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

applications of equivariant degree for gradient maps to symmetric newtonian systems abstract we consider g s 1 with being a finite group for which the complete euler ring structure in u g is described the multiplication tables for d 6 s 4 and a 5 are provided in the appendix the equivariant degree for g orthogonal maps is constructed using the primary equivariant degree with one free parameter we show that the g orthogonal degree extends the degree for g gradient maps in the case of g s 1 introduced by g ba in k g ba w krawcewicz j wu an equivariant degree with applications to symmetric bifurcation problems i construction of the degree bull london math soc 69 1994 377 398 the computational results obtained are applied to a symmetric autonomous newtonian system for which we study the existence of 2 periodic solutions for some concrete cases we present the symmetric classification of the solution set for the systems considered

superconvergence in a dpg method for an ultra weak formulation abstract in this work we study a dpg method for an ultra weak variational formulation of a reaction diffusion problem we improve existing a priori convergence results by sharpening an approximation result for the numerical flux by duality arguments we show that higher convergence rates for the scalar field variable are obtained if the polynomial order of the corresponding approximation space is increased by one furthermore we introduce a simple elementwise postprocessing of the solution and prove superconvergence numerical experiments indicate that the obtained results are valid beyond the underlying model problem

extended discontinuous galerkin methods for two phase flows the spatial discretization summary this work discusses a discontinuous galerkin dg discretization for two phase flows the fluid interface is represented by a level set and the dg approximation space is adapted such that jumps and kinks in pressure and velocity fields can be approximated sharply this adaption of the original dg space which can be performed on the fly for arbitrary interface shapes is referred to as extended discontinuous galerkin by combining this ansatz with a special quadrature technique one can regain spectral convergence properties for low regularity solutions which is demonstrated by numerical examples this work focuses on the aspects of spatial discretization and special emphasis is devoted on how to overcome problems related to quadrature small cut cells and condition number of linear systems temporal discretization will be discussed in future works copyright 2016 john wiley sons ltd

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

on the dimension and the index of the solution set of nonlinear equations we study the covering dimension and the index of the solution set to multiparameter nonlinear and semilinear operator equations involving fredholm maps of positive index the classes of maps under consideration are pseudo a proper and either approximation essential or equivariant approximation essential applications are given to semilinear elliptic bvp s

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

a projection based error analysis of hdg methods we introduce a new technique for the error analysis of hybridizable discontinuous galerkin hdg methods the technique relies on the use of a new projection whose design is inspired by the form of the numerical traces of the methods this renders the analysis of the projections of the discretization errors simple and concise by showing that these projections of the errors are bounded in terms of the distance between the solution and its projection our studies of influence of the stabilization parameter are reduced to local analyses of approximation by the projection we illustrate the technique on a specific hdg method applied to a model second order elliptic problem

duality and operator algebras automatic weak continuity and applications abstract we investigate some subtle and interesting phenomena in the duality theory of operator spaces and operator algebras and give several applications of the surprising fact that certain maps are always weak continuous on dual spaces in particular if x is a subspace of a c algebra a and if a a satisfies ax x then we show that the function x ax on x is automatically weak continuous if either a x is a dual operator space or b a x x and x is a dual banach space these results hinge on a generalization to banach modules of tomiyama s famous theorem on contractive projections onto a c subalgebra applications include a new characterization of the weakly closed possibly nonunital and nonselfadjoint operator algebras and a generalization of the theory of w modules to the framework of modules over such algebras we also give a banach module characterization of weakly closed spaces of operators which are invariant under the action of a von neumann algebra

computation of the equivariant 1 stem abstract let g be a compact lie group in this paper combining a short exact sequence obtained by balanov and krawcewicz with some additional topological techniques we complete the computation of the first equivariant stem 1 g st using the exact sequence and a property of nonabelian connected compact lie groups whose proof was suggested to us by r oliver we show that this group is finite if and only if g is finite

global estimates for mixed methods for second order elliptic equations global error estimates in l2 92 l 9 and h s q 2 in r2 or a re derived for a mixed finite element method for the dirichlet problem for the elliptic operator lp div a grad p bp cp based on the raviart thomas nedelec space vh x wh c h div 92 x l2 g2 optimal order estimates are obtained for the approximation of p and the associated velocity field u a grad p bp in l2 g2 and h s 92 0 6 s k 1 and if q c r2 forp in lv2

