on the p and hp extension of nedelec s curl conforming elements abstract we establish interpolation error estimates for the hp extension of nedelec s curl and divergence conforming three dimensional finite elements using these interpolation estimates we investigate the use of higher order edge finite elements for the linear three dimensional magnetostatic problem our estimates show that the standard mixed method for the magnetostatic problem can be approximated to optimal order in both h and p

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

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

error estimates for the combinedh andp versions of the finite element method in theh version of the finite element method convergence is achieved by refining the mesh while keeping the degree of the elements fixed on the other hand thep version keeps the mesh fixed and increases the degree of the elements in this paper we prove estimates showing the simultaneous dependence of the order of approximation on both the element degrees and the mesh in addition it is shown that a proper design of the mesh and distribution of element degrees lead to a better than polynomial rate of convergence with respect to the number of degrees of freedom even in the presence of corner singularities numerical results comparing theh version p version and combinedh p version for a one dimensional problem are presented

connected components of prym eigenform loci in genus three this paper is devoted to the classification of connected components of prym eigenform loci in the strata mathcal h 2 2 mathrm odd and mathcal h 1 1 2 of the bundle of abelian differentials omega mathcal m 3 over mathcal m 3 these loci discovered by mcmullen duke math j 133 569 590 2006 are mathrm gl 2 mathbb r invariant submanifolds of complex dimension 3 of omega mathcal m g that project to the locus of riemann surfaces whose jacobian variety has a factor admitting real multiplication by some quadratic order mathcal o d these algebraic varieties are not necessarily irreducible the main result we show is that for each discriminant d the corresponding locus has one component if d equiv 0 4 text mod 8 two components if d equiv 1 text mod 8 and is empty if d equiv 5 text mod 8 our result contrasts with the case of prym eigenform loci in the strata mathcal h 1 1 in genus 2 studied by mcmullen ann math 2 165 2 397 456 2007 which is connected for every discriminant d

hp adaptive finite elements in electromagnetics a model problem for the steady state form of maxwell s equations is considered the problem is formulated in a weak form using a lagrange multiplier laying down a foundation for a general class of novel hp adaptive fe approximations a convergence proof for curvilinear elements is presented the proposed method is illustrated and verified by a series of 2d experiments which include elements with curved boundaries and nonhomogeneous media

on the implementation of mixed methods as nonconforming methods for second order elliptic problems in this paper we show that mixed finite element methods for a fairly general second order elliptic problem with variable coefficients can be given a nonmixed formulation lower order terms are treated so our results apply also to parabolic equations we define an approximation method by incorporating some projection operators within a standard galerkin method which we call a projection finite element method it is shown that for a given mixed method if the projection method s finite element space m h satisfies three conditions then the two approximation methods are equivalent these three conditions can be simplified for a single element in the case of mixed spaces possessing the usual vector projection operator we then construct appropriate nonconforming spaces m h for the known triangular and rectangular elements the lowest order raviart thomas mixed solution on rectangular finite elements in r 2 and r 3 on simplices or on prisms is then implemented as a nonconforming method modified in a simple and computationally trivial manner this new nonconforming solution is actually equivalent to a postprocessed version of the mixed solution a rearrangement of the computation of the mixed method solution through this equivalence allows us to design simple and optimal order multigrid methods for the solution of the linear system

source coding and graph entropies a sender wants to accurately convey information to a receiver who has some possibly related data we study the expected number of bits the sender must transmit for one and for multiple instances in two communication scenarios and relate this number to the chromatic and korner 1973 entropies of a naturally defined graph

the method of types information theory the method of types is one of the key technical tools in shannon theory and this tool is valuable also in other fields in this paper some key applications are presented in sufficient detail enabling an interested nonspecialist to gain a working knowledge of the method and a wide selection of further applications are surveyed these range from hypothesis testing and large deviations theory through error exponents for discrete memoryless channels and capacity of arbitrarily varying channels to multiuser problems while the method of types is suitable primarily for discrete memoryless models its extensions to certain models with memory are also discussed

