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491
Unified analysis of discontinuous Galerkin methods for elliptic problems
 SIAM J. Numer. Anal
, 2001
"... Abstract. We provide a framework for the analysis of a large class of discontinuous methods for secondorder 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 ..."
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Cited by 519 (31 self)
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Abstract. We provide a framework for the analysis of a large class of discontinuous methods for secondorder 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.
Essentially nonoscillatory and weighted essentially nonoscillatory schemes for hyperbolic conservation laws
, 1998
"... In these lecture notes we describe the construction, analysis, and application of ENO (Essentially NonOscillatory) and WENO (Weighted Essentially NonOscillatory) schemes for hyperbolic conservation laws and related HamiltonJacobi equations. ENO and WENO schemes are high order accurate nite di ere ..."
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Cited by 275 (28 self)
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In these lecture notes we describe the construction, analysis, and application of ENO (Essentially NonOscillatory) and WENO (Weighted Essentially NonOscillatory) schemes for hyperbolic conservation laws and related HamiltonJacobi 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 selfcontained. It is our hope that with these notes and with the help of the quoted references, the readers can understand the algorithms and code
The development of discontinuous Galerkin methods
, 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 ..."
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Cited by 181 (19 self)
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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 NavierStokes equations, and HamiltonJacobilike equations.
Discontinuous HpFinite Element Methods For AdvectionDiffusion Problems
 SIAM J. Numer. Anal
, 2000
"... We consider the hpversion of the discontinuous Galerkin finite element method for secondorder partial differential equations with nonnegative characteristic form. This class of equations includes secondorder elliptic and parabolic equations, first)rder hyperbolic equations, as well as problems of ..."
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Cited by 101 (12 self)
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We consider the hpversion of the discontinuous Galerkin finite element method for secondorder partial differential equations with nonnegative characteristic form. This class of equations includes secondorder elliptic and parabolic equations, first)rder hyperbolic equations, as well as problems of mixed hyperbolicellipticparabolic type. Our main concern is the error analysis of the method in the absence of streamlinediffusion stabilization. In the hyperbolic case, an hpoptimal error bound is derived. In the selfadjoint elliptic case, an error bound that is h)ptimal and 1 psuboptimal by a power of p is obtained. These estimates are then combined to deduce an error bound in the general case. For elementwise analytic solutions the method exhibits exponential rates of convergence under prefinement. The theoretical results are illustrated by numerical experiments. Key words. hpfinite element methods, discontinuous Galerkin methods, PDEs with nonneg ative characteristic form AMS subject classifications. 65N12, 65N15, 65N30 1.
Weighted Essentially NonOscillatory Schemes on Triangular Meshes
 J. Comput. Phys
, 1998
"... In this paper we construct high order weighted essentially nonoscillatory (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 ..."
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Cited by 70 (12 self)
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In this paper we construct high order weighted essentially nonoscillatory (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.
Reduced basis method for finite volume approximations of parametrized linear evolution equations
 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. RBmethods have so far mainly been applied to finite element sch ..."
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Cited by 66 (24 self)
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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. RBmethods 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 aposteriori 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 multiplesimulation requests. We introduce a new offline basisgeneration algorithm based on our a posteriori error estimator which combines ideas from existing approaches. Numerical experiments for an instationary convectiondiffusion problem demonstrate the efficient applicability of the approach. 1
SpaceTime Discontinuous Galerkin Finite Element Methods
"... In these notes an introduction is given to spacetime discontinuous Galerkin (DG) finite element methods for hyperbolic and parabolic conservation laws on time dependent domains. The spacetime DG discretization is explained in detail, including the definition of the numerical fluxes and stabilizati ..."
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Cited by 53 (4 self)
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In these notes an introduction is given to spacetime discontinuous Galerkin (DG) finite element methods for hyperbolic and parabolic conservation laws on time dependent domains. The spacetime DG discretization is explained in detail, including the definition of the numerical fluxes and stabilization operators necessary to maintain stable and nonoscillatory solutions. In addition, a pseudotime integration method for the solution of the algebraic equations resulting from the DG discretization and the relation between the spacetime 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
 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 quadraturefree explicit singlestep scheme of arbitrary order of accuracy in space and time on Cartesian and tr ..."
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Cited by 50 (7 self)
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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 quadraturefree explicit singlestep scheme of arbitrary order of accuracy in space and time on Cartesian and triangular meshes. The ADERDG scheme does not need more memory than a first order explicit Euler timestepping scheme. This becomes possible because of an extensive use of the governing equations inside the numerical scheme. In the nonlinear case, quadrature of the ADERDG 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.
A Hybridizable Discontinuous Galerkin Method for the Compressible Euler and NavierStokes Equations
"... In this paper, we present a Hybridizable Discontinuous Galerkin (HDG) method for the solution of the compressible Euler and NavierStokes equations. The method is devised by using the discontinuous Galerkin approximation with a special choice of the numerical fluxes and weakly imposing the continuit ..."
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Cited by 49 (13 self)
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In this paper, we present a Hybridizable Discontinuous Galerkin (HDG) method for the solution of the compressible Euler and NavierStokes 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.
Adaptive mesh methods for one and twodimensional hyperbolic conservation laws
 SIAM J. Numer. Anal
"... Abstract. We develop efficient moving mesh algorithms for one and twodimensional hyperbolic systems of conservation laws. The algorithms are formed by two independent parts: PDE evolution and meshredistribution. The first part can be any appropriate highresolution scheme, and the second part is ..."
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Cited by 45 (8 self)
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Abstract. We develop efficient moving mesh algorithms for one and twodimensional hyperbolic systems of conservation laws. The algorithms are formed by two independent parts: PDE evolution and meshredistribution. The first part can be any appropriate highresolution scheme, and the second part is based on an iterative procedure. In each iteration, meshes are first redistributed by an equidistribution principle, and then on the resulting new grids the underlying numerical solutions are updated by a conservativeinterpolation formula proposed in this work. The iteration for the meshredistribution at a given time step is complete when the meshes governed by a nonlinear equation reach the equilibrium state. The main idea of the proposed method is to keep the massconservation of the underlying numerical solution at each redistribution step. In one dimension, we can show that the underlying numerical approximation obtained in the meshredistribution part satisfies the desired TVD property, which guarantees that the numerical solution at any time level is TVD, provided that the PDE solver in the first part satisfies such a property. Several test problems in one and two dimensions are computed using the proposed moving mesh algorithm. The computations demonstrate that our methods are efficient for solving problems with shock discontinuities, obtaining the same resolution with a much smaller number of grid points than the uniform mesh approach.