discontinuous galerkin finite element methods for second order hyperbolic problems abstract we prove a priori and a posteriori error estimates for a finite element method for linear second order hyperbolic equations based on a space time finite element discretization with the basis functions being continuous in space and discontinuous in time

a space time finite element method for the wave equation a numerical method is introduced which uses finite elements in time and space simultaneously to solve the wave equation an error analysis for this scheme is presented and the results of several computations this scheme is accurate for time slabs of arbitrary thickness without introducing extra least squares or stabilization terms

discontinuous galerkin methods for ordinary differential equations a class of galerkin methods derived from discontinuous piecewise polynomial spaces is analyzed for polynomials of degree k these methods lead to a family of one step schemes generating approximations up to order 2k 2 for the solution of an ordinary differential equation

a space time finite element method for the exterior acoustics problem in this paper the development of a space time finite element method for the solution of the transient acoustics problem in exterior domains is discussed the space time formulation for the exterior acoustics problem is obtained from a time discontinuous galerkin variational equation for coupled structural acoustics specialized to the case of zero normal velocities on the wet surface i e a rigid scatterer the formulation employs a finite computational acoustic domain surrounding the scatterer and incorporates high order time dependent nonreflecting radiation boundary conditions on the fluid truncation boundary as natural boundary conditions in the space time variational equation i e they are enforced weakly in both space and time the result is an algorithm for direct transient analysis of acoustic radiation and scattering with the desired combination of good stability and high accuracy the method is especially useful for the application of adaptive solution strategies for transient acousti

discontinuous galerkin finite element method for the wave equation the symmetric interior penalty discontinuous galerkin finite element method is presented for the numerical discretization of the second order wave equation the resulting stiffness matrix is symmetric positive definite and the mass matrix is essentially diagonal hence the method is inherently parallel and leads to fully explicit time integration when coupled with an explicit time stepping scheme optimal a priori error bounds are derived in the energy norm and the l 2 norm for the semidiscrete formulation in particular the error in the energy norm is shown to converge with the optimal order cal o h min s ell with respect to the mesh size h the polynomial degree ell and the regularity exponent s of the continuous solution under additional regularity assumptions the l 2 error is shown to converge with the optimal order cal o h min s ell numerical results confirm the expected convergence rates and illustrate the versatility of the method

global existence of solutions to the incompressible navier stokes vlasov equations in a time dependent domain in this article we prove the existence of global weak solutions for the in compressible navier stokes vlasov system in a three dimensional time dependent domain with absorption boundary conditions for the kinetic part this model arises from the study of respiratory aerosol in the human airways the proof is based on a regularization and approximation strategy designed for our time dependent framework

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

shape representation by multiscale contour approximation an approach is presented for deriving qualitative descriptions of contours containing structures at different unknown scales the descriptions are in terms of straight arcs curved arcs with sign of curvature corners and points delimiting the arcs inflexion points and transitions from straight to curved furthermore the tangents at these points are derived the approach is based on the construction of a hierarchic family of polygons having the scale space property of causality structure can only disappear as scale goes from fine to coarse using the principle that structures that are stable over scale represent significant properties the features of the descriptive representations are then derived

structural dynamic analysis by a time discontinuous galerkin finite element method this paper studies a time discontinuous galerkin finite element method for structural dynamic problems by which both displacements and velocities are approximated as piecewise linear functions in the time domain and may be discontinuous at the discrete time levels a new iterative solution algorithm which involves only one factorization for each fixed time step size and a few iterations at each step is presented for solving the resulted system of coupled equations by using the jumps of the displacements and the velocities in the total energy norm as error indicators an adaptive time stepping procedure for selecting the proper time step size is described numerical examples including both single dof and multi dof problems are used to illustrate the performance of these algorithms comparisons with the exact results and or the results by the newmark integration scheme are given it is shown that the time discontinuous galerkin finite element method discussed in this study possesses good accuracy third order and stability properties its numerical implementation is not difficult and the higher computational cost needed in each time step is compensated by use of a larger time step size

optimal discontinuous galerkin methods for wave propagation we have developed and analyzed a new class of discontinuous galerkin methods dg which can be seen as a compromise between standard dg and the finite element fe method in the way that it is explicit like standard dg and energy conserving like fe in the literature there are many methods that achieve some of the goals of explicit time marching unstructured grid energy conservation and optimal higher order accuracy but as far as we know only our new algorithms satisfy all the conditions we propose a new stability requirement for our dg the stability analysis is based on the careful selection of the two fe spaces which verify the new stability condition the convergence rate is optimal with respect to the order of the polynomials in the fe spaces moreover the convergence is described by a series of numerical experiments

