Results 1 - 10 of 508

Unified analysis of discontinuous Galerkin methods for elliptic problems

by Douglas N. Arnold, Franco Brezzi, Bernardo Cockburn, L. Donatella Marini - SIAM J. Numer. Anal , 2001
"... Abstract. We provide a framework for the analysis of a large class of discontinuous methods for second-order elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed over the past three decades for the numerical treatment ..."
Abstract - Cited by 525 (31 self) - Add to MetaCart
Abstract. We provide a framework for the analysis of a large class of discontinuous methods for second-order elliptic problems. It allows for the understanding and comparison of most of the discontinuous Galerkin methods that have been proposed over the past three decades for the numerical treatment of elliptic problems.
(Show Context)

Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws

by Chi-wang Shu , 1998
"... In these lecture notes we describe the construction, analysis, and application of ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes for hyperbolic conservation laws and related Hamilton-Jacobi equations. ENO and WENO schemes are high order accurate nite di ere ..."
Abstract - Cited by 270 (26 self) - Add to MetaCart
In these lecture notes we describe the construction, analysis, and application of ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes for hyperbolic conservation laws and related Hamilton-Jacobi equations. ENO and WENO schemes are high order accurate nite di erence schemes designed for problems with piecewise smooth solutions containing discontinuities. The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible. ENO and WENO schemes have been quite successful in applications, especially for problems containing both shocks and complicated smooth solution structures, such as compressible turbulence simulations and aeroacoustics. These lecture notes are basically self-contained. It is our hope that with these notes and with the help of the quoted references, the readers can understand the algorithms and code
(Show Context)

The development of discontinuous Galerkin methods

by Bernardo Cockburn , George E. Karniadakis, Chi-Wang Shu , 1999
"... In this paper, we present an overview of the evolution of the discontinuous Galerkin methods since their introduction in 1973 by Reed and Hill, in the framework of neutron transport, until their most recent developments. We show how these methods made their way into the main stream of computational ..."
Abstract - Cited by 182 (20 self) - Add to MetaCart
In this paper, we present an overview of the evolution of the discontinuous Galerkin methods since their introduction in 1973 by Reed and Hill, in the framework of neutron transport, until their most recent developments. We show how these methods made their way into the main stream of computational fluid dynamics and how they are quickly finding use in a wide variety of applications. We review the theoretical and algorithmic aspects of these methods as well as their applications to equations including nonlinear conservation laws, the compressible Navier-Stokes equations, and Hamilton-Jacobi-like equations.

Discontinuous Hp-Finite Element Methods For Advection-Diffusion Problems

by Paul Houston, Christoph Schwab, Endre Süli, Endre S/li - SIAM J. Numer. Anal , 2000
"... We consider the hp-version of the discontinuous Galerkin finite element method for second-order partial differential equations with nonnegative characteristic form. This class of equations includes second-order elliptic and parabolic equations, first)rder hyperbolic equations, as well as problems of ..."
Abstract - Cited by 101 (12 self) - Add to MetaCart
We consider the hp-version of the discontinuous Galerkin finite element method for second-order partial differential equations with nonnegative characteristic form. This class of equations includes second-order elliptic and parabolic equations, first)rder hyperbolic equations, as well as problems of mixed hyperbolic-elliptic-parabolic type. Our main concern is the error analysis of the method in the absence of streamline-diffusion stabilization. In the hyperbolic case, an hpoptimal error bound is derived. In the self-adjoint elliptic case, an error bound that is h)ptimal and 1 p-suboptimal by a power of p is obtained. These estimates are then combined to deduce an error bound in the general case. For element-wise analytic solutions the method exhibits exponential rates of convergence under p-refinement. The theoretical results are illustrated by numerical experiments. Key words. hp-finite element methods, discontinuous Galerkin methods, PDEs with nonneg- ative characteristic form AMS subject classifications. 65N12, 65N15, 65N30 1.

Weighted Essentially Non-Oscillatory Schemes on Triangular Meshes

by C. Hu, Changqing Hu, Changqing Hu, Chi-wang Shu, Chi-wang Shu - J. Comput. Phys , 1998
"... In this paper we construct high order weighted essentially non-oscillatory (WENO) schemes on two dimensional unstructured meshes (triangles) in the finite volume formulation. We present third order schemes using a combination of linear polynomials, and fourth order schemes using a combination of qua ..."
Abstract - Cited by 72 (12 self) - Add to MetaCart
In this paper we construct high order weighted essentially non-oscillatory (WENO) schemes on two dimensional unstructured meshes (triangles) in the finite volume formulation. We present third order schemes using a combination of linear polynomials, and fourth order schemes using a combination of quadratic polynomials. Numerical examples are shown to demonstrate the accuracies and robustness of the methods for shock calculations.
(Show Context)

