vibration of circular plates on a free fluid surface effect of surface waves the effect of free surface waves on free vibrations of circular plates resting on a free fluid surface is studied the solution is achieved by using a perturbation technique and the hankel transformation method which give a couple of dual integral equations of titchmarsh type the fluid is considered inviscid and incompressible and the velocity potential describes its motion the kirchhoff theory of plates is used to model the elastic thin plate the theory is suitable for all axisymmetric plate boundary conditions numerical results are given in non dimensional form for modes up to five nodal circles and diameters for clamped simply supported and free edge plates to be ready to use in applications the effect of free surface waves on the plate s natural frequencies is significant only when bluging and sloshing modes of the system have close natural frequencies

stable galerkin reduced order models for linearized compressible flow the galerkin projection procedure for construction of reduced order models of compressible flow is examined as an alternative discretization of the governing differential equations the numerical stability of galerkin models is shown to depend on the choice of inner product for the projection for the linearized euler equations a symmetry transformation leads to a stable formulation for the inner product boundary conditions for compressible flow that preserve stability of the reduced order model are constructed preservation of stability for the discrete implementation of the galerkin projection is made possible using a piecewise smooth finite element basis stability of the reduced order model using this approach is demonstrated on several model problems where a suitable approximation basis is generated using proper orthogonal decomposition of a transient computational fluid dynamics simulation

tent pitching and trefftz dg method for the acoustic wave equation abstract we present a space time trefftz discontinuous galerkin method for approximating the acoustic wave equation semi explicitly on tent pitched meshes dg trefftz methods use discontinuous test and trial functions which solve the wave equation locally tent pitched meshes allow us to solve the equation elementwise allowing locally optimal advances in time the method is implemented in ngsolve solving the space time elements in parallel whenever possible insights into the implementational details are given including the case of propagation in heterogeneous media

berezin transform on real bounded symmetric domains let d be a bounded symmetric domain in a complex vector space vc with a real form v and d d v g k be the real bounded symmetric domain in the real vector space v we construct the berezin kernel and consider the berezin transform on the l2 space on d the corresponding representation of g is then unitarily equivalent to the restriction to g of a scalar holomorphic discrete series of holomorphic functions on d and is also called the canonical representation we find the spectral symbol of the berezin transform under the irreducible decomposition of the l2 space introduction the main purpose of the present paper is to calculate the spectral symbol of the berezin transform on real bounded symmetric domains to explain our results and motivations we let d be the unit disk in the complex plane with the lebesgue measure dm z we consider the weighted bergman space h 1 of holomorphic functions on d square integrable with respect to the measure 1 z 2 2dm z it has up to some positive constant the reproducing kernel kw z k z w 1 zw moreover the group gc su 1 1 of fractional transformations of d acts on the space h via f z 7 f gz g z 2 and it forms an irreducible unitary projective representation consider the subgroup so 1 1 consisting of transformations of the form z 7 az b bz a with a b r and a b 1 thus it is of interest to study the irreducible decomposition of the weighted bergman space under the subgroup g so 1 1 for that purpose we consider the unit interval d d r 1 1 as a trivial symmetric space g k so 1 1 1 and the restriction of holomorphic functions in h to the interval d more precisely consider r h c d rf x f x 1 x 2 x d let l d d 0 be the l space on d with the so 1 1 invariant measure d 0 x dx 1 x2 whose decomposition under so 1 1 can be done via the mellin transform see below the restriction r is a bounded operator from h into the space l d d 0 with dense image and intertwines the respective actions of so 1 1 received by the editors january 16 2000 and in revised form october 10 2000 2000 mathematics subject classification primary 22e46 43a85 32m15 53c35

on the accuracy of the finite element method plus time relaxation if u denotes a local spatial average of u then u u u is the associated fluctuation consider a time relaxation term added to the usual finite element method the simplest case for the model advection equation u t a u x t is u h t a u h v h u h v h f x t v h we analyze the error in this and more importantly higher order extensions and show that the added time relaxation term not only suppresses excess energy in marginally resolved scales but also increases the accuracy of the resulting finite element approximation

a space time trefftz discontinuous galerkin method for the acoustic wave equation in first order formulation we introduce a space time trefftz discontinuous galerkin method for the first order transient acoustic wave equations in arbitrary space dimensions extending the one dimensional scheme of kretzschmar et al ima j numer anal 36 1599 1635 2016 test and trial discrete functions are space time piecewise polynomial solutions of the wave equations we prove well posedness and a priori error bounds in both skeleton based and mesh independent norms the space time formulation corresponds to an implicit time stepping scheme if posed on meshes partitioned in time slabs or to an explicit scheme if posed on tent pitched meshes we describe two trefftz polynomial discrete spaces introduce bases for them and prove optimal high order h convergence bounds