a space time formulation for multiscale phenomena abstract we develop subgrid scale models for a class of nonsymmetric linear evolution operators by applying the variational multiscale method in space time the results generalize those of hughes 14 which were confined to the steady case the subgrid scale models are shown to be a paradigm for bubble function finite element methods and provide a theoretical and practical framework for the development of so called stabilized methods

application of an ultra weak variational formulation of elliptic pdes to the two dimensional helmholtz problem a new technique to solve elliptic linear pdes called ultra weak variational formulation uwvf in this paper is introduced in b despres c r acad sci paris 318 1994 pp 939 944 this paper is devoted to an evaluation of the potentialities of this technique it is applied to a model wave problem the two dimensional helmholtz problem the new method is presented in three parts following the same style of presentation as the classical one of the finite elements method even though they are definitely conceptually different methods the first part is committed to the variational formulation and to the continuous problem the second part defines the discretization process using a galerkin procedure the third part actually studies the efficiency of the technique from the order of convergence point of view this is achieved using theoretical proofs and a series of numerical experiments in particular it is proven and shown the order of convergence is lower bounded by a linear function of the number of degrees of freedom an application to scattering problems is presented in a fourth part

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

stabilization mechanisms in discontinuous galerkin finite element methods in this paper we propose a new general framework for dg methods which allows to uncover a basic mechanism that ensures suitable stability properties of the methods we show that such a mechanism is essentially the same for apparently unrelated stabilizations such as jump penalty upwinding and hughes franca type residual based stabilizations

three dimensional discontinuous galerkin elements with plane waves and lagrange multipliers for the solution of mid frequency helmholtz problems recently a discontinuous galerkin finite element method with plane wave basis functions and lagrange multiplier degrees of freedom was proposed for the efficient solution in two dimensions of helmholtz problems in the mid frequency regime in this paper this method is extended to three dimensions and several new elements are proposed computational results obtained for several wave guide and acoustic scattering model problems demonstrate one to two orders of magnitude solution time improvement over the higher order galerkin method

the discontinuous enrichment method for multiscale analysis computation naturally separates scales of a problem according to the mesh size a variety of improved numerical methods are described by multiscale considerations differing in the treatment of the unresolved fine scales the discontinuous enrichment method provides a unique multiscale approach to computation by employing fine scales that contain solutions of the homogeneous partial differential equation in a discontinuous framework the method thus combines relative ease of implementation with improved numerical performance these properties are demonstrated for both multiscale wave and transport problems pointing to the potential of considerable savings in computational resources

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

a discontinuous galerkin method with lagrange multipliers for the solution of helmholtz problems in the mid frequency regime we present a discontinuous galerkin method dgm for the solution of the helmholtz equation in the mid frequency regime our approach is based on the discontinuous enrichment method in which the standard polynomial field is enriched within each finite element by a non conforming field that contains free space solutions of the homogeneous partial differential equation to be solved hence for the helmholtz equation the enrichment field is chosen here as the superposition of plane waves we enforce a weak continuity of these plane waves across the element interfaces by suitable lagrange multipliers preliminary results obtained for two dimensional model problems discretized by uniform meshes reveal that the proposed dgm enables the development of elements that are far more competitive than both the standard linear and the standard quadratic galerkin elements for the discretization of helmholtz problems

the discontinuous enrichment method we propose a finite element based discretization method in which the standard polynomial field is enriched within each element by a nonconforming field that is added to it the enrichment contains free space solutions of the homogeneous differential equation that are not represented by the underlying polynomial field continuity of the enrichment across element interfaces is enforced weakly by lagrange multipliers in this manner we expect to attain high coarse mesh accuracy without significant degradation of conditioning assuring good performance of the computation at any mesh resolution examples of application to acoustics and advection diffusion are presented

time finite element methods for structural dynamics summary time finite element methods are developed for the equations of structural dynamics the approach employs the time discontinuous galerkin method and incorporates stabilizing terms having least squares form these enable a general convergence theorem to be proved in a norm stronger than the energy norm results are presented from finite difference analyses of the time discontinuous galerkin and least squares methods with various temporal interpolations and commonly used finite difference methods for structural dynamics these results show that for particular interpolations the time finite element method exhibits improved accuracy and stability