an enhanced solution is required extensive numerical results are provided to demonstrate the performance of the present method

the derivation of hybridizable discontinuous galerkin methods for stokes flow in this paper we introduce a new class of discontinuous galerkin methods for the stokes equations the main feature of these methods is that they can be implemented in an efficient way through a hybridization procedure which reduces the globally coupled unknowns to certain approximations on the element boundaries we present four ways of hybridizing the methods which differ by the choice of the globally coupled unknowns classical methods for the stokes equations can be thought of as limiting cases of these new methods

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

the nonlinear geometry of linear programming i affine and projective scaling trajectories this series of papers studies a geometric structure underlying karmarkar s projective scaling algorithm for solving linear programming problems a basic feature of the projective scaling algorithm is a vector field depending on the objective function which is defined on the interior of the polytope of feasible solutions of the linear program the geometric structure studied is the set of trajectories obtained by integrating this vector field which we call ptrajectories we also study a related vector field the affine scaling vector field and its associated trajectories called a trajectories the affine scaling vector field is associated to another linear programming algorithm the affine scaling algorithm affine and projective scaling vector fields are each defined for linear programs of a special form called strict standard form and canonical form respectively this paper derives basic properties of p trajectories and a trajectones it reviews the projective and affine scaling algorithms defines the projective and affine scaling vector fields and gives differential equations for p trajectories and a trajectories it shows that projective transformations map p trajectories into p trajectories it presents karmarkar s interpretation of a trajectories as steepest descent paths of the objective function c x with respect to the riemannian geometry ds2 z dx dx x2 restricted to the relative interior of the polytope of feasible solutions p trajectories of a canonical form linear program are radial projections of a trajectories of an associated standard form linear program as a consequence there is a polynomial time linear programming algorithm using the affine scaling vector field of this associated linear program this algorithm is essentially karmarkar s algorithm these trajectories are studied in subsequent papers by two nonlinear changes of variables called legendre transform coordinates and projective legendre transform coordinates respectively it will be shown that p trajectories have an algebraic and a geometric interpretation they are algebraic curves and they are geodesics actually distinguished chords of a geometry isometric to a hilbert geometry on a polytope combinatorially dual to the polytope of feasible solutions the a trajectories of strict standard form linear programs have similar interpretations they are algebraic curves and are geodesics of a geometry isometric to euclidean geometry received by the editors july 28 1986 and in revised form september 28 1987 and march 21 1988 1980 mathematics subject classification 1985 revision primary 90c05 secondary 52a40 34a34 research of the first author partially supported by onr contract n00014 87 k0214 d 1989 american mathematical society 0002 9947 89 1 00 25 per page

limiting behavior of the affine scaling continuous trajectories for linear programming problems we consider the continuous trajectories of the vector field induced by the primal affine scaling algorithm as applied to linear programming problems in standard form by characterizing these trajectories as solutions of certain parametrized logarithmic barrier families of problems we show that these trajectories tend to an optimal solution which in general depends on the starting point by considering the trajectories that arise from the lagrangian multipliers of the above mentioned logarithmic barrier families of problems we show that the trajectories of the dual estimates associated with the affine scaling trajectories converge to the so called centered optimal solution of the dual problem we also present results related to asymptotic direction of the affine scaling trajectories we briefly discuss how to apply our results to linear programs formulated in formats different from the standard form finally we extend the results to the primal dual amne scaling algorithm

