using a dpg method to validate dma experimental calibration of viscoelastic materials abstract a discontinuous petrov galerkin dpg method is used to solve the time harmonic equations of linear viscoelasticity it is based on a broken primal variational formulation which is very similar to the classical primal variational formulation used in galerkin methods but has additional interface variables at the boundaries of the mesh elements both the classical and broken formulations are proved to be well posed in the infinite dimensional setting and the resulting discretization is proved to be stable a full h p convergence analysis is also included and the analysis is verified using computational simulations the method is particularly useful as it carries its own natural arbitrary p a posteriori error estimator which is fundamental for solving problems with localized solution features this proves to be useful when validating calibration models of dynamic mechanical analysis dma experiments indeed different dma experiments of epoxy and silicone resins were successfully validated to within 5 of the quantity of interest using the numerical method

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

first order least squares method with weakly imposed boundary condition for convection dominated diffusion problems we present and analyze a first order least squares method for convection dominated diffusion problems which provides robust l 2 a priori error estimate for the scalar variable even if the given data f l 2 the novel theoretical approach is to rewrite the method in the framework of discontinuous petrov galerkin dpg method and then show numerical stability by using a key equation discovered by gopalakrishnan and qiu 2014 this new approach gives an alternative way to do numerical analysis for least squares methods for a large class of differential equations we also show that the condition number of the global matrix is independent of the diffusion coefficient a key feature of the method is that there is no stabilization parameter chosen empirically in addition dirichlet boundary condition is weakly imposed numerical experiments verify our theoretical results and in particular show our way of weakly imposing dirichlet boundary condition is essential to the design of least squares methods numerical solutions on subdomains away from interior layers or boundary layers have remarkable accuracy even on coarse meshes which are unstructured quasi uniform

a first order system least squares method for the helmholtz equation we present a first order system least squares fosls method for the helmholtz equation at high wave number k which always leads to a hermitian positive definite algebraic system by utilizing a non trivial solution decomposition to the dual fosls problem which is quite different from that of the standard finite element methods we give an error analysis to the h p version of the fosls method where the dependence on the mesh size h the approximation order p and the wave number k is given explicitly in particular under some assumption of the boundary of the domain the l 2 norm error estimate of the scalar solution from the fosls method is shown to be quasi optimal under the condition that k h p is sufficiently small and the polynomial degree p is at least o log k numerical experiments are given to verify the theoretical results

an analysis of the practical dpg method we give a complete error analysis of the discontinuous petrov galerkin dpg method accounting for all the approximations made in its practical implementation specifi cally we consider the dpg method that uses a trial space consisting of polynomials of degree p on each mesh element earlier works showed that there is a trial to test operator t which when applied to the trial space defines a test space that guarantees stability in dpg formulations this operator t is local it can be applied element by element however an infinite dimensional problem on each mesh element needed to be solved to apply t in prac tical computations t is approximated using polynomials of some degree r p on each mesh element we show that this approximation maintains optimal convergence rates provided that r 2 p n where n is the space dimension two or more for the laplace equation we also prove a similar result for the dpg method for linear elasticity remarks on the conditioning of the stiffness matrix in dpg methods are also included

grothendieck rings of basic classical lie superalgebras the grothendieck rings of nite dimensional representations of the basic classical lie superalgebras are explicitly described in terms of the corresponding generalized root systems we show that they can be interpreted as the subrings in the weight group rings invariant under the action of certain groupoids called super weyl groupoids

constructively well posed approximation methods with unity inf sup and continuity constants for partial differential equations starting from the generalized lax milgram theorem and from the fact that the approximation error is minimized when the continuity and inf sup constants are unity we develop a theory that provably delivers well posed approximation methods with unity continuity and inf sup constants for numerical solution of linear partial differential equations we demonstrate our single framework theory on scalar hyperbolic equations to constructively derive two different hp finite element methods the first one coincides with a least squares discontinuous galerkin method and the other appears to be new both methods are proven to be trivially well posed with optimal hpconvergence rates the numerical results show that our new discontinuous finite element method namely a discontinuous petrov galerkin method is more accurate has optimal convergence rate and does not seem to have unphysical diffusion compared to the upwind discontinuous galerkin method