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

an hp error analysis of mitc plate elements we give an hp error analysis of several rectangular families of finite elements for the reissner mindlin plate bending equations we consider both the original mitc families k j bathe f brezzi and m fortin internat j numer methods engrg 28 1989 pp 1787 1801 and some new ones introduced in this paper for the deflection and rotation we give error estimates which are optimal with respect to the mesh size h and optimal up to ordo p varepsilon varepsilon 0 arbitrary with respect to the polynomial degree p we also obtain estimates for the error in the shear force calculated via two different methods our analysis utilizes some recent results of ours for the mixed method for the stokes problem as well as hp interpolation estimates for mixed methods for second order elliptic equations in this regard we derive new hp results in this paper for the brezzi douglas fortin marini spaces and improve upon previous estimates for the brezzi douglas marini spaces

inf sup conditions for twofold saddle point problems necessary and sufficient conditions for existence and uniqueness of solutions are developed for twofold saddle point problems which arise in mixed formulations of problems in continuum mechanics this work extends the classical saddle point theory to accommodate nonlinear constitutive relations and the twofold saddle structure application to problems in incompressible fluid mechanics employing symmetric tensor finite elements for the stress approximation is presented

multigrid in a weighted space arising from axisymmetric electromagnetics consider the space of two dimensional vector functions whose components and curl are square integrable with respect to the degenerate weight given by the radial variable this space arises naturally when modeling electromagnetic problems under axial symmetry and performing a dimension reduction via cylindrical coordinates we prove that if the original three dimensional domain is convex then the multigrid v cycle applied to the inner product in this space converges provided certain modern smoothers are used for the convergence analysis we first prove several intermediate results e g the approximation properties of a commuting projector in weighted norms and a superconvergence estimate for a dual mixed method in weighted spaces the uniformity of the multigrid convergence rate with respect to mesh size is then established theoretically and illustrated through numerical experiments

analysis of variable degree hdg methods for convection diffusion equations part ii semimatching nonconforming meshes in this paper we provide a projection based analysis of the hversion of the hybridizable discontinuous galerkin methods for convectiondiffusion equations on semimatching nonconforming meshes made of simplexes the degrees of the piecewise polynomials are allowed to vary from element to element we show that for approximations of degree k on all elements the order of convergence of the error in the diffusive flux is k 1 and that that of a projection of the error in the scalar unknown is 1 for k 0 and k 2 for k 0 we also show that for the variable degree case the projection of the error in the scalar variable is h times the projection of the error in the vector variable provided a simple condition is satisfied for the choice of the degree of the approximation on the elements with hanging nodes these results hold for any bounded irregularity index of the nonconformity of the mesh moreover our analysis can be extended to hypercubes

de rham diagram for hp finite element spaces abstract we prove that the hp finite elements for h curl spaces introduced in 1 fit into a general de rham diagram involving hp approximations the corresponding interpolation operators generalize the notion of hp interpolation introduced in 2 and are different from the classical operators of nedelec and raviart thomas

a combined mixed and discontinuous galerkin method for compressible miscible displacement problem in porous media in this paper we present a numerical scheme for solving the coupled system of compressible miscible displacement problem in porous media the flow equation is solved by the mixed finite element method and the transport equation is approximated by a discontinuous galerkin method the scheme is continuous in time and a priori hp error estimates is presented

hp approximation theory for bdfm and rt finite elements on quadrilaterals we study approximation properties of hp finite element subspaces of boldsymbol mathsf h mathop rm div omega and boldsymbol mathsf h mathop rm rot omega on a polygonal domain omega using brezzi douglas fortin marini bdfm or raviart thomas rt elements approximation theoretic results are derived for the hp version finite element method on non quasi uniform meshes of quadrilateral elements with hanging nodes for functions belonging to weighted sobolev spaces boldsymbol mathsf h omega s ell omega and the countably normed spaces pmb cal b w ell omega these results culminate in a proof of the characteristic exponential convergence property of the hp version finite element method on suitably designed meshes under similar conditions needed for the analysis of the boldsymbol mathsf h 1 omega case by way of illustration exponential convergence rates are deduced for mixed hp approximation of flow in porous media