the nonlinear geometry of linear programming ii legendre transform coordinates and central trajectories karmarkar s projective scaling algorithm for solving linear programming problems associates to each objective function a vector field defined in the interior of the polytope of feasible solutions of the problem this paper studies the set of trajectories obtained by integrating this vector field called p trajectories as well as a related set of trajectories called a trajectories the a trajectories arise from another linear programming algorithm the affine scaling algorithm the affine and projective scaling vector fields are each defined for linear programs of a special form called standard form and canonical form respectively these trajectories are studied using a nonlinear change of variables called legendre transform coordinates which is a projection of the gradient of a logarithmic barrier function the legendre transform coordinate mapping is given by rational functions and its inverse mapping is algebraic it depends only on the constraints of the linear program and is a one to one mapping for canonical form linear programs when the polytope of feasible solutions is bounded there is a unique point mapping to zero called the center the a trajectories of standard form linear programs are linearized by the legendre transform coordinate mapping when the polytope of feasible solutions is bounded they are the complete set of geodesics of a riemannian geometry isometric to euclidean geometry each a trajectory is part of a real algebraic curve each p trajectory for a canonical form linear program lies in a plane in legendre transform coordinates the p trajectory through 0 in legendre transform coordinates called the central p trajectory is part of a straight line and is contained in the a trajectory through 0 called the central a trajectory each p trajectory is part of a real algebraic curve the central a trajectory is the locus of centers of a family of linear programs obtained by adding an extra equality constraint of the form c x u it is also the set of minima of a parametrized family of logarithmic barrier functions power series expansions are derived for the central a trajectory which is also the central p trajectory these power series have a simple recursive form and are useful in developing higher order analogues of karmarkar s algorithm a trajectories are defined for a general linear program using this definition it is shown that the limit point x 0 of a central a trajectory on the boundary of the feasible solution polytope p is the center of the unique face of p containing x 0 in its relative interior received by the editors october 8 1986 and in revised form june 9 1987 and march 25 1988 1980 mathematics subject classification 1985 revision primary 90c05 secondary 52a40 34a34 the first author was partially supported by onr contract n00014 87 k0214 1989 american mathematical society 0002 9947 89 1 00 25 per page

an analysis of the embedded discontinuous galerkin method for second order elliptic problems the embedded discontinuous galerkin methods are obtained from hybridizable discontinuous galerkin methods by a simple change of the space of the hybrid unknown in this paper we consider embedded methods for second order elliptic problems obtained from hybridizable discontinuous methods by changing the space of the hybrid unknown from discontinuous to continuous functions this change results in a significantly smaller stiffness matrix whose size and sparsity structure coincides with those of the stiffness matrix of the statically condensed continuous galerkin method it is shown that this computational advantage has to be balanced against the fact that the approximate solutions for the scalar variable and its flux lose each a full order of convergence indeed we prove that if polynomials of degree k ge1 are used for the original hybridizable discontinuous galerkin method its approximations to the scalar variable and its flux converge with order k 2 and k 1 respectively whereas those of the corresponding embedded discontinuous galerkin method converge with orders k 1 and k respectively only we also provide numerical results comparing the relative efficiency of the methods

an analysis of monotonicity conditions in the mixed hybrid finite element method on unstructured triangulations in this paper we consider the mixed hybrid finite element method on unstructured triangular grids and evaluate its monotonicity properties by using a non standard set of basis functions for the velocity approximation space the mixed hybrid discretization of the steady state diffusion equation produces a system matrix that depends only on the inner product of the outward normals to the edges of the triangulation and not on the choice of the velocity space basis this property is used to study the characteristics of the system matrix it is well known that this matrix is of type m if the angles of the triangulation are not bigger than 2 an m matrix has a nonnegative inverse i e all the elements are nonnegative this implies the existence of a discrete maximum principle and thus monotonicity of the discretization we show that when the triangulation is of delaunay type and satisfies the property that no circumcenters of boundary elements with dirichlet conditions lie outside the domain the inverse of the final matrix is always positive even in the presence of obtuse angles

superconvergence of triangular raviart thomas mixed finite element methods for a bilinear constrained optimal control problem in this paper we analyze the superconvergence of the bilinear constrained elliptic optimal control problem by triangular raviart thomas mixed finite element methods the state and the co state are approximated by the order k 1 raviart thomas mixed finite elements and the control is approximated by piecewise constant functions we obtain the superconvergence property between average l 2 projection and the approximation of the control variable and the convergence order is h 2 two numerical examples are presented for illustrating the superconvergence results

quadratic convergence in a primal dual method we show that the mizuno todd ye o nl iteration predictor corrector primal dual interior point algorithm for linear programming is quadratically convergent our proof does not assume that the problems be nondegenerate we do not assume that the iterate generated by the algorithm be convergent an assumption common to all previous asymptotic convergence analysis

multiscale hybrid mixed method this work presents a priori and a posteriori error analyses of a new multiscale hybrid mixed method mhm for an elliptic model specially designed to incorporate multiple scales into the construction of basis functions this finite element method relaxes the continuity of the primal variable through the action of lagrange multipliers while assuring the strong continuity of the normal component of the flux dual variable as a result the dual variable which stems from a simple postprocessing of the primal variable preserves local conservation we prove existence and uniqueness of a solution for the mhm method as well as optimal convergence estimates of any order in the natural norms also we propose a face residual a posteriori error estimator and prove that it controls the error of both variables in the natural norms several numerical tests assess the theoretical results