on the stability of a family of finite element methods for hyperbolic problems we consider a family of tensor product finite element methods for hyperbolic equations in rn n 2 which are explicit and generate a continuous approximate solution the base case n 2 an extension of the box scheme to higher order is due to winther who proved stability and optimal order convergence by means of a simple counterexample we show that for linear approximation with n 3 the corresponding methods are unstable

symplectic hamiltonian hdg methods for wave propagation phenomena abstract we devise the first symplectic hamiltonian hybridizable discontinuous galerkin hdg methods for the acoustic wave equation we discretize in space by using a hamiltonian hdg scheme that is an hdg method which preserves the hamiltonian structure of the wave equation and in time by using symplectic diagonally implicit and explicit partitioned runge kutta methods the fundamental feature of the resulting scheme is that the conservation of a discrete energy which is nothing but a discrete version of the original hamiltonian is guaranteed we present numerical experiments which indicate that the method achieves optimal approximations of order k 1 in the l 2 norm when polynomials of degree k 0 and runge kutta time marching methods of order k 1 are used in addition by means of post processing techniques and by increasing the order of the runge kutta method to k 2 we obtain superconvergent approximations of order k 2 in the l 2 norm for the displacement and the velocity we also present numerical examples that corroborate that the methods conserve energy and that they compare favorably with dissipative hdg schemes of similar accuracy properties for long time simulations

on the behavior over long time intervals of finite difference and finite element approximations to first order hyperbolic systems abstract standard error estimates for approximations to hyperbolic equations are only valid over a finite time interval this paper considers the behavior of the error in the two most common discretization schemes over the entire interval 0 t finite difference approximations are analyzed for approximating the cauchy problem for hyperbolic systems it is recalled that if the system is dissipative error estimates that are global in t can be obtained for strictly conservative systems the role of local energy decay is pointed out and used to obtain global in t local in x error estimates finite element methods are considered for nonlinear initial boundary value problems for hyperbolic systems it is proven that if the nonlinear term is locally lipschitz and the equation is dissipative when linearized about the true solution then global in t convergence follows with the expected rates

a note on the convergence of the discontinuous galerkin method for a scalar hyperbolic equation a numerical example is presented which shows that the known l 2 error estimate left u u h right leqq ch k 1 2 left u right k 1 for the discontinuous galerkin method cannot be improved within the class of quasi uniform meshes not even for smooth exact solutions u

branching laws for minimal holomorphic representations abstract in this paper we study the branching law for the restriction from su n m to so n m of the minimal representation in the analytic continuation of the scalar holomorphic discrete series we identify the group decomposition with the spectral decomposition of the action of the casimir operator on the subspace of s o n o m invariants the plancherel measure of the decomposition defines an l 2 space of functions for which certain continuous dual hahn polynomials furnish an orthonormal basis it turns out that the measure has point masses precisely when n m 2 under these conditions we construct an irreducible representation of so n m identify it with a parabolically induced representation and construct a unitary embedding into the representation space for the minimal representation of su n m

explicit finite element methods for symmetric hyperbolic equations a family of explicit space time finite element methods for the initial boundary value problem for linear symmetric hyperbolic systems of equations is described and analyzed the method generalizes the discontinuous galerkin method and as is typical for this method obtains error estimates of order o h n 1 2 for approximations by polynomials of degree le n

convergence criteria for systems of non linear elliptic boundary value partial differential equations abstract this paper deals with convergence criteria for a special system of non linear elliptic boundary value partial differential equations a fixed point algorithm is used which iteratively solves one linearized elliptic partial differential equation at a time we establish conditions which help predict the convergence of the fixed point algorithm we prove under reasonable hypotheses that the algorithm converges experimental results are reported which show that algorithm converges in wide variety of cases and the convergence is well correlated with the theoretical conditions introduced in this paper

berezin kernels of tube domains abstract we study the berezin kernel decomposition into positive definite spherical functions on real forms of tube domains in other words we obtain for all tube domains associated to causal pairs g h the decomposition of holomorphic discrete series representations of g when restricted to the fully restrictive subgroup h