multiscale hybrid mixed method for the stokes and brinkman equations the method abstract the multiscale hybrid mixed mhm method is extended to the stokes and brinkman equations with highly heterogeneous coefficients the approach is constructive we first propose an equivalent dual hybrid formulation of the original problem using a coarse partition of the heterogeneous domain faces may be not aligned with jumps in the data then the exact velocity and the pressure are characterized as the solution of a global face problem and the solutions of local independent stokes or brinkman problems at the continuous level owing to this decomposition the one level mhm method stems from the standard galerkin approach for the lagrange multiplier space basis functions are responsible for upscaling the unresolved scales of the medium into the global formulation they are the exact solution of the local problems with prescribed neumann boundary conditions on faces driven by the lagrange multipliers we make the mhm method effective by adopting the unusual stabilized finite element method to solve the local problems approximately as such equal order interpolation turns out to be an option for the velocity the pressure and the lagrange multipliers the numerical solutions share the important properties of the continuum such as local equilibrium with respect to external forces and local mass conservation several academic and highly heterogeneous tests infer that the method achieves super convergence for the velocity as well optimal convergence for the pressure and also for the stress tensor in their natural norms

the dpg star method abstract this article introduces the dpg star from now on denoted dpg finite element method it is a method that is in some sense dual to the discontinuous petrov galerkin dpg method the dpg methodology can be viewed as a means to solve an overdetermined discretization of a boundary value problem in the same vein the dpg methodology is a means to solve an underdetermined discretization these two viewpoints are developed by embedding the same operator equation into two different saddle point problems the analyses of the two problems have many common elements comparison to other methods in the literature round out the newly garnered perspective notably dpg and dpg methods can be seen as generalizations of l l and least squares methods respectively a priori error analysis and a posteriori error control for the dpg method are considered in detail reports of several numerical experiments are provided which demonstrate the essential features of the new method a notable difference between the results from the dpg and dpg analyses is that the convergence rates of the former are limited by the regularity of an extraneous lagrange multiplier variable

eliminating gibbs phenomena a non linear petrov galerkin method for the convection diffusion reaction equation abstract in this article we consider the numerical approximation of the convection diffusion reaction equation one of the main challenges of designing a numerical method for this problem is that boundary layers occurring in the convection dominated case can lead to non physical oscillations in the numerical approximation often referred to as gibbs phenomena the idea of this article is to consider the approximation problem as a residual minimization in dual norms in l q type sobolev spaces with 1 q we then apply a non standard non linear petrov galerkin discretization that is applicable to reflexive banach spaces such that the space itself and its dual are strictly convex similar to discontinuous petrov galerkin methods this method is based on minimizing the residual in a dual norm replacing the intractable dual norm by a suitable discrete dual norm gives rise to a non linear inexact mixed method this generalizes the petrov galerkin framework developed in the context of discontinuous petrov galerkin methods to more general banach spaces for the convection diffusion reaction equation this yields a generalization of a similar approach from the l 2 setting to the l q setting a key advantage of considering a more general banach space setting is that in certain cases the oscillations in the numerical approximation vanish as q tends to 1 as we will demonstrate using a few simple numerical examples

a dual petrovgalerkin finite element method for the convectiondiffusion equation we present a minimum residual finite element method based on a dual petrovgalerkin formulation for convectiondiffusion problems in a higher order adaptive continuous galerkin setting the method borrows concepts from both the discontinuous petrovgalerkin dpg method by demkowicz and gopalakrishnan 2011 and the method of variational stabilization by cohen dahmen and welper 2012 and it can also be interpreted as a variational multiscale method in which the fine scales are defined through a dual orthogonality condition a key ingredient in the method is the proper choice of dual norm used to measure the residual and we present two choices which are observed to be robust in both convection and diffusion dominated regimes as well as a proof of stability for quasi uniform meshes and a method for the weak imposition of boundary conditions numerically obtained convergence rates in 2d are reported and benchmark numerical examples are given to illustrate the behavior of the method