hybridized globally divergence free ldg methods part i the stokes problem we devise and analyze a new local discontinuous galerkin ldg method for the stokes equations of incompressible fluid flow this optimally convergent method is obtained by using an ldg method to discretize a vorticity velocity formulation of the stokes equations and by applying a new hybridization to the resulting discretization one of the main features of the hybridized method is that it provides a globally divergence free approximate velocity without having to construct globally divergence free finite dimensional spaces only elementwise divergence free basis functions are used another important feature is that it has significantly less degrees of freedom than all other ldg methods in the current literature in particular the approximation to the pressure is only defined on the faces of the elements on the other hand we show that as expected the condition number of the schur complement matrix for this approximate pressure is of order h 2 in the mesh size h finally we present numerical experiments that confirm the sharpness of our theoretical a priori error estimates

block iterative solvers for higher order finite volume methods recently new higher order finite volume methods fvm were introduced in z cai j douglas m park development and analysis of higher order finite volume methods over rectangles for elliptic equations adv comput math 19 2003 3 33 where the linear system derived by the hybridization with lagrange multiplier satisfying the flux consistency condition is reduced to a linear system for a pressure variable by an appropriate quadrature rule we study the convergence of an iterative solver for this linear system the conjugate gradient cg method is a natural choice to solve the system but it seems slow possibly due to the non diagonal dominance of the system in this paper we propose block iterative methods with a reordering scheme to solve the linear system derived by the higher order fvm and prove their convergence with a proper ordering each block subproblem can be solved by fast methods such as the multigrid mg method the numerical experiments show that these block iterative methods are much faster than cg

a conservative and monotone mixed hybridized finite element approximation of transport problems in heterogeneous domains in this article we discuss the numerical approximation of transport phenomena occurring at material interfaces between physical subdomains with heterogenous properties the model in each subdomain consists of a partial differential equation with diffusive convective and reactive terms the coupling between each subdomain being realized through an interface transmission condition of robin type the numerical approximation of the problem in the two dimensional case is carried out through a dual mixed hybridized finite element method with numerical quadrature of the mass flux matrix the resulting method is a conservative finite volume scheme over triangular grids for which a discrete maximum principle is proved under the assumption that the mesh is of delaunay type in the interior of the domain and of weakly acute type along the domain external boundary and internal interface the stability accuracy and robustness of the proposed method are validated on several numerical examples motivated by applications in biology electrophysiology and neuroelectronics

convergence and optimality of adaptive mixed finite element methods the convergence and optimality of adaptive mixed finite element methods for the poisson equation are established in this paper the main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonality a quasi orthogonality property is proved using the fact that the error is orthogonal to the divergence free subspace while the part of the error that is not divergence free can be bounded by the data oscillation using a discrete stability result this discrete stability result is also used to get a localized discrete upper bound which is crucial for the proof of the optimality of the adaptive approximation

an ol n 3 potential reduction algorithm for linear programming we describe a primal dual potential function for linear programming phi x s rho ln x t s sum limits j 1 n ln x j s j where n x is the primal variable ands is the dual slack variable as a result we develop an interior point algorithm seeking reductions in the potential function with rho n sqrt n neither tracing the central path nor using the projective transformation the algorithm converges to the optimal solution set in o sqrt n l iterations and uses o n3l total arithmetic operations we also suggest a practical approach to implementing the algorithm

breaking spaces and forms for the dpg method and applications including maxwell equations discontinuous petrov galerkin dpg methods are made easily implementable using broken test spaces i e spaces of functions with no continuity constraints across mesh element interfaces broken spaces derivable from a standard exact sequence of first order unbroken sobolev spaces are of particular interest a characterization of interface spaces that connect the broken spaces to their unbroken counterparts is provided stability of certain formulations using the broken spaces can be derived from the stability of analogues that use unbroken spaces this technique is used to provide a complete error analysis of dpg methods for maxwell equations with perfect electric boundary conditions the technique also permits considerable simplifications of previous analyses of dpg methods for other equations reliability and efficiency estimates for an error indicator also follow finally the equivalence of stability for various formulations of the same maxwell problem is proved including the strong form the ultraweak form and various forms in between