Reduced basis method for finite volume approximations of parametrized linear evolution equations

by Bernard Haasdonk, Mario Ohlberger - M2AN, Math. Model. Numer. Anal
"... The model order reduction methodology of reduced basis (RB) techniques offers efficient treatment of parametrized partial differential equations (P 2 DEs) by providing both approximate solution procedures and efficient error estimates. RB-methods have so far mainly been applied to finite element sch ..."
Abstract - Cited by 64 (24 self) - Add to MetaCart
The model order reduction methodology of reduced basis (RB) techniques offers efficient treatment of parametrized partial differential equations (P 2 DEs) by providing both approximate solution procedures and efficient error estimates. RB-methods have so far mainly been applied to finite element schemes for elliptic and parabolic problems. In the current study we extend the methodology to general evolution schemes such as finite volume schemes for parabolic and hyperbolic evolution equations. The new theoretic contributions are the formulation of a reduced basis approximation scheme for general evolution problems and the derivation of rigorous a-posteriori error estimates in various norms. Algorithmically, an offline/online decomposition of the scheme and the error estimators is realized. This is the basis for a rapid online computation in case of multiple-simulation requests. We introduce a new offline basis-generation algorithm based on our a posteriori error estimator which combines ideas from existing approaches. Numerical experiments for an instationary convection-diffusion problem demonstrate the efficient applicability of the approach. 1
(Show Context)

Space-Time Discontinuous Galerkin Finite Element Methods

by J. J. W. van der Vegt
"... In these notes an introduction is given to space-time discontinuous Galerkin (DG) finite element methods for hyperbolic and parabolic conservation laws on time dependent domains. The space-time DG discretization is explained in detail, including the definition of the numerical fluxes and stabilizati ..."
Abstract - Cited by 52 (4 self) - Add to MetaCart
In these notes an introduction is given to space-time discontinuous Galerkin (DG) finite element methods for hyperbolic and parabolic conservation laws on time dependent domains. The space-time DG discretization is explained in detail, including the definition of the numerical fluxes and stabilization operators necessary to maintain stable and non-oscillatory solutions. In addition, a pseudo-time integration method for the solution of the algebraic equations resulting from the DG discretization and the relation between the space-time DG method and an arbitrary Lagrangian Eulerian approach are discussed. Finally, a brief overview of some applications to aerodynamics is given.

Arbitrary high order discontinuous Galerkin schemes

by M. Dumbser, C.-D. Munz - GOUDON & E. SONNENDRUCKER EDS). IRMA SERIES IN MATHEMATICS AND THEORETICAL PHYSICS , 2005
"... In this paper we apply the ADER one step time discretization to the Discontinuous Galerkin framework for hyperbolic conservation laws. In the case of linear hyperbolic systems we obtain a quadrature-free explicit single-step scheme of arbitrary order of accuracy in space and time on Cartesian and tr ..."
Abstract - Cited by 50 (7 self) - Add to MetaCart
In this paper we apply the ADER one step time discretization to the Discontinuous Galerkin framework for hyperbolic conservation laws. In the case of linear hyperbolic systems we obtain a quadrature-free explicit single-step scheme of arbitrary order of accuracy in space and time on Cartesian and triangular meshes. The ADER-DG scheme does not need more memory than a first order explicit Euler time-stepping scheme. This becomes possible because of an extensive use of the governing equations inside the numerical scheme. In the nonlinear case, quadrature of the ADER-DG scheme in space and time is performed with Gaussian quadrature formulae of suitable order of accuracy. We show numerical convergence results for the linearized Euler equations up to 10th order of accuracy in space and time on Cartesian and triangular meshes. Numerical results for the nonlinear Euler equations up to 6th order of accuracy in space and time are provided as well. In this paper we also show the possibility of applying a linear reconstruction operator of the order 3N + 2 to the degrees of freedom of the DG method resulting in a numerical scheme of the order 3N + 3 on Cartesian grids where N is the order of the original basis functions before reconstruction.
(Show Context)

A Hybridizable Discontinuous Galerkin Method for the Compressible Euler and Navier-Stokes Equations

by J. Peraire, N. C. Nguyen, B. Cockburn
"... In this paper, we present a Hybridizable Discontinuous Galerkin (HDG) method for the solution of the compressible Euler and Navier-Stokes equations. The method is devised by using the discontinuous Galerkin approximation with a special choice of the numerical fluxes and weakly imposing the continuit ..."
Abstract - Cited by 49 (12 self) - Add to MetaCart
In this paper, we present a Hybridizable Discontinuous Galerkin (HDG) method for the solution of the compressible Euler and Navier-Stokes equations. The method is devised by using the discontinuous Galerkin approximation with a special choice of the numerical fluxes and weakly imposing the continuity of the normal component of the numerical fluxes across the element interfaces. This allows the approximate conserved variables defining the discontinuous Galerkin solution to be locally condensed, thereby resulting in a reduced system which involves only the degrees of freedom of the approximate traces of the solution. The HDG method inherits the geometric flexibility and arbitrary high order accuracy of Discontinuous Galerkin methods, but offers a significant reduction in the computational cost as well as improved accuracy and convergence properties. In particular, we show that HDG produces optimal converges rates for both the conserved quantities as well as the viscous stresses and the heat fluxes. We present some numerical results to demonstrate the accuracy and convergence properties of the method. I.

Locally divergence-free discontinuous Galerkin methods for the Maxwell Equations

by Bernardo Cockburn , Fengyan Li , Chi-Wang Shu - J. Comput. Phys
"... Abstract 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 h ..."
Abstract - Cited by 47 (4 self) - Add to MetaCart
Abstract 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 þ 1)th order of accuracy when piecewise polynomials of degree k are used.
(Show Context)