Discontinuous Galerkin finite element methods for second order hyperbolic problems

Johnson, Claes

doi:10.1016/0045-7825(93)90170-3

Cited by 221 publications

(185 citation statements)

References 8 publications

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“…For each slab S n , we consider a mesh. For h > 0 let T n h be a triangulation of the slab S n into triangle K, satisfying as usual the minimum angle condition [30]. The triangulation of S n may be chosen independently of that of S n−1 , but for the sake of simplicity it must satisfy quasi-uniformity conditions for finite element meshes [7,8,9,10].…”

Section: E65mentioning

confidence: 99%

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The new streamline diffusion for the 3D coupled Schrodinger equations with a cross-phase modulation

Rostamy¹

2013

ANZIAMJ

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We study the new streamline diffusion finite element method for treating the three dimensional coupled nonlinear Schrödinger equation. We derive stability estimates and optimal convergence rates. Moreover, an a priori error estimate is obtained and we compare the corresponding optimal convergence rate for popular numerical methods such as conservative finite difference, semi-implicit finite difference, semi-discrete finite element and the time-splitting spectral method. We justify the advantage of the streamline diffusion method versus the some numerical methods with some examples. Test problems are presented to verify the efficiency and accuracy of the method. The results reveal that the proposed scheme is very effective, convenient and quite accurate for such considered problems rather than other methods.

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Section: E65mentioning

confidence: 99%

“…Streamline diffusion methods have been considered by many [2,3,4,5,6,12,13,14,15,16,17,18,25,26,27,28,29,30]. These methods are alternative numerical approaches to the classical numerical methods such as the finite element and boundary element method.…”

Section: Introductionmentioning

confidence: 99%

The new streamline diffusion for the 3D coupled Schrodinger equations with a cross-phase modulation

Rostamy¹

2013

ANZIAMJ

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“…Let Y be the solution of (13), and y the CTG approximation defined by (14). Then, assuming Y 0 and F have the indicated regularity, for…”

Section: Theoremmentioning

confidence: 99%

“…In this connection see also [3]. For three other nonclassical finite element treatments of the wave equation, see Babuška and Janik [2], Johnson [14], and Richter [17].…”

Section: Introductionmentioning

confidence: 99%

“…Strauss and Vazquez [18] note that certain explicit methods fail while an implicit energy-preserving scheme gives a sensible approximation to the generalized Klein-Gordon equation. In addition, the variational formulation of space-time finite element methods seems to facilitate the derivation of a posteriori error estimates, which may serve as the basis for rational adaptive grid refinement (see [14] for an example for the wave equation). Based on these observations, we consider the CTG method a viable approach to many wave problems, and hope that the analysis presented here for the linear wave equation will lay the foundation for future work on the sort of nonlinear problems mentioned above.…”

Section: Introductionmentioning

confidence: 99%

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A continuous space-time finite element method for the wave equation

1996

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Abstract. We consider a finite element method for the nonhomogeneous second-order wave equation, which is formulated in terms of continuous approximation functions in both space and time, thereby giving a unified treatment of the spatial and temporal discretizations. Our analysis uses primarily energy arguments, which are quite common for spatial discretizations but not for time.We present a priori nodal (in time) superconvergence error estimates without any special time step restrictions. Our method is based on tensor-product spaces for the full discretization.

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Structural Dynamic Analysis by a Time-Discontinuous Galerkin Finite Element Method

Wiberg

1996

Int. J. Numer. Meth. Engng.

103

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SUMMARYThis 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.

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Discontinuous Galerkin finite element methods for second order hyperbolic problems

Cited by 221 publications

References 8 publications

The new streamline diffusion for the 3D coupled Schrodinger equations with a cross-phase modulation

The new streamline diffusion for the 3D coupled Schrodinger equations with a cross-phase modulation

A continuous space-time finite element method for the wave equation

Structural Dynamic Analysis by a Time-Discontinuous Galerkin Finite Element Method

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