an optimal order error estimate for the discontinuous galerkin method in this paper a new approach is developed for analyzing the discontinuous galerkin method for hyperbolic equations for a model problem in r2 the method is shown to converge at a rate 0 hn l when applied with nth degree polynomial approxi mations over a semiuniform triangulation assuming sufficient regularity in the solution

a continuous wavelet galerkin method for the linear wave equation we consider the continuous space time galerkin method for the linear second order wave equation proposed by french and peterson in 1996 a bottleneck for this approach is how to solve the discrete problems effectively in this paper we tackle this bottleneck by essentially employing wavelet bases in space we show how to decouple the corresponding linear system and we prove that the resulting subsystems can be uniformly preconditioned by simple diagonal preconditioners leading to efficient iterative solutions

sufficient condition of stability of finite element method for symmetric t hyperbolic systems with constant coefficients in this note finite element method is applied to solve the symmetric t hyperbolic system with dissipative boundary condition and its stability is proved in two dimensional space complex program is developed for the numerical solution of the mixed problem in simple connected region on the uniform grid delphi 7 is used for the code of the complex program numerical results are in line with the theoretical findings

tensor products of holomorphic representations and bilinear differential operators let h be the weighted bergman space on a bounded symmetric domain d g k it has analytic continuation in the weight and for in the so called wallach set h still forms unitary irreducible projective representations of g we give the irreducible decomposition of the tensor product h 1 h 2 of the representations for any two unitary weights and we find the highest weight vectors of the irreducible components we find also certain bilinear differential intertwining operators realizing the decomposition and they generalize the classical transvectants in invariant theory of sl 2 c as applications we find a generalization of the bol s lemma and we characterize the multiplication operators by the coordinate functions on the quotient space of the tensor product h 1 h 2 modulo the subspace of functions vanishing of certain degree on the diagonal

high accuracy finite element methods for positive symmetric systems abstract a nonstandard type least squares finite element method is proposed for the solution of first order positive symmetric systems this method gives optimal accuracy in a norm similar to the h1 norm when a regularity condition holds it is optimal in l2 as well otherwise it gives errors suboptimal by only h 1 2 where h is the mesh diameter thus it has greater accuracy than usual finite element finite difference or least squares methods for such problems in addition the spectral condition number of the associated linear system is only o h 1 vs o h 2 for the usual least squares methods thus the method promises to be an efficient high accuracy method for hyperbolic systems such as maxwell s equations it is also equally promising for mixed type equations that have a formulation as a positive symmetric system

branching laws for discrete wallach points we consider the projective representations of the group of holomorphic automorphisms of a symmetric tube domain v i that are obtained by analytic continuation of the holomorphic discrete series for a representation corresponding to a discrete point in the wallach set we find the decomposition under restriction to the identity component of gl using riesz distributions an explicit intertwining operator is constructed as an analytic continuation of an integral operator the density of the plancherel measure involves quotients of functions and the c function for a symmetric cone of smaller rank

hydroelastic vibration of circular plates abstract this paper is concerned with the virtual mass effect due to the presence of water on the natural frequencies of circular plates which are placed into a hole of an infinite rigid wall with one side exposed to water this classical problem was solved by lamb but he provided limited theoretical results to obtain the complete theoretical results the so called non dimensionalized added virtual mass incremental factors are obtained by employing the integral transformation technique in conjunction with the fourier bassel series approach it is found that the non dimensionalized added virtual mass incremental factors for circular plates vibrating in an infinite rigid wall with one side exposed to water are larger than those for circular plates which rest on a free surface this is due to the fact that the rigid wall constrains the motion of the water thus resulting in an increase in the kinetic energy of the water it is also observed that the effect of water on the natural frequencies generally decreases with order

degenerate principal series representations and their holomorphic extensions let x h l be an irreducible real bounded symmetric domain realized as a real form in an hermitian symmetric domain d g k the intersection s of the shilov boundary of d with x defines a distinguished subset of the topological boundary of x and is invariant under h it can be realized as s h p for certain parabolic subgroup p of h we study the spherical representations of h induced from p we find formulas for the spherical functions in terms of the macdonald hypergeometric function this generalizes the earlier result of faraut koranyi for hermitian symmetric spaces d we consider a class of h invariant integral intertwining operators from the representations on l2 s to the holomorphic representations of g restricted to h we construct a new class of complementary series for the groups h so n m su n m with n m 2 and sp n m with n m 1 we realize them as discrete components in the branching rule of the analytic continuation of the holomorphic discrete series of g su n m su n m su n m and su 2n 2m respectively

