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 hybridizable discontinuous galerkin method for stokes flow abstract in this paper we introduce a hybridizable discontinuous galerkin method for stokes flow the method is devised by using the discontinuous galerkin methodology to discretize a velocity pressure gradient formulation of the stokes system with appropriate choices of the numerical fluxes and by applying a hybridization technique to the resulting discretization one of the main features of this approach is that it reduces the globally coupled unknowns to the numerical trace of the velocity and the mean of the pressure on the element boundaries thereby leading to a significant reduction in the size of the resulting matrix moreover by using an augmented lagrangian method the globally coupled unknowns are further reduced to the numerical trace of the velocity only another important feature is that the approximations of the velocity pressure and gradient converge with the optimal order of k 1 in the l 2 norm when polynomials of degree k 0 are used to represent the approximate variables based on the optimal convergence of the hdg method we apply an element by element postprocessing scheme to obtain a new approximate velocity which converges with order k 2 in the l 2 norm for k 1 the postprocessing performed at the element level is less expensive than the solution procedure numerical results are provided to assess the performance of the method

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

enumerating representations in finite wreath products abstract let be a group finite or infinite h a finite group and let rn denote the sequence h sn of symmetric wreath products as well as certain variants of it including in particular h an and wn the weyl group of type dn we compute the exponential generating function for the number hom rn of representations in rn and for some refinements of this sequence under very mild finiteness assumptions on always met for instance if is finitely generated this generalizes in a uniform way the connection between the problem of counting finite index subgroups in a group and the enumeration of actions on finite sets on the one hand and the recent results of chigira concerning solutions of the equation xm 1 in the groups h sn h an and wn on the other we also study the asymptotics of the function hom g h sn for arbitrary finite groups g and h

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

enumerating representations in finite wreath products ii explicit formulae abstract we present several contributions to the enumerative theory of wreath product representations developed in a previous paper by the first named author adv in math 153 2000 118 154 theorem 3 1 of the present paper establishes an explicit formula for one of the key ingredients in the description of the corresponding generating functions given in muller 2000 the exterior function building on theorem 1 in muller 2000 and the latter result we derive explicit formulae for the exponential generating function of the series hom r n in the case where is dihedral or a finite abelian group and the representation sequence r n is any of h sn or h an with a fixed finite group h or the sequence hom g wn moreover we verify a conjecture concerning the asymptotic behaviour of the sequence hom g wn for finite groups g made in muller 2000 in the case when g is dihedral or abelian

on 2d elliptic discontinuous galerkin methods we discuss the discretization using discontinuous galerkin dg formulation of an elliptic poisson problem two commonly used dg schemes are investigated the original average flux proposed by bassi and rebay j comput phys 1997 131 267 and the local discontinuous galerkin ldg siam j numer anal 1998 35 2440 2463 scheme in this paper we expand on previous expositions discontinuous galerkin methods theory computation and applications springer berlin 2000 135 146 siam j sci comput 2002 24 2 524 547 int j numer meth engng 2003 58 2 1119 1148 by adopting a matrix based notation with a view to highlighting the steps required in a numerical implementation of the dg method through consideration of standard c0 type expansion bases as opposed to elementally orthogonal expansions with the matrix formulation we are able to apply static condensation techniques to improve efficiency of the direct solver when high order expansions are adopted the use of c0 type expansions also permits the direct enforcement of dirichlet boundary conditions through a lifting approach where the ldg flux does not require further stabilization in our construction we also adopt a formulation of the continuous dg fluxes that permits a more general interpretation of their numerical implementation in particular it allows us to determine the conditions under which the ldg method provides a near local stencil finally a study of the conditioning and the size of the null space of the matrix systems resulting from the dg discretization of the elliptic problem is undertaken copyright 2005 john wiley sons ltd

the hp local discontinuous galerkin method for low frequency time harmonic maxwell equations the local discontinuous galerkin method for the numerical approximation of the time harmonic maxwell equations in a low frequency regime is introduced and analyzed topologically nontrivial domains and heterogeneous media are considered containing both conducting and insulating materials the presented method involves discontinuous galerkin discretizations of the curl curl and grad div operators derived by introducing suitable auxiliary variables and so called numerical fluxes an hp analysis is carried out and error estimates that are optimal in the meshsize h and slightly suboptimal in the approximation degree p are obtained