a weakly compressible hybridizable discontinuous galerkin formulation for fluid structure interaction problems abstract a scheme for the solution of fluid structure interaction fsi problems with weakly compressible flows is proposed in this work a novel hybridizable discontinuous galerkin hdg method is derived for the discretization of the fluid equations while the standard continuous galerkin cg approach is adopted for the structural problem the chosen hdg solver combines robustness of discontinuous galerkin dg approaches in advection dominated flows with higher order accuracy and efficient implementations two coupling strategies are examined in this contribution namely a partitioned dirichlet neumann scheme in the context of hybrid hdg cg discretizations and a monolithic approach based on nitsche s method exploiting the definition of the numerical flux and the trace of the solution to impose the coupling conditions numerical experiments show optimal convergence of the hdg and cg primal and mixed variables and superconvergence of the postprocessed fluid velocity the robustness and the efficiency of the proposed weakly compressible formulation in comparison to a fully incompressible one are also highlighted on a selection of two and three dimensional fsi benchmark problems

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

symmetric functional differential equations and neural networks with memory we establish an analytic local hopf bifurcation theorem and a topological global hopf bifurcation theorem to detect the existence and to describe the spatial temporal pattern the asymptotic form and the global continuation of bifurcations of periodic wave solutions for functional differential equations in the presence of symmetry we apply these general results to obtain the coexistence of multiple large amplitude wave solutions for the delayed hopfield cohen grossberg model of neural networks with a symmetric circulant connection matrix

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

local analysis of hdg methods for convection dominated diffusion problems in this paper we establish local error analysis of hybridizable discontinuous galerkin hdg methods for convection dominated diffusion equation in two types of subdomains away from the layers the first subdomain is of o log 1 h h 1 2 away from the interior layers and of o log 1 h h away from the boundary layers and the second one is of o away from the outflow of the boundary our local error bound for the first subdomain can be improved by a factor of log 1 h we use weighted estimates to prove the first result by constructing new weight functions and the inf sup condition of the bilinear form to prove the second one

an hdg method for distributed control of convection diffusion pdes abstract we propose a hybridizable discontinuous galerkin hdg method to approximate the solution of a distributed optimal control problem governed by an elliptic linear convection diffusion pde we use degree k polynomials to approximate the state adjoint state their fluxes and the optimal control and we show the approximations converge with order k 1 in the l 2 norm finally we use a simple element by element postprocessing scheme to obtain new superconvergent approximations of the state dual state and the control we show the postprocessed variables converge with order k 2 in the l 2 norm we present 2d and 3d numerical experiments to illustrate our theoretical results

a space time hybridizable discontinuous galerkin method for incompressible flows on deforming domains we present the first space time hybridizable discontinuous galerkin hdg finite element method for the incompressible navier stokes and oseen equations major advantages of a space time formulation are its excellent capabilities of dealing with moving and deforming domains and grids and its ability to achieve higher order accurate approximations in both time and space by simply increasing the order of polynomial approximation in the space time elements our formulation is related to the hdg formulation for incompressible flows introduced recently in e g n c nguyen j peraire b cockburn a hybridizable discontinuous galerkin method for stokes flow comput methods appl mech eng 199 2010 582 597 however ours is inspired in typical dg formulations for compressible flows which allow for a more straightforward implementation another difference is the use of polynomials of fixed total degree with space time hexahedral and quadrilateral elements instead of simplicial elements we present numerical experiments in order to assess the quality of the performance of the methods on deforming domains and to experimentally investigate the behavior of the convergence rates of each component of the solution with respect to the polynomial degree of the approximations in both space and time

hopf bifurcations of functional differential equations with dihedral symmetries abstract we discuss the joint impact of temporal delay and spatial dihedral symmetries on the occurence and multiplicity of hopf bifurcations for a system of fdes by applying the equivariant degree theory we establish a result on the existence of multiple branches of nonconstant periodic solutions and classify their symmetries general results are illustrated by a ring of identical oscillators with identical coupling between adjacent cells

duality and operator algebras ii operator algebras as banach algebras abstract we answer by counterexample several questions concerning algebras of operators on a hilbert space the answers add further weight to the thesis that for many purposes such algebras ought to be studied in the framework of operator spaces as opposed to that of banach spaces and banach algebras in particular the nonselfadjoint analogue of a w algebra resides naturally in the category of dual operator spaces as opposed to dual banach spaces we also show that an automatic w continuity result in the preceding paper of the authors is sharp