adaptive spacetime discontinuous galerkin method for hyperbolic advection diffusion with a non negativity constraint summary applications where the diffusive and advective time scales are of similar order give rise to advection diffusion phenomena that are inconsistent with the predictions of parabolic fickian diffusion models non fickian diffusion relations can capture these phenomena and remedy the paradox of infinite propagation speeds in fickian models in this work we implement a modified frame invariant form of cattaneo s hyperbolic diffusion relation within a spacetime discontinuous galerkin advection diffusion model an h adaptive spacetime meshing procedure supports an asynchronous patch by patch solution procedure with linear computational complexity in the number of spacetime elements this localized solver enables the selective application of optimization algorithms in only those patches that require inequality constraints to ensure a non negative concentration solution in contrast to some previous methods we do not modify the numerical fluxes to enforce non negative concentrations thus the element wise conservation properties that are intrinsic to discontinuous galerkin models are defined with respect to physically meaningful riemann fluxes on the element boundaries we present numerical examples that demonstrate the effectiveness of the proposed model and we explore the distinct features of hyperbolic advection diffusion response in subcritical and supercritical flows copyright 2015 john wiley sons ltd

an analysis of the discontinuous galerkin method for a scalar hyperbolic equation we prove lp stability and error estimates for the discontinuous galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain using finite element analysis techniques we obtain l2 estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials lp estimates for p 2 are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree the latter estimates are proved by combining finite difference and finite element analysis techniques

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

a tent pitching scheme motivated by friedrichs theory certain friedrichs systems can be posed on hilbert spaces normed with a graph norm functions in such spaces arising from advective problems are found to have traces with a weak continuity property at points where the inflow and outflow boundaries meet motivated by this continuity property an explicit space time finite element scheme of the tent pitching type with spaces that conform to the continuity property is designed numerical results for a model one dimensional wave propagation problem are presented

stable and unstable numerical boundary conditions for galerkin approximations to hyperbolic systems abstract the galerkin method for first order hyperbolic systems is considered this method is seen to be unstable in an example because of improper treatment of the boundary conditions in the discrete problem the galerkin method is modified to be stable for any well posed initial boundary value problem in one dimension it is shown that the previous estimates of the error in this modified method can be improved when the system admits an energy conserving norm a system in two space dimensions in a region with a corner is considered it is shown how the construction of stable methods for such problems can be reduced to a very simple linear programming problem

riemann solutions and spacetime discontinuous galerkin method for linear elastodynamic contact abstract we derive riemann solutions for stick slip and separation contact modes in a linear elastic material with an isotropic coulomb friction relation and explore their numerical implementation the riemann solutions preserve the characteristic structure of the underlying elastodynamic system and imply dynamic contact conditions that are distinct from the quasi static conditions used in some numerical models nonphysical discontinuities in the standard coulomb model at stick slip transitions can cause contact mode chatter in numerical simulations we restate the coulomb relation to remove these artificial discontinuities and eliminate the need for algorithmic remedies discontinuous response at abrupt separation to contact transitions is physically reasonable and we propose a regularization scheme to address this case we implement the riemann contact solutions within an adaptive spacetime discontinuous galerkin sdg code and report numerical results that demonstrate the model s efficacy

transcending classical invariant theory let sp2 r sp2 sp be the real symplectic group of rank n let sp denote the two fold cover of sp there is a unitary representation constructed by shale sh and weil wa of sp that is of considerable interest in the theory of automorphic forms we denote this representation by co and call it the oscillator representation the purpose of this paper is to establish about c a fact that should be useful for clarifying the structure of spaces of automorphic forms constructed using co we begin by formulating the result let e be a reductive subgroup of sp and let e denote the inverse image of e in sp denote by r e the set of infinitesimal equivalence classes of continuous irreducible admissible representations of e on locally convex spaces let co be the smooth representation of sp associated to co let co be realized on a hilbert space and let the subspace of smooth vectors on which space co is defined be written denote by r9 e co the set of elements of e which are realized as quotients by co e invariant closed subspaces of t g consider a reductive dual pair g g c sp h2 it is not hard to show that g and g commute with one another hence we may regard co i g g as a representation of gx g it is well known f that r gx g s9 g xr9 g the identification associates to p e r g and p e i g the tensor product p x p we note that p x p is not defined as a topological vector space nevertheless the infinitesimal equivalence class of p op is well defined select p x p e g x g suppose that in fact p p e 9 g g c then clearly p e r g c and p e 9 g c hence g co defines the graph of a correspondence between certain subsets of r9 g cv and j g co in fact the situation is quite precise