an implicit high order hybridizable discontinuous galerkin method for the incompressible navier stokes equations we present an implicit high order hybridizable discontinuous galerkin method for the steady state and time dependent incompressible navier stokes equations the method is devised by using the discontinuous galerkin discretization for a velocity gradient pressure velocity formulation of the incompressible navier stokes equations with a special choice of the numerical traces the method possesses several unique features which distinguish itself from other discontinuous galerkin methods first it reduces the globally coupled unknowns to the approximate trace of the velocity and the mean of the pressure on element boundaries thereby leading to a significant reduction in the degrees of freedom moreover if the augmented lagrangian method is used to solve the linearized system the globally coupled unknowns become the approximate trace of the velocity only second it provides for smooth viscous dominated problems approximations of the velocity pressure and velocity gradient which converge with the optimal order of k 1 in the l2 norm when polynomials of degree k 0 are used for all components of the approximate solution and third it displays superconvergence properties that allow us to use the above mentioned optimal convergence properties to define an element by element postprocessing scheme to compute a new and better approximate velocity indeed this new approximation is exactly divergence free h div conforming and converges with order k 2 for k 1 and with order 1 for k 0 in the l2 norm moreover a novel and systematic way is proposed for imposing boundary conditions for the stress viscous stress vorticity and pressure which are not naturally associated with the weak formulation of the method this can be done on different parts of the boundary and does not result in the degradation of the optimal order of convergence properties of the method extensive numerical results are presented to demonstrate the convergence and accuracy properties of the method for a wide range of reynolds numbers and for various polynomial degrees

the relationship between terminal state constraints and penalties for the discrete time lqp problem associated with the adjustment of accelerometer data abstract one of the methods of adjusting and integrating acceleration data is to cast it within the discrete time optimal control framework of minimizing the sum of squares of adjustments here regarded as controls subject to the linear dynamics of double integration excessive deviations in terminal velocity and displacement the states could be avoided by adding to the sum of squares a terminal penalty weighting such deviations in this paper a procedure using terminal state constraints is developed as an alternative to terminal penalties to prevent drift in the terminal states the explicit relationship between the controls for these two alternatives are derived in particular expressions for the dependence of the controls on terminal weights are obtained and used to account for the insensitivity of the state and control trajectories to certain weights while the focus of this paper is on the dynamics associated with the double integration of acceleration it is shown that the approach has wider application to other lqp problems

a hybridizable discontinuous galerkin method combined to a schwarz algorithm for the solution of 3d time harmonic maxwell s equation a schwarz type domain decomposition method is presented for the solution of the system of 3d time harmonic maxwell s equations we introduce a hybridizable discontinuous galerkin hdg scheme for the discretization of the problem based on a tetrahedrization of the computational domain the discrete system of the hdg method on each subdomain is solved by an optimized sparse direct lu factorization solver the solution of the interface system in the domain decomposition framework is accelerated by a krylov subspace method the formulation and the implementation of the resulting dd hdg domain decomposed hybridizable discontinuous galerkin method are detailed numerical results show that the resulting dd hdg solution strategy has an optimal convergence rate and can save both cpu time and memory cost compared to a classical upwind flux based dd dg domain decomposed discontinuous galerkin approach

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

nodal high order methods on unstructured grids we present a convergent high order accurate scheme for the solution of linear conservation laws in geometrically complex domains as our main example we include a detailed development and analysis of a scheme for the time domain solution of maxwell s equations in a three dimensional domain the fully unstructured spatial discretization is made possible by the use of a high order nodal basis employing multivariate lagrange polynomials defined on the triangles and tetrahedra while the equations themselves are satisfied in a discontinuous galerkin form with the boundary conditions being enforced weakly through a penalty term accuracy stability and convergence of the semidiscrete approximation to maxwell s equations is established rigorously and bounds on the growth of the global divergence error are provided concerns related to efficient implementations are discussed in detail this sets the stage for the presentation of examples verifying the theoretical results and illustrating the versatility flexibility and robustness when solving two and three dimensional benchmark problems in computational electromagnetics pure scattering as well as penetration is discussed and high parallel performance of the scheme is demonstrated

the enumeration of coverings of closed orientable euclidean manifolds g3 and g5 abstract there are only 10 euclidean forms that is flat closed three dimensional manifolds six are orientable and four are non orientable the aim of this paper is to describe all types of n fold coverings over the orientable euclidean manifolds g 3 and g 5 and calculate the numbers of non equivalent coverings of each type the manifolds g 3 and g 5 are uniquely determined among other forms by their homology groups h 1 g 3 z 3 z and h 1 g 5 z we classify subgroups in the fundamental groups 1 g 3 and 1 g 5 up to isomorphism given index n we calculate the numbers of subgroups and the numbers of conjugacy classes of subgroups for each isomorphism type and provide the dirichlet generating functions for the above sequences