a conservative discontinuous galerkin semi implicit formulation for the navier stokes equations in nonhydrostatic mesoscale modeling a discontinuous galerkin dg finite element formulation is proposed for the solution of the compressible navier stokes equations for a vertically stratified fluid which are of interest in mesoscale nonhydrostatic atmospheric modeling the resulting scheme naturally ensures conservation of mass momentum and energy a semi implicit time integration approach is adopted to improve the efficiency of the scheme with respect to the explicit runge kutta time integration strategies usually employed in the context of dg formulations a method is also presented to reformulate the resulting linear system as a pseudo helmholtz problem in doing this we obtain a dg discretization closely related to those proposed for the solution of elliptic problems and we show how to take advantage of the numerical integration rules required in all dg methods for the area and flux integrals to increase the efficiency of the solution algorithm the resulting numerical formulation is then validated on a collection of classical two dimensional test cases including density driven flows and mountain wave simulations the performance analysis shows that the semi implicit method is indeed superior to explicit methods and that the pseudo helmholtz formulation yields further efficiency improvements

an a priori error analysis of the local discontinuous galerkin method for elliptic problems in this paper we present the first a priori error analysis for the local discontinuous galerkin ldg method for a model elliptic problem for arbitrary meshes with hanging nodes and elements of various shapes we show that for stabilization parameters of order one the l2 norm of the gradient and the l2 norm of the potential are of order k and k 1 2 respectively when polynomials of total degree at least k are used if stabilization parameters of order h 1 are taken the order of convergence of the potential increases to k 1 the optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains

primal stabilized hybrid and dg finite element methods for the linear elasticity problem abstract primal stabilized hybrid finite element methods for the linear elasticity problem are proposed consisting of locally discontinuous galerkin problems in the primal variable coupled to a global problem in the multiplier which is identified with the trace of the displacement field numerical analysis covering both continuous or discontinuous interpolations of the multiplier shows that the proposed formulation preserves the main properties of the associate dg method such as consistency stability boundedness and optimal rates of convergence in the energy norm and in the l 2 norm for adjoint consistent formulations convergence studies confirm the optimal rates of convergence predicted by the numerical analysis presented here and a local post processing technique is proposed to recover stress approximations with improved rates of convergence in h div norm

optimal convergence of the original dg method for the transport reaction equation on special meshes we show that the approximation given by the original discontinuous galerkin method for the transport reaction equation in d space dimensions is optimal provided the meshes are suitably chosen the l 2 norm of the error is of order k 1 when the method uses polynomials of degree k these meshes are not necessarily conforming and do not satisfy any uniformity condition they are required only to be made of simplexes each of which has a unique outflow face we also find a new element by element postprocessing of the derivative in the direction of the flow which superconverges with order k 1

an operator space characterization of dual operator algebras this paper is devoted to dual algebras that is w closed algebras of bounded operators on hilbert space we give abstract characterizations of unital dual algebras resp dual algebras up to w homeomorphic unital complete isometry resp complete isomorphism the results we obtain are natural w versions of analogous characterizations due to blecher ruan sinclair and blecher respectively in the category of operator algebras

an explicit hybridizable discontinuous galerkin method for the acoustic wave equation abstract we present an explicit hybridizable discontinuous galerkin hdg method for numerically solving the acoustic wave equation the method is fully explicit high order accurate in both space and time and coincides with the classic discontinuous galerkin dg method with upwinding fluxes for a particular choice of its stabilization function this means that it has the same computational complexity as other explicit dg methods however just as its implicit version it provides optimal convergence of order k 1 for all the approximate variables including the gradient of the solution and when the time stepping method is of order k 2 it displays a superconvergence property which allow us by means of local postprocessing to obtain new improved approximations of the scalar field variables at any time levels for which an enhanced accuracy is required in particular the new approximations converge with order k 2 in the l 2 norm for k 1 these properties do not hold for all numerical fluxes indeed our results show that when the hdg numerical flux is replaced by the lax friedrichs flux the above mentioned superconvergence properties are lost although some are recovered when the lax friedrichs flux is used only in the interior of the domain finally we extend the explicit hdg method to treat the wave equation with perfectly matched layers we provide numerical examples to demonstrate the performance of the proposed method

analysis of an hdg method for linear elasticity summary we present the first a priori error analysis for the first hybridizable discontinuous galerkin method for linear elasticity proposed in internat j numer methods engrg 80 2009 no 8 1058 1092 we consider meshes made of polyhedral shape regular elements of arbitrary shape and show that whenever piecewise polynomial approximations of degree k 0 are used and the exact solution is smooth enough the antisymmetric part of the gradient of the displacement converges with order k the stress and the symmetric part of the gradient of the displacement converge with order k 1 2 and the displacement converges with order k 1 we also provide numerical results showing that the orders of convergence are actually sharp copyright 2014 john wiley sons ltd