plancherel formula for berezin deformation of l 2 on riemannian symmetric space consider natural representations of the pseudounitary group u p q in the space of holomorphic functions on the cartan domain hermitian symmetric space u p q u p u q berezin representations of o p q are the restrictions of such representations to the subgroup o p q we obtain the explicit plancherel formula for the berezin representations the support of the plancherel measure is a union of many series of representations the density of the plancherel measure on each piece of the support is an explicit product of functions we also show that the berezin representations give an interpolation between l2 on noncompact symmetric space o p q o p o q and l2 on compact symmetric space o p q o p o q

laplace and segal bargmann transforms on hermitian symmetric spaces and orthogonal polynomials let d g k be a complex bounded symmetric domain of tube type in a complex jordan algebra v and let dr j d d be its real form in a formally real euclidean jordan algebra j v dr h l is a bounded realization of the symmetric cone in j we consider representations of h that are gotten by the generalized segal bargmann transform from a unitary g space of holomorphic functions on d to an l2 space on dr we prove that in the unbounded realization the inverse of the unitary part of the restriction map is actually the laplace transform we find the extension to d of the spherical functions on dr and find their expansion in terms of the l spherical polynomials on d which are jack symmetric polynomials we prove that the coefficients are orthogonal polynomials in an l2 space the measure being the harish chandra plancherel measure multiplied by the symbol of the berezin transform we prove the difference equation and recurrence relation for those polynomials by considering the action of the lie algebra and the cayley transform on the polynomials on d finally we use the laplace transform to study generalized laguerre functions on symmetric cones

solution of linear elastodynamics problems with space time finite elements on structured and unstructured meshes abstract a new implicit approach for the solution of linear elastodynamics problems on structured and unstructured meshes with space time finite elements is suggested new weak formulations are based on time continuous tcg and discontinuous tdg galerkin methods it was shown that for structured meshes the new tcg method has a higher order of accuracy by a factor of two than that of the standard tcg method at the same number of degrees of freedom the accuracy of the new tcg method is one order higher than the accuracy of the standard tdg method the introduction of new weighting functions for the weak formulations of the tcg and tdg methods allows control of the high frequency behavior without degradation of accuracy of the methods a variable number of elements in the time direction as well as a variable order of polynomial time approximations of finite elements for time slabs allows the implementation of a coupled space time adaptive procedure using structured or unstructured meshes numerical examples for 1 d problems demonstrate the effectiveness of the technique

branching coefficients of holomorphic representations and segal bargmann transform let d g k be a complex bounded symmetric domain of tube type in a jordan algebra vc and let d h l d v be its real form in a jordan algebra v vc the analytic continuation of the holomorphic discrete series on dopf forms a family of interesting representations of g we consider the restriction on d and the branching rule under h of the scalar holomorphic representations the unitary part of the restriction map gives then a generalization of the segal bargmann transform the group l is a spherical subgroup of k and we find a canonical basis of l invariant polynomials in the components of the schmid decomposition and we express them in terms of the jack symmetric polynomials we prove that the segal bargmann transform of those l invariant polynomials are under the spherical transform on d multi variable wilson type polynomials and we give a simple alternative proof of their orthogonality relation we find the expansion of the spherical functions on d when extended to a holomorphic function in a neighborhood of 0 d in terms of the l spherical holomorphic polynomials on d the coefficients being the wilson polynomials

effect of fluid depth on the hydroelastic vibration of free edge circular plate abstract this paper is concerned with the effect of water depth on the free vibration of free edge circular plates resting on a free fluid surface the problem addressed is formulated by using the hankel transformation method which leads to dual integral equations the solution of dual integral equations is solved numerically by using fourier bessel series the fluid is assumed to be inviscid and incompressible the kirchhoff theory of plates is used to model the elastic thin plate numerical results are given in non dimensional form for free edge circular plates in order to be ready to use in applications to validate the theoretical results experiments were carried out experimental results are in good agreement with theoretical results it is found that the effect of the fluid depth can be neglected when the fluid depth is greater than the diameter of the circular plates but becomes significant as the water depth decreases it is also observed in the experimental results that the fluid damping increases as the water depth decreases