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

a priori and computable a posteriori error estimates for an hdg method for the coercive maxwell equations abstract in this paper we present and analyze a hybridizable discontinuous galerkin hdg method for a mixed curl curl formulation of the steady state coercive maxwell equations with a discrete sobolev embedding type estimates for the discontinuous polynomials we provide a priori error estimates for the electric field and the lagrange multiplier in the energy norm with the smooth or minimal regularity assumption on the exact solution we have optimal convergence rate for the electric field and the lagrange multiplier in the energy norm the a priori error estimate for the electric field in the l 2 norm is also obtained by the duality argument and the approximation is also optimal for the electric field in the l 2 norm moreover by employing suitable helmholtz decompositions of the error together with the upper bound estimate for the lagrange multiplier we provide a computable residual based a posteriori error estimator which is derived based on the error measured in terms of a mesh dependent energy norm the efficiency of the a posteriori error estimator is also established three dimensional numerical results testing the performance of the a priori and a posteriori error estimates for the maxwell equations are given

hierarchal vector basis functions of arbitrary order for triangular and tetrahedral finite elements new vector finite elements are proposed for electromagnetics the new elements are triangular or tetrahedral edge elements tangential vector elements of arbitrary polynomial order they are hierarchal so that different orders can be used together in the same mesh and p adaption is possible they provide separate representation of the gradient and rotational parts of the vector field explicit formulas are presented for generating the basis functions to arbitrary order the basis functions can be used directly or after a further stage of partial orthogonalization to improve the matrix conditioning matrix assembly for the frequency domain curl curl equation is conveniently carried out by means of universal matrices application of the new elements to the solution of a parallel plate waveguide problem demonstrates the expected convergence rate of the phase of the reflection coefficient for tetrahedral elements to order 4 in particular the full order elements have only the same asymptotic convergence rate as elements with a reduced gradient space such as the whitney element however further tests reveal that the optimum balance of the gradient and rotational components is problem dependent

p adic estimates of the number of permutation representations abstract let p be a prime and let g be a finite abelian p group of type 1 2 with 1 2 and i s set u max 1 s 1 2 and v s u for each nonnegative integer n let h n g be the number of homomorphisms from g to the symmetric group s n on n letters except for the case where p 2 and u v 0 v 1 the kronecker delta or p 3 and u v 1 there exist p adic analytic functions f r x for r 0 1 p u 1 1 and a polynomial x with integer coefficients such that for any nonnegative integer y h p u 1 y r g p j 1 u p j u v y f r y j 1 y j and ord p h p u 1 y r g j 1 u p j u v y ord p f r y if p 2 3 0 and u v 1 or if p 3 and u v 1 then h n g has analogous properties under the assumption that 3 0 some results for the number of permutation representations of g in the wreath product of a cyclic group of order p with s n are also presented

a hybridizable discontinuous galerkin method for steady state convection diffusion reaction problems in this article we propose a novel discontinuous galerkin method for convection diffusion reaction problems characterized by three main properties the first is that the method is hybridizable this renders it efficiently implementable and competitive with the main existing methods for these problems the second is that when the method uses polynomial approximations of the same degree for both the total flux and the scalar variable optimal convergence properties are obtained for both variables this is in sharp contrast with all other discontinuous methods for this problem the third is that the method exhibits superconvergence properties of the approximation to the scalar variable this allows us to postprocess the approximation in an element by element fashion to obtain another approximation to the scalar variable which converges faster than the original one in this paper we focus on the efficient implementation of the method and on the validation of its computational performance with this aim e

hierarchic finite element bases on unstructured tetrahedral meshes the problem of constructing hierarchic bases for finite element discretization of the spaces h1 h curl h div and l2 on tetrahedral elements is addressed a simple and efficient approach to ensuring conformity of the approximations across element interfaces is described hierarchic bases of arbitrary polynomial order are presented it is shown how these may be used to construct finite element approximations of arbitrary non uniform local order approximation on unstructured meshes of curvilinear tetrahedral elements

hdg pod reduced order model of the heat equation abstract we propose a new hybridizable discontinuous galerkin hdg model order reduction technique based on proper orthogonal decomposition pod we consider the heat equation as a test problem and prove error bounds that converge to zero as the number of pod modes increases we present 2d and 3d numerical results to illustrate the convergence analysis