a robust petrov galerkin discretisation of convection diffusion equations a petrov galerkin discretisation is studied of an ultra weak variational formulation of the convection diffusion equation in a mixed form to arrive at an implementable method the truly optimal test space has to be replaced by its projection onto a finite dimensional test search space to prevent that this latter space has to be taken increasingly large for vanishing diffusion a formulation is constructed that is well posed in the limit case of a pure transport problem numerical experiments show approximations that are very close to the best approximations to the solution from the trial space uniformly in the size of the diffusion term

robust coupling of dpg and bem for a singularly perturbed transmission problem abstract we consider a transmission problem consisting of a singularly perturbed reaction diffusion equation on a bounded domain and the laplacian in the exterior connected through standard transmission conditions we establish a dpg scheme coupled with galerkin boundary elements for its discretization and prove its robustness for the field variables in so called balanced norms our coupling scheme is the one from fuhrer et al 2016 adapted to the singularly perturbed case by using the scheme from heuer and karkulik 2015 essential feature of our method is that optimal test functions have to be computed only locally we report on various numerical experiments in two dimensions

the dpg method for the stokes problem we discuss well posedness and convergence theory for the dpg method applied to a general system of linear partial differential equations pdes and specialize the results to the classical stokes problem the stokes problem is an iconic troublemaker for standard bubnov galerkin methods if discretizations are not carefully designed they may exhibit non convergence or locking by contrast dpg does not require us to treat the stokes problem in any special manner we illustrate and confirm our theoretical convergence estimates with numerical experiments

a robust dpg method for convection dominated diffusion problems ii adjoint boundary conditions and mesh dependent test norms we introduce a dpg method for convection dominated diffusion problems the choice of a test norm is shown to be crucial to achieving robust behavior with respect to the diffusion parameter demkowicz and heuer 2011 18 we propose a new inflow boundary condition which regularizes the adjoint problem allowing the use of a stronger test norm the robustness of the method is proven and numerical experiments demonstrate the method s robust behavior

construction of dpg fortin operators revisited abstract we construct a general family of dpg fortin operators for the exact energy spaces defined on a tetrahedral element

locally conservative discontinuous petrov galerkin finite elements for fluid problems we develop a locally conservative formulation of the discontinuous petrov galerkin finite element method dpg for convection diffusion type problems using lagrange multipliers to exactly enforce conservation over each element we provide a proof of convergence as well as extensive numerical experiments showing that the method is indeed locally conservative we also show that standard dpg while not guaranteed to be conservative is nearly conservative for many of the benchmarks considered the new method preserves many of the attractive features of dpg but turns the normally symmetric positive definite dpg system into a saddle point problem

discrete least squares finite element methods abstract a finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators 1 the differential operator defining the spatial boundary value problem and 2 a riesz map on the test space the resulting linear system is overdetermined two different approaches for solving the system are suggested although others are discussed 1 solving the associated normal equation with linear solvers for symmetric positive definite systems e g cholesky factorization and 2 solving the overdetermined system with orthogonalization algorithms e g qr factorization the finite element assembly algorithm for each of these approaches is described in detail the normal equation approach is usually faster for direct solvers and requires less storage the second approach reduces the condition number of the system by a power of two and is less sensitive to round off error the rectangular stiffness matrix of second approach is demonstrated to have condition number o h 1 for a variety of formulations of poisson s equation the stiffness matrix from the normal equation approach is demonstrated to be related to the monolithic stiffness matrices of least squares finite element methods and it is proved that the two are identical in some cases an example with poisson s equation indicates that the solutions of these two different linear systems can be nearly indistinguishable if round off error is not an issue and rapidly converge to each other the orthogonalization approach is suggested to be beneficial for problems which induce poorly conditioned linear systems experiments with poisson s equation in single precision arithmetic as well as the linear acoustics problem near resonance in double precision arithmetic verify this conclusion the methodology described here is an outgrowth of the discontinuous petrov galerkin dpg methodology of demkowicz and gopalakrishnan 2015 the strength of dpg is highlighted throughout however the majority of theory presented is more general extensions to constrained minimization principles are also considered throughout but are not analyzed in experiments