a space time discontinuous galerkin method for the solution of the wave equation in the time domain in recent years the focus of research in the field of computational acoustics has shifted to the medium frequency regime and multiscale wave propagation this has led to the development of new concepts including the discontinuous enrichment method its basic principle is the incorporation of features of the governing partial differential equation in the approximation in this contribution this concept is adapted for the simulation of transient problems governed by the wave equation we present a space time discontinuous galerkin method with lagrange multipliers where the shape approximation in space and time is based on solutions of the homogeneous wave equation the use of hierarchical wave like basis functions is enabled by means of a variational formulation that allows for discontinuities in both the spatial and the temporal discretizations numerical examples in one space dimension demonstrate the outstanding performance of the proposed method compared with conventional space time finite element methods copyright 2008 john wiley sons ltd

construction of energy stable projection based reduced order models an approach for building energy stable galerkin reduced order models roms for linear hyperbolic or incompletely parabolic systems of partial differential equations pdes using continuous projection is developed this method is an extension of earlier work by the authors specific to the equations of linearized compressible inviscid flow the key idea is to apply to the pdes a transformation induced by the lyapunov function for the system and to build the rom in the transformed variables for linear problems the desired transformation is induced by a special inner product termed the symmetry inner product which is derived herein for several systems of physical interest connections are established between the proposed approach and other stability preserving model reduction methods giving the paper a review flavor more specifically it is shown that a discrete counterpart of this inner product is a weighted l 2 inner product obtained by solving a lyapunov equation first proposed by rowley et al and termed herein the lyapunov inner product comparisons between the symmetry inner product and the lyapunov inner product are made and the performance of roms constructed using these inner products is evaluated on several benchmark test cases

function spaces and reproducing kernels on bounded symmetric domains there are several natural hilbert spaces of holomorphic functions on a bounded symmetric domain such are the bergman type spaces on which the holomorphic discrete series operates and the hardy type spaces which are related to the analytic continuation of the holomorphic discrete series also closely related is the bergmann space of entire functions on the ambient which arises as the closure of the polynomials with respect to a natural inner product the space of holomorphic polynomials decomposes into irreducible subspaces under the action of the isotropy group k of the domain the main facts about this decomposition were proved by schmid 21 for another proof see 22 each irreducible subspace contains a unique normalized l invariant spherical polynomial where l is the isotropy group of the shilov boundary in k our first main result is the explicit computation of the norms of the spherical polynomials with respect to each of the hilbert spaces considered for the domains of classical type they were considered by hua 9 and for some of the hilbert spaces this was done before by upmeier 24 using different methods for certain others there are partial results in 22 we are able to do this in a fairly simple unified way by making strong use of gindikin s generalized gamma function s next we obtain a description of the reproducing kernels of the k irreducible subspaces in each of our hilbert spaces and an expansion in terms of these for

optimal numerical parameterization of discontinuous galerkin method applied to wave propagation problems this paper deals with the high order discontinuous galerkin dg method for solving wave propagation problems first we develop a one dimensional dg scheme and numerically compute dissipation and dispersion errors for various polynomial orders an optimal combination of time stepping scheme together with the high order dg spatial scheme is presented it is shown that using a time stepping scheme with the same formal accuracy as the dg scheme is too expensive for the range of wave numbers that is relevant for practical applications an efficient implementation of a high order dg method in three dimensions is presented using 1d convergence results we further show how to adequately choose elementary polynomial orders in order to equi distribute a priori the discretization error we also show a straightforward manner to allow variable polynomial orders in a dg scheme we finally propose some numerical examples in the field of aero acoustics

stable galerkin methods for hyperbolic systems this paper considers the stability of the galerkin method for first order hyperbolic systems just as in finite difference theory many galerkin methods stable and convergent for the cauchy problem are unstable for hyperbolic boundary value problems because of improper treatment of the boundary conditions the usual galerkin method is seen to be unstable for a linear constant coefficient well posed hyperbolic system on 0 1 a method of treating boundary conditions is then proposed which will yield a stable and convergent method for any well posed linear hyperbolic system in one dimension this idea generalizes in a natural way to problems in more than one space dimension examples in one and two dimensions are considered this paper extends the work of gunzburger math comp 31 1977 pp 671 675 to the general hyperbolic system

galerkin methods for two point boundary value problems for first order systems galerkin methods for two point boundary value problems for first order systems are studied for a model problem rigorous upper and lower estimates upon the error are derived these estimates show the previous work to be sharp in some cases and pessimistic in others a nonstandard least squares type of method is then analyzed this method gives the optimal accuracy of the usual least squares method when the system does not satisfy a regularity assumption the nonstandard method gives slightly more accurate answers measured in l 2 the linear system arising from the nonstandard method has condition number o h 1 contrasted with o h 2 for the usual least squares method

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

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