classification and statistics of finite index subgroups in free products abstract the principal theme of this paper is the enumeration of finite index subgroups in a free product of finite groups under various restrictions on the isomorphism type of in particular we completely resolve the realization asymptotic and distribution problems for free products of cyclic groups of prime order prior to this work these questions were wide open even in the case of the classical modular group this complex of problems usually referred to as poincare klein problem originally arose around 1880 out of the work of klein and poincare on automorphic functions and related number theory but has also grown roots in geometric function theory and more recently in the theory of subgroup growth ideas and techniques from the theory of generalized permutation representations an enumerative theory of wreath product representations recently developed by the first named author play a fundamental role here other tools come from analytic number theory combinatorics and probability theory

hybridizable discontinuous galerkin projection methods for navier stokes and boussinesq equations schemes for the incompressible navier stokes and boussinesq equations are formulated and derived combining the novel hybridizable discontinuous galerkin hdg method a projection method and implicit explicit runge kutta imex rk time integration schemes we employ an incremental pressure correction and develop the corresponding hdg finite element discretization including consistent edge space fluxes for the velocity predictor and pressure correction we then derive the proper forms of the element local and hdg edge space final corrections for both velocity and pressure including the hdg rotational correction we also find and explain a consistency relation between the hdg stability parameters of the pressure correction and velocity predictor we discuss and illustrate the effects of the time splitting error we then detail how to incorporate the hdg projection method time split within standard imex rk time stepping schemes our high order hdg projection schemes are implemented for arbitrary mixed element unstructured grids with both straight sided and curved meshes in particular we provide a quadrature free integration method for a nodal basis that is consistent with the hdg method to prevent numerical oscillations we develop a selective nodal limiting approach its applications show that it can stabilize high order schemes while retaining high order accuracy in regions where the solution is sufficiently smooth we perform spatial and temporal convergence studies to evaluate the properties of our integration and selective limiting schemes and to verify that our solvers are properly formulated and implemented to complete these studies and to illustrate a range of properties for our new schemes we employ an unsteady tracer advection benchmark a manufactured solution for the steady diffusion and stokes equations and a standard lock exchange boussinesq problem

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

a phase based hybridizable discontinuous galerkin method for the numerical solution of the helmholtz equation we introduce a new hybridizable discontinuous galerkin hdg method for the numerical solution of the helmholtz equation over a wide range of wave frequencies our approach combines the hdg methodology with geometrical optics in a fashion that allows us to take advantage of the strengths of these two methodologies the phase based hdg method is devised as follows first we enrich the local approximation spaces with precomputed phases which are solutions of the eikonal equation in geometrical optics second we propose a novel scheme that combines the hdg method with ray tracing to compute multivalued solution of the eikonal equation third we utilize the proper orthogonal decomposition to remove redundant modes and obtain locally orthogonal basis functions which are then used to construct the global approximation spaces of the phase based hdg method and fourth we propose an appropriate choice of the stabilization parameter to guarantee stability and accuracy for the proposed method numerical experiments presented show that optimal orders of convergence are achieved that the number of degrees of freedom to achieve a given accuracy is independent of the wave number and that the number of unknowns required to achieve a given accuracy with the proposed method is orders of magnitude smaller than that with the standard finite element method

a discontinuous hp finite element method for convection diffusion problems this paper presents a new method which exhibits the best features of both finite volume and finite element techniques special attention is given to the issues of conservation flexible accuracy and stability the method is elementwise conservative the order of polynomial approximation can be adjusted element by element and the stability is not based on the introduction of artificial diffusion but on the use of a very particular finite element formulation with discontinuous basis functions the method supports h p and hp approximations and can be applied to any type of meshes including non matching grids a priori error estimates and numerical experiments on representative model problems indicate that the method is robust and capable of delivering high accuracy

higher order interpolatory vector bases for computational electromagnetics low order vector basis functions compatible with the nedelec 1980 representations are widely used for electromagnetic field problems higher order functions are receiving wider application but their development is hampered by the complex procedures used to generate them and lack of a consistent notation for both elements and bases in this paper fully interpolatory higher order vector basis functions of the nedelec type are defined in a unified and consistent manner for the most common element shapes it is shown that these functions can be obtained as the product of zeroth order nedelec representations and interpolatory polynomials with specially arranged arrays of interpolation points the completeness properties of the vector functions are discussed and expressions for the vector functions of arbitrary polynomial order are presented sample numerical results confirm the faster convergence of the higher order functions

