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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 525 (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.
Adaptive Discontinuous Galerkin Finite Element Methods for Compressible Fluid Flows
 SIAM J. Sci. Comput
"... this paper is to discuss the a posteriori error analysis and adaptive mesh design for discontinuous Galerkin finite element approximations to systems of conservation laws. In Section 2, we introduce the model problem and formulate its discontinuous Galerkin finite element approximation. Section 3 is ..."
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Cited by 122 (17 self)
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this paper is to discuss the a posteriori error analysis and adaptive mesh design for discontinuous Galerkin finite element approximations to systems of conservation laws. In Section 2, we introduce the model problem and formulate its discontinuous Galerkin finite element approximation. Section 3 is devoted to the derivation of weighted a posteriori error bounds for linear functionals of the solution. Finally, in Section 4 we present some numerical examples to demonstrate the performance of the resulting adaptive finite element algorithm. 2 Model problem and discretisation Given an open bounded polyhedral domain fl in lI n, n _> 1, with boundary 0fl, we consider the following problem: find u: fl > lI m, m _> 1, such that div(u) = 0 in , (2.1) where, ,: m __> mxn is continuously differentiable. We assume that the system of conservation laws (2.1) may be supplemented by appropriate initial/boundary conditions. For example, assuming that B(u, y) := EiL1 biVu'(u) has m real eigenvalues and a complete set of linearly independent eigenvectors for all y = (yl,, Yn) C n; then at inflow/outflow boundaries, we require that B(u, n)(u g) = 0, where n denotes the unit outward normal vector to 0fl, B(u, n) is the negative part of B(u, n) and g is a (given) realvalued function. To formulate the discontinuous Galerkin finite element method (DGFEM, for short) for (2.1), we first introduce some notation. Let 7 = {n} be an admissible subdivision of fl into open element domains n; here h is a piecewise constant mesh function with h(x) = diam(n) 2 Houston e al. when x is in element n. For p Iq0, we define the following finite element space n,  {v [L()]": vl [%(n)] " Vn }, where Pp(n) denotes the set of polynomials of degree at most p over n. Given that v [Hi(n)] m for each n...
Stabilization mechanisms in discontinuous Galerkin finite element methods
 Comput. Methods Appl. Mech. Engrg
, 2006
"... In this paper we propose a new general framework for the construction and the analysis of Discontinuous Galerkin (DG) methods which reveals a basic mechanism, responsible for certain distinctive stability properties of DG methods. We show that this mechanism is common to apparently unrelated stabili ..."
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Cited by 44 (6 self)
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In this paper we propose a new general framework for the construction and the analysis of Discontinuous Galerkin (DG) methods which reveals a basic mechanism, responsible for certain distinctive stability properties of DG methods. We show that this mechanism is common to apparently unrelated stabilizations, including jump penalty, upwinding, and Hughes–Franca type residualbased stabilizations.
A discontinuous Galerkin method with nonoverlapping domain decomposition for the Stokes and NavierStokes problems
 Math. Comp
"... Abstract. A family of discontinuous Galerkin finite element methods is formulated and analyzed for Stokes and NavierStokes problems. An infsup condition is established as well as optimal energy estimates for the velocity and L2 estimates for the pressure. In addition, it is shown that the method c ..."
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Cited by 42 (8 self)
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Abstract. A family of discontinuous Galerkin finite element methods is formulated and analyzed for Stokes and NavierStokes problems. An infsup condition is established as well as optimal energy estimates for the velocity and L2 estimates for the pressure. In addition, it is shown that the method can treat a finite number of nonoverlapping domains with nonmatching grids at interfaces. 1.
A CLASS OF DISCONTINUOUS PETROVGALERKIN METHODS. PART I: THE TRANSPORT EQUATION
"... Abstract. Considering a simple model transport problem, we present a new finite element method. While the new method fits in the class of discontinuous Galerkin (DG) methods, it differs from standard DG and streamline diffusion methods, in that it uses a space of discontinuous trial functions tailor ..."
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Cited by 41 (13 self)
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Abstract. Considering a simple model transport problem, we present a new finite element method. While the new method fits in the class of discontinuous Galerkin (DG) methods, it differs from standard DG and streamline diffusion methods, in that it uses a space of discontinuous trial functions tailored for stability. The new method, unlike the older approaches, yields optimal estimates for the primal variable in both the element size h and polynomial degree p, and outperforms the standard upwind DG method. 1.
Plane wave discontinuous Galerkin methods
, 2007
"... Abstract. We are concerned with a finite element approximation for timeharmonic wave propagation governed by the Helmholtz equation. The usually oscillatory behavior of solutions, along with numerical dispersion, render standard finite element methods grossly inefficient already in mediumfrequency ..."