a superconvergent hybridisable discontinuous galerkin method for linear elasticity this is the peer reviewed version of the following article sevilla r giacomini m karkoulias a huerta a a superconvergent hybridisable discontinuous galerkin method for linear elasticity international journal for numerical methods in engineering 12 octubre 2018 vol 116 num 2 p 91 116 which has been published in final form at https doi org 10 1002 nme 5916 this article may be used for non commercial purposes in accordance with wiley terms and conditions for self archiving

continuous discontinuous galerkin methods applied to elasticity problems abstract a new family of coupled continuous discontinuous galerkin formulations is presented and analyzed in this paper these formulations have some distinguishing properties support to all boundary conditions without differentiating whether the condition is dirichlet or not the continuous part of the formulations can use the discontinuous part to have better accuracy and robustness properties and the discontinuous part has the same stabilized properties of common discontinuous galerkin methods a new promising stability parameter is introduced and its effects analyzed in the numerical experiments some important results related to the stability of the formulations associated with the polynomial degree adopted for the continuous component are also shown

a local post processing technique for improving the accuracy in mixed finite element approximations a general simple post processing technique is given in this paper under certain conditions if an algorithm approximating a function u also provides a good approximation for its gradient and for some locally defined projection pu then by this method a better approximation for u can be easily obtained essentially the leading term of the error between u and its original approximation can be computed it is shown how the method may be applied to various mixed type finite element methods

a family of mixed finite elements for the elasticity problem a new mixed finite element formulation for the equations of linear elasticity is considered in the formulation the variables approximated are the displacement the unsymmetric stress tensor and the rotation the rotation act as a lagrange multiplier introduced in order to enforce the symmetry of the stress tensor based on this formulation a new family of both two and three dimensional mixed methods is defined optimal error estimates which are valid uniformly with respect to the poisson ratio are derived finally a new postprocessing scheme for improving the displacement is introduced and analyzed

the staggered dg method is the limit of a hybridizable dg method we show in the framework of steady state diffusion boundary value problems that the staggered discontinuous galerkin sdg method siam j numer anal 47 2009 pp 3820 3848 can be obtained from a hybridizable discontinuous galerkin hdg method siam j numer anal 47 2009 pp 1319 1365 by setting its stabilization function to zero at some suitably chosen element faces and by letting it go to infinity at all the remaining others we then show that this point of view allows the sdg method to immediately acquire new properties all inherited from the hdg methods namely their efficient implementation by hybridization their postprocessings and their superconvergence properties

an upwind like discontinuous galerkin method for hyperbolic systems we investigate an upwind like dg method for solving first order hyperbolic problems written as the friedrichs systems under certain condition this dg scheme may be semi explicit such that the discrete equations can be solved layer by layer we give the stability analysis and error estimate of order k 1 2 in the dg norm in particular for some hyperbolic systems we show that the convergence rate is of order k 1 in the l 2 norm if the q k elements are used on rectangular meshes finally we provide some numerical experiments to illustrate the theoretical analysis

superconvergence of the numerical traces of discontinuous galerkin and hybridized methods for convection diffusion problems in one space dimension in this paper we uncover and study a new superconvergence property of a large class of finite element methods for one dimensional convection diffusion problems this class includes discontinuous galerkin methods defined in terms of numerical traces discontinuous petrov galerkin methods and hybridized mixed methods we prove that the so called numerical traces of both variables superconverge at all the nodes of the mesh provided that the traces are conservative that is provided they are single valued in particular for is local discontinuous galerkin method we show that the superconvergence is order 2p 1 when polynomials of degree at most p are used extensive numerical results verifying our theoretical results are displayed

a priori and a posteriori error analyses of an hdg method for the brinkman problem abstract in this paper we introduce and analyze a hybridizable discontinuous galerkin hdg method for the linear brinkman model of porous media flow in two and three dimensions and with non homogeneous dirichlet boundary conditions we consider a fully mixed formulation in which the main unknowns are given by the pseudostress the velocity and the trace of the velocity whereas the pressure is easily recovered through a simple postprocessing we show that the corresponding continuous and discrete schemes are well posed in particular we use the projection based error analysis in order to derive a priori error estimates furthermore we develop a reliable and efficient residual based a posteriori error estimator and propose the associated adaptive algorithm for our hdg approximation finally several numerical results illustrating the performance of the method confirming the theoretical properties of the estimator and showing the expected behavior of the adaptive refinements are presented

optimal convergence of the original dg method on special meshes for variable transport velocity we prove optimal convergence rates for the approximation provided by the original discontinuous galerkin method for the transport reaction problem this is achieved in any dimension on meshes related in a suitable way to the possibly variable velocity carrying out the transport thus if the method uses polynomials of degree k the l 2 norm of the error is of order k 1 moreover we also show that by means of an element by element postprocessing a new approximate flux can be obtained which superconverges with order k 1