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

a discontinuous galerkin method with lagrange multipliers for spatially dependent advection diffusion problems abstract a higher accuracy discontinuous galerkin method with lagrange multipliers dglm is presented for the solution of the advection diffusion equation with a spatially varying advection field in the high peclet number regime where the classical polynomial finite element method fem produces spurious oscillations in the solution at practical mesh resolutions the proposed dglm method is based on discontinuous polynomial shape functions that are attached to an element rather than its nodes it overcomes the aforementioned spurious oscillation issue by enriching these functions with approximate free space solutions of homogeneous equations derived from an asymptotic analysis of the governing partial differential equation inspired by prandtl s boundary layer theory these enrichment functions are capable of resolving exponential parabolic and corner boundary layers at relatively coarse mesh resolutions the proposed method enforces a weak continuity of the solution approximation across inter element boundaries using polynomial lagrange multipliers which makes it a hybrid method however unlike other hybrid methods it operates directly on the second order form of the advection diffusion equation and does not require any stabilization its intrinsic performance and its superiority over the higher order polynomial fem are demonstrated for several test problems at peclet numbers ranging from one thousand to one billion

superconvergent discontinuous galerkin methods for second order elliptic problems we identify discontinuous galerkin methods for second order elliptic problems in several space dimensions having superconvergence properties similar to those of the raviart thomas and the brezzi douglas marini mixed methods these methods use polynomials of degree k 0 for both the potential as well as the flux we show that the approximate flux converges in l2 with the optimal order of k 1 and that the approximate potential and its numerical trace superconverge in l2 like norms to suitably chosen projections of the potential with order k 2 we also apply element by element postprocessing of the approximate solution to obtain new approximations of the flux and the potential the new approximate flux is proven to have normal components continuous across inter element boundaries to converge in l2 with order k 1 and to have a divergence converging in l2 also with order k 1 the new approximate potential is proven to converge with order k 2 in l2 numerical experiments validating these theoretical results are presented

hybridizable discontinuous galerkin methods for the time harmonic maxwell s equations we present two hybridizable discontinuous galerkin hdg methods for the numerical solution of the time harmonic maxwell s equations the first hdg method explicitly enforces the divergence free condition and thus necessitates the introduction of a lagrange multiplier it produces a linear system for the degrees of freedom of the approximate traces of both the tangential component of the vector field and the lagrange multiplier the second hdg method does not explicitly enforce the divergence free condition and thus results in a linear system for the degrees of freedom of the approximate trace of the tangential component of the vector field only for both hdg methods the approximate vector field converges with the optimal order of k 1 in the l 2 norm when polynomials of degree k are used to represent all the approximate variables we propose elementwise postprocessing to obtain a new h c u r l conforming approximate vector field which converges with order k 1 in the h c u r l norm we present extensive numerical examples to demonstrate and compare the performance of the hdg methods

high order implicit hybridizable discontinuous galerkin methods for acoustics and elastodynamics we present a class of hybridizable discontinuous galerkin hdg methods for the numerical simulation of wave phenomena in acoustics and elastodynamics the methods are fully implicit and high order accurate in both space and time yet computationally attractive owing to their following distinctive features first they reduce the globally coupled unknowns to the approximate trace of the velocity which is defined on the element faces and single valued thereby leading to a significant saving in the computational cost in addition all the approximate variables including the approximate velocity and gradient converge with the optimal order of k 1 in the l 2 norm when polynomials of degree k 0 are used to represent the numerical solution and when the time stepping method is accurate with order k 1 when the time stepping method is of order k 2 superconvergence properties allows us by means of local postprocessing to obtain better yet inexpensive approximations of the displacement and velocity 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 when k 1 for both acoustics and elastodynamics extensive numerical results are provided to illustrate these distinctive features

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

hessian based model reduction for large scale systems with initial condition inputs reduced order models that are able to approximate output quantities of interest of high fidelity computational models over a wide range of input parameters play an important role in making tractable large scale optimal design optimal control and inverse problem applications we consider the problem of determining a reduced model of an initial value problem that spans all important initial conditions and pose the task of determining appropriate training sets for reduced basis construction as a sequence of optimization problems we show that under certain assumptions these optimization problems have an explicit solution in the form of an eigenvalue problem yielding an efficient model reduction algorithm that scales well to systems with states of high dimension furthermore tight upper bounds are given for the error in the outputs of the reduced models the reduction methodology is demonstrated for a large scale contaminant transport problem