analysis for quadrilateral mitc elements for the reissner mindlin plate problem the present paper is made up of two parts in the first part we study the mathematical stability and convergence of the quadrilateral mitc elements for the reissner mindlin plate problem in an abstract setting we generalize the brezzi bathe fortin conditions to the quadrilateral mitc elements by weakening the second and fourth conditions under these conditions we show the well posedness of the discrete problem and establish an abstract error estimate in the energy norm the conclusion of this part is sparsity in the mathematical research of the quadrilateral mitc elements in the sense that one only needs to check these five conditions in the second part we extend four families of rectangular mitc elements of stenberg and suri to the quadrilateral meshes we prove that these quadrilateral elements satisfy the generalized brezzi bathe fortin conditions from the first part we develop the h p error estimates in both energy and l 2 norm for these quadrilateral elements for the first three families of quadrilateral elements the error estimates indicate that their convergent rates in both energy and l 2 norm depend on the mesh distortion parameter a we can get optimal error estimates for them provided that a 1 in addition we show the optimal convergence rates in energy norm uniformly in a for the fourth family of quadrilateral elements like their rectangular counterparts these quadrilateral elements are locking free

modeling of electromagnetic absorption scattering problems using hp adaptive finite elements a model problem for the steady state form of maxwell s equations is considered the problem is formulated in a weak form using a lagrange multiplier laying down a foundation for a general class of novel hp adaptive fe approximations a convergence proof for affine elements is presented the proposed method is illustrated and verified with a series of 2d experiments including elements with curved boundaries and nonhomogeneous media

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

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

maxwell eigenvalues and discrete compactness in two dimensions abstract we present an elementary proof of the discrete compactness result for a general class of hp finite elements introduced in 1 2 we discuss h convergence of 2d elements only and in this context the results are not new as the analysis of h curl conforming elements for maxwell s equations can be reduced to the long known results for raviart thomas elements 3 the work is based on the result of kikuchi 4 5 for nedelec s edge triangular elements of the lowest order and presents an alternative to techniques presented in 3 6 in particular the present version does not use an inverse inequality argument and therefore is valid for h adaptive meshes we conclude the presentation with a number of 2d computational experiments including nonconvex domains

a multiscale hybrid method for darcy s problems using mixed finite element local solvers abstract multiscale hybrid mixed mhm method refers to a numerical technique targeted to approximate systems of differential equations with strongly varying solutions for fluid flow normal fluxes multiplier over macro element boundaries and coarse piecewise constant potential approximations in each macro element are computed upscaling then small details are resolved by local problems using fine representations inside the macro elements setting the multiplier as neumann boundary conditions downscaling in this work a variant of the method is developed denoted by mhm h div adopting mixed finite elements at the downscaling stage instead of continuous finite elements used in all previous publications of the method thus this alternative mhm method inherits improvements typical of mixed methods as better flux accuracy and local mass conservation at the micro scale level inside the macro elements which are important properties for multi phase flows in rough heterogeneous media different two scale stable space settings are considered vector face functions are supposed to have normal components restricted to a given finite dimensional trace space defined over the macro element boundaries in each macro element the internal flux components with vanishing normal traces and the potential approximations may be enriched in different extents with respect to internal mesh size internal polynomial degree or both the choice being determined by the problem at hands a unified general error analysis of the mhm h div method is presented for all these two scale space scenarios both mhm versions are compared for 2d test problems with smooth solutions for convergence rates verification and for darcy s flow in heterogeneous media mhm h div 3d simulations are presented for a known singular darcy s solution using adaptive macro partitions and for an oscillatory permeability scenario

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

dilworth rate a generalization of witsenhausen s zero error rate for directed graphs we investigate a communication setup where a source output is sent through a free noisy channel first and an additional codeword is sent through a noiseless but expensive channel later with the help of the second message the decoder should be able to decide with zero error whether its decoding of the first message was error free this scenario leads to the definition of a digraph parameter that generalizes witsenhausen s zero error rate for directed graphs we investigate this new parameter for some specific directed graphs and explore its relations to other digraph parameters such as sperner capacity and dichromatic number we also look at the natural variant of the above problem where the decoder should decode the first message with zero error not only decide whether its earlier decoding was correct in this case the witsenhausen rate of an appropriately defined undirected graph turns out to be the relevant parameter

connected components of the moduli spaces of abelian differentials with prescribed singularities consider the moduli space of pairs c where c is a smooth compact complex curve of a given genus and is a holomorphic 1 form on c with a given list of multiplicities of zeroes we describe connected components of this space this classification is important in the study of dynamics of interval exchange transformations and billiards in rational polygons and in the study of geometry of translation surfaces