a geometric multigrid preconditioning strategy for dpg system matrices abstract the discontinuous petrov galerkin dpg methodology of demkowicz and gopalakrishnan 2010 2011 guarantees the optimality of the solution in an energy norm and provides several features facilitating adaptive schemes a key question that has not yet been answered in general though there are some results for poisson e g is how best to precondition the dpg system matrix so that iterative solvers may be used to allow solution of large scale problems in this paper we detail a strategy for preconditioning the dpg system matrix using geometric multigrid which we have implemented as part of camellia roberts 2014 2016 and demonstrate through numerical experiments its effectiveness in the context of several variational formulations we observe that in some of our experiments the behavior of the preconditioner is closely tied to the discrete test space enrichment we include experiments involving adaptive meshes with hanging nodes for lid driven cavity flow demonstrating that the preconditioners can be applied in the context of challenging problems we also include a scalability study demonstrating that the approach and our implementation scales well to many mpi ranks

the dpg methodology applied to different variational formulations of linear elasticity abstract the flexibility of the dpg methodology is exposed by solving the linear elasticity equations under different variational formulations including some with non symmetric functional settings different infinite dimensional trial and test spaces the family of formulations presented is proved to be mutually ill or well posed when using traditional energy spaces on the whole domain moreover they are shown to remain well posed when using broken energy spaces and interface variables four variational formulations are solved in 3d using the dpg methodology numerical evidence is given for both smooth and singular solutions and the expected convergence rates are observed

analysis of the dpg method for the poisson equation we give an error analysis of the recently developed dpg method applied to solve the poisson equation and a convection diffusion problem we prove that the method is quasioptimal error estimates in terms of both the mesh size h and the polynomial degree p for various element shapes can be derived from our results results of extensive numerical experiments are also presented

a unified discontinuous petrov galerkin method and its analysis for friedrichs systems we propose a unified discontinuous petrov galerkin dpg framework with optimal test functions for friedrichs like systems which embrace a large class of elliptic parabolic and hyperbolic partial differential equations pdes the well posedness i e existence uniqueness and stability of the dpg solution is established on a single abstract dpg formulation and two abstract dpg methods corresponding to two different but equivalent norms are devised we then apply the single dpg framework to several linear ized pdes including but not limited to scalar transport laplace diffusion convection diffusion convection diffusion reaction linear ized continuum mechanics e g linear ized elasticity a version of linearized navier stokes equations etc time domain acoustics and a version of the maxwell s equations the results show that we not only recover several existing dpg methods but also discover new dpg methods for both pdes currently considered in the dpg community and new ones as

a numerical study of the pollution error and dpg adaptivity for long waveguide simulations abstract high frequency wave propagation has many important applications in acoustics elastodynamics and electromagnetics unfortunately the finite element discretization for these problems suffers from significant numerical pollution errors that increase with the wavenumber it is critical to control these errors to obtain a stable and accurate method we study the effect of pollution for very long waveguide problems in the context of robust discontinuous petrov galerkin dpg finite element discretizations our numerical experiments show that the pollution primarily has a diffusive effect causing energy loss in the dpg method while phase errors appear less significant we report results for 3d vectorial time harmonic maxwell problems in waveguides with more than 8000 wavelengths our results corroborate previous analysis for the galerkin discretization of the helmholtz operator by melenk and sauter 2011 additionally we discuss adaptive refinement strategies for multi mode fiber waveguides where the propagating transverse modes must be resolved sufficiently our study shows the applicability of the dpg error indicator to this class of problems finally we illustrate the importance of load balancing in these simulations for distributed memory parallel computing

polynomial extension operators part ii consider the tangential trace of a vector polynomial on the surface of a tetrahedron we construct an extension operator that extends such a trace function into a polynomial on the tetrahedron this operator can be continuously extended to the trace space of h mathbf curl furthermore it satisfies a commutativity property with an extension operator we constructed in part i of this series such extensions are a fundamental ingredient of high order finite element analysis

a locking free hp dpg method for linear elasticity with symmetric stresses we present two new methods for linear elasticity that simultaneously yield stress and displacement approximations of optimal accuracy in both the mesh size h and polynomial degree p this is achieved within the recently developed discontinuous petrov galerkin dpg framework in this framework both the stress and the displacement approximations are discontinuous across element interfaces we study locking free convergence properties and the interrelationships between the two dpg methods