symmetric and conforming mixed finite elements for plane elasticity using rational bubble functions we construct stable conforming and symmetric finite elements for the mixed formulation of the linear elasticity problem in two dimensions in our approach we add three divergence free rational functions to piecewise polynomials to form the stress finite element space the relation with the elasticity elements and a class of generalized c 1 c 1 zienkiewicz elements is also discussed

analysis of a family of hdg methods for second order elliptic problems in this paper we analyze a family of hybridizable discontinuous galerkin hdg methods for second order elliptic problems in two and three dimensions the methods use piecewise polynomials of degree k 0 for both the flux and numerical trace and piecewise polynomials of degree k 1 for the potential we establish error estimates for the numerical flux and potential under the minimal regularity condition moreover we construct a local postprocessing for the flux which produces a numerical flux with better conservation numerical experiments in two space dimensions confirm our theoretical results

incompressible finite elements via hybridization part ii the stokes system in three space dimensions in this paper we introduce a new and efficient way to compute exactly divergence free velocity approximations for the stokes equations in two space dimensions we begin by considering a mixed method that provides an exactly divergence free approximation of the velocity and a continuous approximation of the vorticity we then rewrite this method solely in terms of the tangential fluid velocity and the pressure on mesh edges by means of a new hybridization technique this novel formulation bypasses the difficult task of constructing an exactly divergence free basis for velocity approximations moreover the discrete system resulting from our method has fewer degrees of freedom than the original mixed method since the pressure and the tangential velocity variables are defined just on the mesh edges once these variables are computed the velocity approximation satisfying the incompressibility condition exactly as well as the continuous numerical approximation of the vorticity can at once be obtained locally moreover a discontinuous numerical approximation of the pressure within elements can also be obtained locally we show how to compute the matrix system for our tangential velocity pressure formulation on general meshes and present in full detail such computations for the lowest order case of our method

hybridization and postprocessing techniques for mixed eigenfunctions we introduce hybridization and postprocessing techniques for the raviart thomas approximation of second order elliptic eigenvalue problems hybridization reduces the raviart thomas approximation to a condensed eigenproblem the condensed eigenproblem is nonlinear but smaller than the original mixed approximation we derive multiple iterative algorithms for solving the condensed eigenproblem and examine their interrelationships and convergence rates an element by element postprocessing technique to improve accuracy of computed eigenfunctions is also presented we prove that a projection of the error in the eigenspace approximation by the mixed method of any order superconverges and that the postprocessed eigenfunction approximations converge faster for smooth eigenfunctions numerical experiments using a square and an l shaped domain illustrate the theoretical results

unified primal formulation based a priori and a posteriori error analysis of mixed finite element methods we derive in this paper a unified framework for a priori and a posteriori error analysis of mixed finite element discretizations of second order elliptic problems it is based on the classical primal weak formulation the postprocessing of the potential proposed in t arbogast and z chen on the implementation of mixed methods as nonconforming methods for second order elliptic problems math comp 64 1995 943 972 and the discrete friedrichs inequality our analysis in particular avoids any explicit use of the uniform discrete inf sup condition and in a straightforward manner and under minimal necessary assumptions known convergence and superconvergence results are recovered the same framework then turns out to lead to optimal a posteriori energy error bounds in particular estimators for all families and orders of mixed finite element methods on grids consisting of simplices or rectangular parallelepipeds are derived they give a guaranteed and fully computable upper bound on the energy error represent error local lower bounds and are robust under some conditions on the diffusion dispersion tensor they are thus suitable for both overall error control and adaptive mesh refinement moreover the developed abstract framework and a posteriori error estimates are quite general and apply to any locally conservative method we finally prove that in parallel and simultaneously in converse to galerkin finite element methods under some circumstances the weak solution is the orthogonal projection of the postprocessed mixed finite element approximation onto the h 1 0 space and also establish several links between mixed finite element approximations and some generalized weak solutions

limiting behavior of weighted central paths in linear programming we study the limiting behavior of the weighted central paths x s 0 in linear programming at both 0 and we establish the existence of a partition b n of the index set 1 n such thatx i ands j as fori b andj n andx n s b converge to weighted analytic centers of certain polytopes for allk 1 we show that thekth order derivativesx k ands k converge when 0 and consequently the derivatives of each order are bounded in the interval 0 we calculate the limiting derivatives explicitly and establish the surprising result that all higher order derivatives k 2 converge to zero when