an implicit hybridized discontinuous galerkin method for the 3d time domain maxwell equations we present a time implicit hybridizable discontinuous galerkin hdg method for numerically solving the system of three dimensional 3d time domain maxwell equations this method can be seen as a fully implicit variant of classical so called dgtd discontinuous galerkin time domain methods that have been extensively studied during the last 10 years for the simulation of time domain electromagnetic wave propagation the proposed method has been implemented for dealing with general 3d problems discretized using unstructured tetrahedral meshes we provide numerical results aiming at assessing its numerical convergence properties by considering a model problem on one hand and its performance when applied to more realistic problems we also include some performance comparisons with a centered flux time implicit dgtd method

on wohlfahrt series and wreath products abstract suppose that a group a contains only a finite number of subgroups of index d for each positive integer d let g s n be the wreath product of a finite group g with the symmetric group s n on 1 n for each positive integer n let k n be a subgroup of g s n containing the commutator subgroup of g s n if the sequence k n 0 satisfies a certain compatible condition then the exponential generating function n 0 hom a k n x n g n n of the sequence hom a k n 0 takes the form of a sum of exponential functions

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

a discontinuous galerkin method with lagrange multiplier for hyperbolic conservation laws with boundary conditions we introduce a discontinuous galerkin method with lagrange multiplier dglm to approximate the solution to the hyperbolic conservation laws with boundary conditions lagrange multipliers are introduced on the edge face of the element via weak divergence wang and ye 2014 the final global system has reduced numbers of unknowns of the standard dg methods numerical fluxes from finite volume difference method are not considered for the time discretization backward euler difference method is used stability of the approximate solution is proved in energy norm discontinuity of the solution is allowed in the error analysis local error estimates of o hr 12 t with pr e elements rd 12 are derived where h and t are the maximum diameter of the elements and time steps respectively and d is the dimension of the spatial domain the high order approximation is obtained under an appropriate condition on the stabilizing parameter it is shown that the method preserves the property of the local mass conservation an explanation on algorithmic aspects is given

a model and variance reduction method for computing statistical outputs of stochastic elliptic partial differential equations we present a model and variance reduction method for the fast and reliable computation of statistical outputs of stochastic elliptic partial differential equations our method consists of three main ingredients 1 the hybridizable discontinuous galerkin hdg discretization of elliptic partial differential equations pdes which allows us to obtain high order accurate solutions of the governing pde 2 the reduced basis method for a new hdg discretization of the underlying pde to enable real time solution of the parameterized pde in the presence of stochastic parameters and 3 a multilevel variance reduction method that exploits the statistical correlation among the different reduced basis approximations and the high fidelity hdg discretization to accelerate the convergence of the monte carlo simulations the multilevel variance reduction method provides efficient computation of the statistical outputs by shifting most of the computational burden from the high fidelity hdg approximation to the reduced basis approximations furthermore we develop a posteriori error estimates for our approximations of the statistical outputs based on these error estimates we propose an algorithm for optimally choosing both the dimensions of the reduced basis approximations and the sizes of monte carlo samples to achieve a given error tolerance we provide numerical examples to demonstrate the performance of the proposed method

statistics of isomorphism types in free products abstract let be a free product of finitely many finite and infinite cyclic groups for a subgroup of finite index given by its coset representation we compute its isomorphism type i e its decomposition as a free product of finite and infinite cyclic groups we determine the set of isomorphism types realized by finite index subgroups the asymptotics of the subgroup numbers with prescribed isomorphism types and the distribution of the isomorphism types among subgroups of fixed index apart from group theoretic arguments the proofs of the present paper make use of asymptotic combinatorial and probabilistic ideas and techniques

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

a locally conservative ldg method for the incompressible navier stokes equations in this paper a new local discontinuous galerkin method for the incompressible stationary navier stokes equations is proposed and analyzed four important features render this method unique its stability its local conservativity its high order accuracy and the exact satisfaction of the incompressibility constraint although the method uses completely discontinuous approximations a globally divergence free approximate velocity in h div is obtained by simple element by element post processing optimal error estimates are proven and an iterative procedure used to compute the approximate solution is shown to converge this procedure is nothing but a discrete version of the classical fixed point iteration used to obtain existence and uniqueness of solutions to the incompressible navier stokes equations by solving a sequence of oseen problems numerical results are shown which verify the theoretical rates of convergence they also confirm the independence of the number of fixed point iterations with respect to the discretization parameters finally they show that the method works well for a wide range of reynolds numbers