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

a low order discontinuous petrov galerkin method for the stokes equations this paper introduces a low order discontinuous petrov galerkin dpg finite element method fem for the stokes equations the ultra weak formulation utilizes piecewise constant and affine ansatz functions and piecewise affine and discontinuous lowest order raviart thomas test search functions this low order discretization for the stokes equations allows for a direct proof of the discrete inf sup condition with explicit constants the general framework of carstensen et al siam j numer anal 52 3 1335 1353 2014 then implies a complete a priori and a posteriori error analysis of the dpg fem in the natural norms numerical experiments investigate the performance of the method and underline its quasi optimal convergence

analysis of the discontinuous petrov galerkin method with optimal test functions for the reissner mindlin plate bending model we analyze the discontinuous petrov galerkin dpg method with optimal test functions when applied to solve the reissner mindlin model of plate bending we prove that the hybrid variational formulation underlying the dpg method is well posed stable with a thickness dependent constant in a norm encompassing the l 2 norms of the bending moment the shear force the transverse deflection and the rotation vector we then construct a numerical solution scheme based on quadrilateral scalar and vector finite elements of degree p we show that for affine meshes the discretization inherits the stability of the continuous formulation provided that the optimal test functions are approximated by polynomials of degree p 3 we prove a theoretical error estimate in terms of the mesh size h and polynomial degree p and demonstrate numerical convergence on affine as well as non affine mesh sequences

the double adaptivity paradigm how to circumvent the discrete inf sup conditions of babuska and brezzi abstract we present an efficient implementation of the double adaptivity algorithm of cohen et al 2012 within the setting of the petrov galerkin method with optimal test functions we apply this method to the ultraweak variational formulation of a general linear variational problem discretized with the standard galerkin finite element method as an example we demonstrate the feasibility of the method in the context of the convection dominated diffusion problem the presented ideas however apply to virtually any well posed system of first order partial differential equations including singular perturbation problems

selection dynamics for deep neural networks abstract this paper presents a partial differential equation framework for deep residual neural networks and for the associated learning problem this is done by carrying out the continuum limits of neural networks with respect to width and depth we study the wellposedness the large time solution behavior and the characterization of the steady states of the forward problem several useful time uniform estimates and stability instability conditions are presented we state and prove optimality conditions for the inverse deep learning problem using standard variational calculus the hamilton jacobi bellmann equation and the pontryagin maximum principle this serves to establish a mathematical foundation for investigating the algorithmic and theoretical connections between neural networks pde theory variational analysis optimal control and deep learning

the discontinuous petrov galerkin methodology for the mixed multiscale finite element method abstract we present the application of the discontinuous petrov galerkin dpg methodology for the mixed multiscale finite element method msfem the msfem upscaling technique relies on incorporating fine scale features through special in a sense optimized for approximability trial functions while the dpg methodology allows for the selection of the optimal test functions to provide stability of the fem approximation the special trial functions are computed online by the solution of local boundary value problems we improved this process using the static condensation that restricted the construction of the functions to the coarse mesh skeleton element interfaces only we have verified by numerical tests that the proposed improvement of msfem reduced both the approximation error and computational cost moreover it simplified the algorithm significantly the key component of this prolongation construction is our novel method for prolongation of both traction and displacement vectors on element edges of arbitrary shape the proposed prolongation operator may be also used in the multigrid solver for direct analysis of composites with varying material parameters using an arbitrary well posed functional setting with or without the dpg methodology

dpg method with optimal test functions for a transmission problem we propose and analyze a numerical method to solve an elliptic transmission problem in full space the method consists of a variational formulation involving standard boundary integral operators on the coupling interface and an ultra weak formulation in the interior to guarantee the discrete inf sup condition the system is discretized by the dpg method with optimal test functions we prove that the principal unknowns are approximated quasi optimally numerical experiments for problems with smooth and singular solutions confirm optimal convergence orders