a raviart thomas mixed finite element scheme for the two dimensional three temperature heat conduction problems summary for the two dimensional three temperature radiative heat conduction problem appearing in the inertial confinement numerical stimulations we choose the freezing coefficient method to linearize the nonlinear equations and initially apply the well known mixed finite element scheme with the lowest order raviart thomas element associated with the triangulation to the linearized equations and obtain the convergence with one order with respect to the space direction for the temperature and flux function approximations and design a simple but efficient algorithm for the discrete system three numerical examples are displayed the former two verify theoretical results and show the super convergence for temperature and flux functions at the barycenter of the element which is helpful for solving the radiative heat conduction problems the third validates the robustness of this scheme with small energy conservative error and one order convergence for the time discretization copyright 2016 john wiley sons ltd

an a priori error estimate for a monotone mixed finite element discretization of a convection diffusion problem we present a local exponential fitting hybridized mixed finite element method for convection diffusion problem on a bounded domain with mixed dirichlet neuman boundary conditions with a new technique that interpretes the algebraic system after static condensation as a bilinear form acting on certain lifting operators we prove an a priori error estimate on the lagrange multipliers that requires minimal regularity while an extension of more classical arguments provide an estimate for the other solution components

locally conservative fluxes for the continuous galerkin method the standard continuous galerkin cg finite element method for second order elliptic problems suffers from its inability to provide conservative flux approximations a much needed quantity in many applications we show how to overcome this shortcoming by using a two step postprocessing the first step is the computation of a numerical flux trace defined on element interfaces and is motivated by the structure of the numerical traces of discontinuous galerkin methods this computation is nonlocal in that it requires the solution of a symmetric positive definite system but the system is well conditioned independently of mesh size so it can be solved at asymptotically optimal cost the second step is a local element by element postprocessing of the cg solution incorporating the result of the first step this leads to a conservative flux approximation with continuous normal components this postprocessing applies for the cg method in its standard form or for a hybridized version of it we present the hybridized version since it allows easy handling of variable degree polynomials and hanging nodes furthermore we provide an a priori analysis of the error in the postprocessed flux approximation and display numerical evidence suggesting that the approximation is competitive with the approximation provided by the raviart thomas mixed method of corresponding degree

a posteriori error analysis of a cell centered finite volume method for semilinear elliptic problems in this paper we conduct a goal oriented a posteriori analysis for the error in a quantity of interest computed from a cell centered finite volume scheme for a semilinear elliptic problem the a posteriori error analysis is based on variational analysis residual errors and the adjoint problem to carry out the analysis we use an equivalence between the cell centered finite volume scheme and a mixed finite element method with special choice of quadrature

linear programming and the newton barrier flow in this note we report a simple characteristic of linear programming central trajectories which has a surprising consequence specifically we show that given a bounded polyhedral setp with nonempty interior the logarithmic barrier function with no objective component induces a vector field of negative newton directions which flows from the center ofp along central trajectories to solutions of every possible linear program onp

an inexpensive method for the evaluation of the solution of the lowest order raviart thomas mixed method for the model problem operatorname div a nabla u f we show how the approximate solution produced by the raviart thomas lecture notes in mathematics 606 springer verlag berlin 1977 mixed method of lowest degree can be obtained from the solution of the p 1 nonconforming galerkin method modified in a virtually cost free manner

symmetric nonconforming mixed finite elements for linear elasticity we present a family of mixed methods for linear elasticity that yield exactly symmetric but only weakly conforming stress approximations the method is presented in both two and three dimensions on triangular and tetrahedral meshes the method is efficiently implementable by hybridization the degrees of freedom of the lagrange multipliers which approximate the displacements at the faces solve a symmetric positive definite system the design and analysis of this method is motivated by a new set of unisolvent degrees of freedom for symmetric polynomial matrices these new degrees of freedom are also used to give a new simple calculation of the dimension of the space of polynomial symmetric matrix fields with vanishing normal traces and zero divergence on a tetrahedron such a dimension count was important in the development of the symmetric h mathrm div conforming methods found in d n arnold g awanou and r winther math comp 77 2008 pp 1229 1251