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Cited by 40 (8 self)
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Abstract. We are concerned with a finite element approximation for timeharmonic wave propagation governed by the Helmholtz equation. The usually oscillatory behavior of solutions, along with numerical dispersion, render standard finite element methods grossly inefficient already in mediumfrequency regimes. As an alternative, methods that incorporate information about the solution in the form of plane waves have been proposed. Among them the ultra weak variational formulation (UWVF) of Cessenat and Despres [O. Cessenat and B. Despres, Application of an ultra weak variational formulation of elliptic PDEs to the twodimensional Helmholtz equation, SIAM J. Numer. Anal., 35 (1998), pp. 255–299.]. We identify the UWVF as representative of a class of Trefftztype discontinuous Galerkin methods that employs trial and test spaces spanned by local plane waves. In this paper we give a priori convergence estimates for the hversion of these plane wave discontinuous Galerkin methods. To that end, we develop new inverse and approximation estimates for plane waves in two dimensions and use these in the context of duality techniques. Asymptotic optimality of the method in a mesh dependent norm can be established. However, the estimates require a minimal resolution of the mesh beyond what it takes to resolve the wavelength. We give numerical evidence that this requirement cannot be dispensed with. It reflects the presence of numerical dispersion. Key words. Wave propagation, finite element methods, discontinuous Galerkin methods, plane waves, ultra weak variational formulation, duality estimates, numerical dispersion AMS subject classifications. 65N15, 65N30, 35J05
Optimal a priori error estimates for the hpversion of the local discontinuous Galerkin method for convectiondiffusion problems
 Math. Comp
"... Abstract. We study the convergence properties of the hpversion of the local discontinuous Galerkin finite element method for convectiondiffusion problems; we consider a model problem in a onedimensional space domain. We allow arbitrary meshes and polynomial degree distributions and obtain upper b ..."
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Cited by 35 (7 self)
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Abstract. We study the convergence properties of the hpversion of the local discontinuous Galerkin finite element method for convectiondiffusion problems; we consider a model problem in a onedimensional space domain. We allow arbitrary meshes and polynomial degree distributions and obtain upper bounds for the energy norm of the error which are explicit in the meshwidth h, in the polynomial degree p, and in the regularity of the exact solution. We identify a special numerical flux for which the estimates are optimal in both h and p. The theoretical results are confirmed in a series of numerical examples. 1.
An overlapping domain decomposition preconditioner for a class of discontinuous Galerkin approximations of advectiondiffusion problems
 Math. Comp
"... Abstract. We consider a scalar advection{diusion problem and a recently proposed discontinuous Galerkin approximation, which employs discontinuous nite element spaces and suitable bilinear forms containing interface terms that ensure consistency. For the corresponding sparse, nonsymmetric linear s ..."
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Cited by 30 (0 self)
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Abstract. We consider a scalar advection{diusion problem and a recently proposed discontinuous Galerkin approximation, which employs discontinuous nite element spaces and suitable bilinear forms containing interface terms that ensure consistency. For the corresponding sparse, nonsymmetric linear system, we propose and study an additive, two{level overlapping Schwarz preconditioner, consisting of a coarse problem on a coarse triangulation and local solvers associated to suitable problems dened on a family of subdomains. This is a generalization of the corresponding overlapping method for approximations on continuous nite element spaces. Related to the lack of continuity of our approximation spaces, some interesting new features arise in our generalization, which have no analog in the conforming case. We prove an upper bound for the number of iterations obtained by using this preconditioner with GMRES, which is independent of the number of degrees of freedom of the original problem and the number of subdomains. The performance of the method is illustrated by several numerical experiments for dierent test problems, using linear nite elements in two dimensions.
A unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty
"... We discuss stabilized Galerkin approximations in a new framework, widening the scope from the usual dichotomy of the discontinuous Galerkin method on the one hand and Petrov– Galerkin methods such as the SUPG method on the other. The idea is to use interior penalty terms as a means of stabilizing th ..."
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Cited by 29 (14 self)
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We discuss stabilized Galerkin approximations in a new framework, widening the scope from the usual dichotomy of the discontinuous Galerkin method on the one hand and Petrov– Galerkin methods such as the SUPG method on the other. The idea is to use interior penalty terms as a means of stabilizing the finite element method using conforming or nonconforming approximation, thus circumventing the need of a Petrov–Galerkintype choice of spaces. This is made possible by adding a higherorder penalty term giving L²control of the jumps in the gradients between adjacent elements. We consider convectiondiffusionreaction problems using piecewise linear approximations and prove optimal order a priori error estimates for two different finite element spaces, the standard H¹conforming space of piecewise linears and the nonconforming space of piecewise linear elements where the nodes are situated at the midpoint of the element sides (the Crouzeix–Raviart element). Moreover, we show how the formulation extends to discontinuous Galerkin interior penalty methods in a natural way by domain decomposition using Nitsche’s method.
CONTINUOUS INTERIOR PENALTY hpFINITE ELEMENT METHODS FOR ADVECTION AND ADVECTIONDIFFUSION EQUATIONS
"... Abstract. A continuous interior penalty hpfinite element method that penalizes the jump of the gradient of the discrete solution across mesh interfaces is introduced. Error estimates are obtained for advection and advectiondiffusion equations. The analysis relies on three technical results that ar ..."
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Cited by 29 (10 self)
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Abstract. A continuous interior penalty hpfinite element method that penalizes the jump of the gradient of the discrete solution across mesh interfaces is introduced. Error estimates are obtained for advection and advectiondiffusion equations. The analysis relies on three technical results that are of independent interest: an hpinverse trace inequality, a local discontinuous to continuous hpinterpolation result, and hperrorestimatesforcontinuousL 2orthogonal projections. 1.