a posteriori error control for dpg methods a combination of ideas in least squares finite element methods with those of hybridized methods recently led to discontinuous petrov galerkin dpg finite element methods they minimize a residual inherited from a piecewise ultraweak formulation in a nonstandard locally computable dual norm this paper establishes a general a posteriori error analysis for the natural norms of the dpg schemes under conditions equivalent to a priori stability estimates it is proven that the locally computable residual norm of any discrete function is a lower and an upper error bound up to explicit data approximation errors the presented abstract framework for a posteriori error analysis applies to known dpg discretizations of laplace and lame equations and to a novel dpg method for the stress velocity formulation of stokes flow with symmetric stress approximations since the error control does not rely on the discrete equations it applies to inexactly computed or otherwise perturbed solutions within the discrete space

a class of discontinuous petrov galerkin methods part i the transport equation considering a simple model transport problem we present a new finite element method while the new method fits in the class of discontinuous galerkin dg methods it differs from standard dg and streamline diffusion methods in that it uses a space of discontinuous trial functions tailored for stability the new method unlike the older approaches yields optimal estimates for the primal variable in both the element size h and polynomial degree p and outperforms the standard upwind dg method

high order polygonal discontinuous petrov galerkin polydpg methods using ultraweak formulations abstract this work represents the first endeavor in using ultraweak formulations to implement high order polygonal finite element methods via the discontinuous petrov galerkin dpg methodology ultraweak variational formulations are nonstandard in that all the weight of the derivatives lies in the test space while most of the trial space can be chosen as copies of l 2 discretizations that have no need to be continuous across adjacent elements additionally the test spaces are broken along the mesh interfaces this allows one to construct conforming polygonal finite element methods termed here as polydpg methods by defining most spaces by restriction of a bounding triangle or box to the polygonal element the only variables that require nontrivial compatibility across elements are the so called interface or skeleton variables which can be defined directly on the element boundaries unlike other high order polygonal methods polydpg methods do not require ad hoc stabilization terms thanks to the crafted stability of the dpg methodology a proof of convergence of the form h p is provided and corroborated through several illustrative numerical examples these include polygonal meshes with n sided convex elements and with highly distorted concave elements as well as the modeling of discontinuous material properties along an arbitrary interface that cuts a uniform grid since polydpg methods have a natural a posteriori error estimator a polygonal adaptive strategy is developed and compared to standard adaptivity schemes based on constrained hanging nodes this work is also accompanied by an open source polydpg software supporting polygonal and conventional elements

note on discontinuous trace approximation in the practical dpg method we analyze a non conforming dpg method with discontinuous trace approximation for the poisson problem in two and three space dimensions we show its well posedness and quasi optimal convergence in the principal unknown numerical experiments confirming the theory have been presented previously

some observations on babu vs ka and brezzi theories summary some observations are made on abstract error estimates for galerkin approximations based on babuska brezzi conditions a basic error estimate due to babuska is sharpened by means of an identity that p i p for any nontrivial idempotent operator p some remarks are also made on the brezzi s theory for mixed variational problems and their galerkin approximations

a primal dpg method without a first order reformulation abstract we show that it is possible to apply the dpg methodology without reformulating a second order boundary value problem into a first order system by considering the simple example of the poisson equation the result is a new weak formulation and a new dpg method for the poisson equation which has no numerical trace variable but has a numerical flux approximation on the element interfaces in addition to the primal interior variable

an adaptive dpg method for high frequency time harmonic wave propagation problems abstract the discontinuous petrov galerkin dpg method for high frequency wave propagation problems is discussed the dpg method with its attractive uniform mesh and wavenumber independent pre asymptotic stability property allows for a fully automatic adaptive h p algorithm that can be initiated from very coarse meshes moreover dpg always delivers a hermitian positive definite system suggesting the use of the conjugate gradient algorithm for its solution we present a new iterative solution scheme which capitalizes on these attractive properties of dpg this novel solver is integrated within the adaptive procedure by constructing a two grid like preconditioner for the conjugate gradient method that exploits information from previous meshes the construction of our preconditioner is discussed and its efficacy is illustrated with an example of a 2d acoustics problem our results show that the proposed iterative algorithm converges at a rate independent of the mesh and the wavenumber

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