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

a hybridized discontinuous petrov galerkin scheme for scalar conservation laws summary we present a hybridized discontinuous petrov galerkin hdpg method for the numerical solution of steady and time dependent scalar conservation laws the method combines a hybridization technique with a local petrov galerkin approach in which the test functions are computed to maximize the inf sup condition since the petrov galerkin approach does not guarantee a conservative solution we propose to enforce this explicitly by introducing a constraint into the local petrov galerkin problem when the resulting nonlinear system is solved using the newton raphson procedure the solution inside each element can be locally condensed to yield a global linear system involving only the degrees of freedom of the numerical trace this results in a significant reduction in memory storage and computation time for the solution of the matrix system albeit at the cost of solving the local petrov galerkin problems however these local problems are independent of each other and thus perfectly scalable we present several numerical examples to assess the performance of the proposed method the results show that the hdpg method outperforms the hybridizable discontinuous galerkin method for problems involving discontinuities moreover for the test case proposed by peterson the hdpg method provides optimal convergence of order k 1 copyright 2012 john wiley sons ltd

hierarchic hp edge element families for maxwell s equations on hybrid quadrilateral triangular meshes we construct and study a set of hierarchic basis functions for the galerkin discretisation of the space h curl 169 suitable for hybrid meshes containing both quadrilateral and triangular elements with arbitrary non uniform order polynomial approximation we investigate the conditioning and dispersive behaviour of the elements in addition numerical examples are shown which demonstrate the accuracy of the space for computing solutions of the time harmonic maxwell s equations that have singularities

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

character theory of symmetric groups subgroup growth of fuchsian groups and random walks we develop a number of statistical aspects of symmetric groups mostly dealing with the distribution of cycles in various subsets of sn asymptotic properties of ordinary characters of symmetric groups and estimates for the multiplicities of root number functions of these groups as main applications we present an estimate for the subgroup growth of an arbitrary fuchsian group a finiteness result for the number of fuchsian presentations of such a group resolving a long standing problem of roger lyndon as well as a proof of a well known conjecture of roichman concerning the mixing time of random walks on symmetric groups

decomposable functors and the exponential principle abstract we develop a combinatorial theory for the operation of forming the logarithm of generating functions associated with covariant functors on the category of finite sets and bijective maps this leads to a rather general and flexible exponential principle for labeled combinatorial structures we also introduce the concept of a topos like category which turns out to be a convenient framework for the discussion of a wide range of applications a few of which are studied in detail

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

counting conjugacy classes of subgroups in a finitely generated group abstract a new general formula is obtained for the number of conjugacy classes of subgroups of given index in an arbitrary finitely generated group

generating functions for permutation representations abstract we discuss the categorical approach to representations in wreath products and generalize the wohlfahrt formula of the exponential generating function for the number of permutation representations

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

an improved iterative hdg approach for partial differential equations abstract we propose and analyze an iterative high order hybridized discontinuous galerkin ihdg discretization for linear partial differential equations we improve our previous work 45 in several directions 1 the improved ihdg approach converges in a finite number of iterations for the scalar transport equation 2 it is unconditionally convergent for both the linearized shallow water system and the convection diffusion equation 3 it has improved stability and convergence rates 4 we uncover a relationship between the number of iterations and time stepsize solution order meshsize and the equation parameters this allows us to choose the time stepsize such that the number of iterations is approximately independent of the solution order and the meshsize and 5 we provide both strong and weak scalings of the improved ihdg approach up to 16 384 cores a connection between ihdg and time integration methods such as parareal and implicit explicit methods are discussed extensive numerical results for linear and nonlinear pdes are presented to verify the theoretical findings

locally divergence free discontinuous galerkin methods for the maxwell equations in this paper we develop the locally divergence free discontinuous galerkin method for numerically solving the maxwell equations the distinctive feature of the method is the use of approximate solutions that are exactly divergence free inside each element as a consequence this method has a smaller computational cost than that of the discontinuous galerkin method with standard piecewise polynomial spaces we show that in spite of this fact it produces approximations of the same accuracy we also show that this method is more efficient than the discontinuous galerkin method using globally divergence free piecewise polynomial bases finally a post processing technique is used to recover 2k 1 th order of accuracy when piecewise polynomials of degree k are used

the application of optimal control methodology to nonlinear programming problems dynamic programming techniques have proven to be more successful than alternative nonlinear programming algorithms for solving many discrete time optimal control problems the reason for this is that because of the stagewise decomposition which characterizes dynamic programming the computational burden grows approximately linearly with the numbern of decision times whereas the burden for other methods tends to grow faster e g n3 for newton s method the idea motivating the present study is that the advantages of dynamic programming can be brought to bear on classical nonlinear programming problems if only they can somehow be rephrased as optimal control problems