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

dynamics of sl2 ℝ over moduli space in genus two this paper classifies orbit closures and invariant measures for the natural action of sl2 r on um2 the bundle of holomorphic 1 forms over the moduli space of riemann surfaces of genus two

three dimensional hierarchical mixed finite element approximations with enhanced primal variable accuracy abstract there are different possibilities of choosing balanced pairs of approximation spaces for dual flux and primal pressure variables to be used in discrete versions of the mixed finite element method for elliptic problems arising in fluid simulations three cases shall be studied and compared for discretized three dimensional formulations based on tetrahedral hexahedral and prismatic meshes the principle guiding the constructions is the property that the divergence of the dual space and the primal approximation space should coincide while keeping the same order of accuracy for the flux variable and varying the accuracy order of the primal variable there is the classic case of b d m k spaces based on tetrahedral meshes and polynomials of total degree k for the dual variable and k 1 for the primal variable showing stable simulations with optimal convergence rates of orders k 1 and k respectively another case is related to r t k and b d f m k 1 spaces for hexahedral and tetrahedral meshes respectively but holding for prismatic elements as well it gives identical approximation order k 1 for both primal and dual variables an improvement in accuracy obtained by increasing the degree of primal functions to k and by enriching the dual space with some properly chosen internal shape functions of degree k 1 while keeping degree k for the border fluxes a new type of approximation is proposed by further incrementing the order of some internal flux functions to k 2 and matching primal functions to k 1 higher than the border fluxes of degree k thus higher convergence rate of order k 2 is obtained for the primal variable using static condensation the global condensed system to be solved in all the cases has same dimension and structure which is proportional to the space dimension of the border fluxes for each element geometry illustrating results comparing the three different space configurations are presented for simulations based on hierarchical high order shape functions for h div conforming spaces which are specially constructed for affine tetrahedral hexahedral and prismatic meshes expected convergence rates are obtained for the flux pressure and divergence variables

the zero undetected error capacity approaches the sperner capacity ahlswede cai and zhang proved that in the noise free limit the zero undetected error capacity is lower bounded by the sperner capacity of the channel graph and they conjectured equality here we derive an upper bound that proves the conjecture

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

graph capacities and zero error transmission over compound channels this correspondence investigates the behavior of the compound channel under a zero error constraint we derive expressions for the capacity when a neither the encoder nor decoder has side information about the channel b when only the encoder has such side information these expressions are given in terms of capacities of appropriately defined sets of graphs we clarify that an earlier treatment of the zero error capacity of a compound channel corresponds to the case where the decoder has side information about the channel we also characterize the minimum asymptotic rate for the source coding dual of the compound channel problem namely source coding with compound side information finally we contrast the zero error and asymptotically vanishing error capacities of the compound channel

prym varieties and teichmuller curves this paper gives a uniform construction of infinitely many primitive teichmuller curves v mg for g 2 3 and 4

the treatment of nonhomogeneous dirichlet boundary conditions by thep version of the finite element method the paper addresses the problem of the implementation of nonhomogeneous essential dirichlet type boundary conditions in thep version of the finite element method

splitting mixed hodge structures over affine invariant manifolds we prove that affine invariant manifolds in strata of flat surfaces are algebraic varieties the result is deduced from a generalization of a theorem of moller namely we prove that the image of a certain twisted abel jacobi map lands in the torsion of a factor of the jacobians this statement can be viewed as a splitting of certain mixed hodge structures

the hp mitc finite element method for the reissner mindlin plate problem the popular mitc finite elements used for the approximation of the reissner mindlin plate are extended to the case where elements of non uniform degree p distribution are used on locally refined meshes such an extension is of particular interest to the hp version and hp adaptive finite element methods a priori error bounds are provided showing that the method is locking free the analysis is based on new approximation theoretic results for non uniform brezzi douglas fortin marini spaces and extends the results obtained in the case of uniform order approximation on globally quasi uniform meshes presented by stenberg and suri siam j numer anal 34 1997 544 numerical examples illustrating the theoretical results and comparing the performance with alternative standard galerkin approaches are presented for two new benchmark problems with known analytic solution including the case where the shear stress exhibits a boundary layer the new method is observed to be locking free and able to provide exponential rates of convergence even in the presence of boundary layers

