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180
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.
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.
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 45 (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.
Multigrid Solution for HighOrder Discontinuous Galerkin . . .
, 2004
"... A highorder discontinuous Galerkin finite element discretization and pmultigrid solution procedure for the compressible NavierStokes equations are presented. The discretization has an elementcompact stencil such that only elements sharing a face are coupled, regardless of the solution space. Thi ..."
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Cited by 45 (16 self)
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A highorder discontinuous Galerkin finite element discretization and pmultigrid solution procedure for the compressible NavierStokes equations are presented. The discretization has an elementcompact stencil such that only elements sharing a face are coupled, regardless of the solution space. This limited coupling maximizes the effectiveness of the pmultigrid solver, which relies on an elementline Jacobi smoother. The elementline Jacobi smoother solves implicitly on lines of elements formed based on the coupling between elements in a p = 0 discretization of the scalar transport equation. Fourier analysis of 2D scalar convectiondiffusion shows that the elementline Jacobi smoother as well as the simpler element Jacobi smoother are stable independent of p and flow condition. Mesh refinement studies for simple problems with analytic solutions demonstrate that the discretization achieves optimal order of accuracy of O(h p+1). A subsonic, airfoil test case shows that the multigrid convergence rate is independent of p but weakly dependent on h. Finally, higherorder is shown to outperform grid refinement in terms of the time required to reach a desired accuracy level.
Contact discontinuity capturing schemes for linear advection and compressible gas dynamics
 J. Sci. Comput
, 2002
"... Abstract We present a nondiffusive and contact discontinuity capturing scheme for linear advection and compressible Euler system. In the case of advection, this scheme is equivalent to the UltraBee limiter of [20], [24]. We prove for the UltraBee scheme a property of exact advection for a large s ..."
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Cited by 44 (6 self)
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Abstract We present a nondiffusive and contact discontinuity capturing scheme for linear advection and compressible Euler system. In the case of advection, this scheme is equivalent to the UltraBee limiter of [20], [24]. We prove for the UltraBee scheme a property of exact advection for a large set of piecewise constant functions. We prove that the numerical error is uniformly bounded in time for such prepared (i.e. piecewise constant) initial data, and state a conjecture of nondiffusion at infinite time based on some local overcompressivity of the scheme for general initial data. We generalize the scheme to compressible gas dynamics and present some numerical results.
Local Discontinuous Galerkin Methods For The Stokes System
, 2000
"... In this paper, we introduce and analyze local discontinuous Galerkin methods for the Stokes system. For arbitrary meshes with hanging nodes and elements of various shapes we derive a priori estimates for the L²norm of the errors in the velocities and the pressure. We show that optimal order estimat ..."
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Cited by 42 (16 self)
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In this paper, we introduce and analyze local discontinuous Galerkin methods for the Stokes system. For arbitrary meshes with hanging nodes and elements of various shapes we derive a priori estimates for the L²norm of the errors in the velocities and the pressure. We show that optimal order estimates are obtained when polynomials of degree k are used for each component of the velocity and polynomials of degree k  1 for the pressure, for any k >= 1. We also consider the case in which all the unknowns are approximated with polynomials of degree k and show that, although the orders of convergence remain the same, the method is more efficient. Numerical experiments verifying these facts are displayed.
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 34 (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.
Discontinuous Galerkin methods for Friedrichs’ symmetric systems
 I. General theory. SIAM J. Numer. Anal
, 2005
"... Abstract. This paper is the second part of a work attempting to give a unified analysis of Discontinuous Galerkin methods. The setting under scrutiny is that of Friedrichs ’ systems endowed with a particular 2×2 structure in which some of the unknowns can be eliminated to yield a system of secondor ..."
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Cited by 32 (13 self)
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Abstract. This paper is the second part of a work attempting to give a unified analysis of Discontinuous Galerkin methods. The setting under scrutiny is that of Friedrichs ’ systems endowed with a particular 2×2 structure in which some of the unknowns can be eliminated to yield a system of secondorder ellipticlike PDE’s for the remaining unknowns. For such systems, a general Discontinuous Galerkin method is proposed and analyzed. The key feature is that the unknowns that can be eliminated at the continuous level can also be eliminated at the discrete level by solving local problems. All the design constraints on the boundary operators that weakly enforce boundary conditions and on the interface operators that penalize interface jumps are fully stated. Examples are given for advection–diffusion–reaction, linear elasticity, and a simplified version of the magnetohydrodynamics equations. Comparisons with wellknown Discontinuous Galerkin approximations for the Poisson equation are presented.
Local Discontinuous Galerkin Methods for Nonlinear Dispersive Equations
, 2003
"... We develop local discontinuous Galerkin (DG) methods for solving nonlinear dispersive partial dierential equations that have compactly supported traveling waves solutions, the socalled "compactons". The schemes we present extend the previous works of Yan and Shu on approximating soluti ..."
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Cited by 31 (5 self)
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We develop local discontinuous Galerkin (DG) methods for solving nonlinear dispersive partial dierential equations that have compactly supported traveling waves solutions, the socalled "compactons". The schemes we present extend the previous works of Yan and Shu on approximating solutions for linear dispersive equations and for certain KdVtype equations. We present
A characterization of hybridized mixed methods for second order elliptic problems
 SIAM J. Numer. Anal
"... This paper is dedicated to Jim Douglas, Jr., on the occasion of his 75 th birthday. Abstract. In this paper, we give a new characterization of the approximate solution given by hybridized mixed methods for secondorder, selfadjoint elliptic problems. We apply this characterization to obtain an expl ..."
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Cited by 29 (7 self)
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This paper is dedicated to Jim Douglas, Jr., on the occasion of his 75 th birthday. Abstract. In this paper, we give a new characterization of the approximate solution given by hybridized mixed methods for secondorder, selfadjoint elliptic problems. We apply this characterization to obtain an explicit formula for the entries of the matrix equation for the Lagrange multiplier unknowns resulting from hybridization. We also obtain necessary and sufficient conditions under which the multipliers of the RaviartThomas and the BrezziDouglasMarini methods of similar order are identical. 1.