a high order discontinuous galerkin method for unsteady advection diffusion problems abstract a high order discontinuous galerkin method with lagrange multipliers is presented for the solution of unsteady advection diffusion problems in the high peclet number regime it operates directly on the second order form of the governing equation and does not require any stabilization its spatial basis functions are chosen among the free space solutions of the homogeneous form of the partial differential equation obtained after time discretization it also features lagrange multipliers for enforcing a weak continuity of the approximated solution across the element interface boundaries this leads to a system of differential algebraic equations which are time integrated by an implicit family of schemes the numerical stability of these schemes and the well posedness of the overall discretization method are supported by a theoretical analysis the performance of this method is demonstrated for various high peclet number constant coefficient model flow problems

integration of accelerometer data filter analysis and design using riccati solutions the adjustment and integration of noisy accelerometer data is formulated as a discrete time optimal control problem which requires the minimization of a quadratic performance measure subject to linear dynamics the algebraic riccati equation associated with this problem is solved by expressing the entries of the riccati matrix explicitly in terms of the weights in the quadratic performance criterion steady state feedback gains are thus easily computed as functions of the weights the frequency domain analysis of the adjustment procedure provides the basis for selecting appropriate weights

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

the compact discontinuous galerkin cdg method for elliptic problems we present a compact discontinuous galerkin cdg method for an elliptic model problem the problem is first cast as a system of first order equations by introducing the gradient of the primal unknown or flux as an additional variable a standard discontinuous galerkin dg method is then applied to the resulting system of equations the numerical interelement fluxes are such that the equations for the additional variable can be eliminated at the element level thus resulting in a global system that involves only the original unknown variable the proposed method is closely related to the local discontinuous galerkin ldg method b cockburn and c w shu siam j numer anal 35 1998 pp 2440 2463 but unlike the ldg method the sparsity pattern of the cdg method involves only nearest neighbors also unlike the ldg method the cdg method works without stabilization for an arbitrary orientation of the element interfaces the computation of the numerical interface fluxes for the cdg method is slightly more involved than for the ldg method but this additional complication is clearly offset by increased compactness and flexibility compared to the br2 f bassi and s rebay j comput phys 131 1997 pp 267 279 and ip j douglas jr and t dupont in computing methods in applied sciences second internat sympos versailles 1975 lecture notes in phys 58 springer berlin 1976 pp 207 216 methods which are known to be compact the present method produces fewer nonzero elements in the matrix and is computationally more efficient

hybridizable discontinuous galerkin method hdg for stokes interface flow abstract in this paper we present a hybridizable discontinuous galerkin hdg method for solving the stokes interface problems with discontinuous viscosity and variable surface tension the jump condition of the stress tensor across the interface is naturally incorporated into the hdg formulation through a constraint on the numerical flux the most important feature of hdg method compared to other dg methods is that it reduces the number of globally coupled unknowns significantly when high order approximate polynomials are used for problems with polygonal interfaces it provides optimal convergence rates of order k 1 in l 2 norm for the velocity pressure and as well as the gradient of velocity furthermore a new approximate velocity can be obtained by an element by element postprocessing which converges with order k 2 in the l 2 norm for stokes interface problems with curved interfaces we use general curvilinear element to ensure the optimal convergence rates an error estimate is given for the approximation of the interface it indicates that curvilinear elements of degree 2 k 1 should be used for optimal convergence rate of order k 1

combined hybridizable discontinuous galerkin hdg and runge kutta discontinuous galerkin rk dg formulations for green naghdi equations on unstructured meshes abstract in this paper we introduce some new high order discrete formulations on general unstructured meshes especially designed for the study of irrotational free surface flows based on partial differential equations belonging to the family of fully nonlinear and weakly dispersive shallow water equations working with a recent family of optimized asymptotically equivalent equations we benefit from the simplified analytical structure of the linear dispersive operators to conveniently reformulate the models as the classical nonlinear shallow water equations supplemented with several algebraic source terms which globally account for the non hydrostatic effects through the introduction of auxiliary coupling variables high order discrete approximations of the main flow variables are obtained with a rk dg method while the trace of the auxiliary variables are approximated on the mesh skeleton through the resolution of second order linear elliptic sub problems with high order hdg formulations the combined use of hybrid unknowns and local post processing significantly helps to reduce the number of globally coupled unknowns in comparison with previous approaches the proposed formulation is then extended to a more complex family of three parameters enhanced green naghdi equations the resulting numerical models are validated through several benchmarks involving nonlinear waves transformations and propagation over varying topographies showing good convergence properties and very good agreements with several sets of experimental data

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100