the optimal convergence of the h p version of the finite element method with quasi uniform meshes in the framework of the jacobi weighted besov spaces we analyze the convergence of the h p version of finite element solutions on quasi uniform meshes and the lower and upper bounds of errors for elliptic problems on polygons both lower and upper bounds are proved to be optimal in h and p which leads to the optimal convergence of the h p version of the finite element method with quasi uniform meshes for elliptic problems on polygons the results proved for the h p version include the h version with quasi uniform meshes and the p version with quasi uniform degrees as two special cases

on computing the pressure by thep version of the finite element method for stokes problem this paper introduces and analyzes two ways of extracting the hydrostatic pressure when solving stokes problem using thep version of the finite element method when one uses a localh 1 projection we show that optimal rates of convergence for the pressure approximation is achieved when the pressure is not inh 1 or the value of the pressure is only needed at a few points one may extract the pressure pointwise using e g a single layer potential recovery negative zero and higher norm estimates for the stokes velocity are derived within the framework of thep version of the f e m

second order nedelec curl conforming prismatic element for computational electromagnetics a systematic approach to obtaining mixed order curl conforming basis functions for a triangular prism is presented focus is made on the second order case space of functions for the prism is given basis functions are obtained as the dual basis with respect to suitably discretized nedelec degrees of freedom functionals acting on elements of the space thus the linear independence of the basis functions is assured while the belonging of the basis to the a priori given space of functions is guaranteed different strategies for the finite element assembly of the basis are discussed numerical results showing the verification procedure of the correctness of the implemented basis functions are given numerical results about sensibility of the condition number of the basis obtained concerning the quality of the elements of the mesh are also shown comparison with other representative sets of basis functions for prisms is included

sparsity optimized high order finite element functions for h div on simplices this paper deals with conforming high order finite element discretizations of the vector valued function space h div in 2 and 3 dimensions a new set of hierarchic basis functions on simplices with the following two main properties is introduced when working with an affine simplicial triangulation first the divergence of the basis functions is l 2 orthogonal and secondly the l 2 inner product of the interior basis functions is sparse with respect to the polynomial degree p the construction relies on a tensor product based construction with properly weighted jacobi polynomials as well as an explicit splitting of the higher order basis functions into solenoidal and non solenoidal ones the basis is suited for fast assembling strategies the general proof of the sparsity result is done by the assistance of computer algebra software tools several numerical experiments document the proved sparsity patterns and practically achieved condition numbers for the parameter dependent div div problem even on curved elements a tremendous improvement in condition numbers is observed the precomputed mass and stiffness matrix entries in general form are available online

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

mixed finite element methods for nonlinear elliptic problems the h p version mixed finite element methods for strongly nonlinear second order elliptic problems are proposed and analyzed existence and uniqueness of the approximate solution are demonstrated using a fixed point argument convergence and stability of the method are proved both with respect to mesh refinement and increase of the degree of the approximating polynomials the analysis is carried out in detail using raviart thomas nedelec spaces as an example numerical results for minimal surface problems are obtained using brezzi douglas marini spaces graphs of the approximate solutions are presented for two different problems

a priori error analysis of stochastic galerkin mixed approximations of elliptic pdes with random data we construct stochastic galerkin approximations to the solution of a first order system of pdes with random coefficients under the standard finite dimensional noise assumption we transform the variational saddle point problem to a parametric deterministic one approximations are constructed by combining mixed finite elements on the computational domain with m variate tensor product polynomials we study the inf sup stability and well posedness of the continuous and finite dimensional problems the regularity of solutions with respect to the m parameters describing the random coefficients and establish a priori error estimates for stochastic galerkin finite element approximations

full wave analysis of dielectric waveguides at a given frequency new variational formulation to compute propagation constants is proposed based on it vector finite element method is proved to exclude spurious modes provided finite elements possess discrete compactness property convergence analysis is conducted in the framework of collectively compact operators reported theoretical results apply to a wide class of vector finite elements including two families of nedelec and their generalization the hp edge elements numerical experiments fully support theoretical estimates for convergence rates

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

a new family of mixed finite elements in ir 3 we introduce two families of mixed finite element on conforming inh div and one conforming inh curl these finite elements can be used to approximate the stokes system

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

the p version of the finite element method in the p version of the finite element method the triangulation is fixed and the degree p of the piecewise polynomial approximation is progressively increased until some desired level of precision is reached in this paper we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation these properties lead to an a priori estimate of the asymptotic rate of convergence of the p version the estimate shows that the p version gives results which are not worse than those obtained by the conventional finite element method called the h version in which h represents the maximum diameter of the elements when quasi uniform triangulations are employed and the basis for comparison is the number of degrees of freedom furthermore in the case of a singularity problem we show under conditions which are usually satisfied in practice that the rate of convergence of the p version is twice that of the h version with quasi uniform mesh inve

the p and h p versions of the finite element method basic principles and properties in the classical form of the finite element method called the hversion piecewise polynomials of fixed degree p are used and the mesh size h is decreased for accuracy in this paper we discuss the fundamental theoretical ideas behind the relatively recent p version and h p version in the p version a fixed mesh is used and p is allowed to increase the h p version combines both approaches the authors describe and explain the basic properties and characteristics of these newer versions especially in areas where their behavior is significantly different from that of the h version simplified proofs of key concepts are included and computational illustrations of several results are provided a benchmark comparison between the various versions in included

variations of hodge structures of a teichmuller curve teichmuller curves are geodesic discs in teichmuller space that project to an algebraic curve in the moduli space mg we show that for all g 2 teichmuller curves map to the locus of real multiplication in the moduli space of abelian varieties observe that mcmullen has shown that precisely for g 2 the locus of real multiplication is stable under the sl2 r action on the tautological bundle mg we also show that teichmuller curves are defined over number fields and we provide a completely algebraic description of teichmuller curves in terms of higgs bundles as a consequence we show that the absolute galois group acts on the set of teichmuller curves

a remark concerning divergence accuracy order for h div conforming finite element flux approximations abstract the construction of finite element approximations in h div usually requires the piola transformation to map vector polynomials from a master element to vector fields in the elements of a partition of the region it is known that degradation may occur in convergence order if non affine geometric mappings are used on this point we revisit a general procedure for the improvement of two dimensional flux approximations discussed in a recent paper of this journal comput math appl 74 2017 3283 3295 the starting point is an approximation scheme which is known to provide l 2 errors with accuracy of order k 1 for sufficiently smooth flux functions and of order r 1 for flux divergence an example is r t k spaces on quadrilateral meshes where r k or k 1 if linear or bilinear geometric isomorphisms are applied furthermore the original space is required to be expressed by a factorization in terms of edge and internal shape flux functions the goal is to define a hierarchy of enriched flux approximations to reach arbitrary higher orders of divergence accuracy r n 1 as desired for any n 1 the enriched versions are defined by adding higher degree internal shape functions of the original family of spaces at level k n while keeping the original border fluxes at level k the case n 1 has been discussed in the mentioned publication for two particular examples general stronger enrichment n 1 shall be analyzed and applied to darcy s flow simulations the global condensed systems to be solved having same dimension and structure of the original scheme

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

on the stability and convergence of higher order mixed finite element methods for second order elliptic problems we investigate the use of higher order mixed methods for second order elliptic problems by establishing refined stability and convergence estimates which take into account both the mesh size h and polynomial degree p our estimates yield asymptotic convergence rates for the p and h p versions of the finite element method they also describe more accurately than previously proved estimates the increased rate of convergence expected when the h version is used with higher order polynomials

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

optimal error estimation for bfh rmcurl conforming p interpolation in two dimensions in this paper we prove an optimal error estimate for the bf h rm curl conforming projection based p interpolation operator introduced in l demkowicz and i babuska siam j numer anal 41 2003 pp 1195 1208 this result is proved on the reference element either triangle or square k for regular vector fields in bf h r rm curl k with arbitrary r 0 the formulation of the result in the bf h rm div conforming setting which is relevant for the analysis of high order boundary element approximations for maxwell s equations is provided as well

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

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

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

billiards and teichmuller curves on hilbert modular surfaces this paper exhibits an infinite collection of algebraic curves isometrically embedded in the moduli space of riemann surfaces of genus two these teichmuller curves lie on hilbert modular surfaces parameterizing abelian varieties with real multiplication explicit examples constructed from l shaped polygons give billiard tables with optimal dynamical properties