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Adjoint consistency analysis of discontinuous Galerkin discretizations
 SIAM J. Numer. Anal
"... Abstract. This paper is concerned with the adjoint consistency of discontinuous Galerkin (DG) discretizations. Adjoint consistency—in addition to consistency—is the key requirement for DG discretizations to be of optimal order in L2 as well as measured in terms of target functionals. We provide a ge ..."
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Cited by 27 (5 self)
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Abstract. This paper is concerned with the adjoint consistency of discontinuous Galerkin (DG) discretizations. Adjoint consistency—in addition to consistency—is the key requirement for DG discretizations to be of optimal order in L2 as well as measured in terms of target functionals. We provide a general framework for analyzing the adjoint consistency of DG discretizations which is also useful for the derivation of adjoint consistent methods. This analysis will be performed for the DG discretizations of the linear advection equation, the interior penalty DG method for elliptic problems, and the DG discretization of the compressible Euler equations. This framework is then used to derive an adjoint consistent DG discretization of the compressible Navier–Stokes equations. Numerical experiments demonstrate the link of adjoint consistency to the accuracy of numerical flow solutions and the smoothness of discrete adjoint solutions.
Quasioptimal convergence rate of an adaptive discontinuous Galerkin method
 SIAM J. NUMER. ANAL
, 2010
"... We analyze an adaptive discontinuous finite element method (ADFEM) for symmetric second order linear elliptic operators. The method is formulated on nonconforming meshes made of simplices or quadrilaterals, with any polynomial degree and in any dimension ≥ 2. We prove that the ADFEM is a contraction ..."
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Cited by 21 (4 self)
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We analyze an adaptive discontinuous finite element method (ADFEM) for symmetric second order linear elliptic operators. The method is formulated on nonconforming meshes made of simplices or quadrilaterals, with any polynomial degree and in any dimension ≥ 2. We prove that the ADFEM is a contraction for the sum of the energy error and the scaled error estimator, between two consecutive adaptive loops. We design a refinement procedure that maintains the level of nonconformity uniformly bounded, and prove that the approximation classes using continuous and discontinuous finite elements are equivalent. The geometric decay and the equivalence of classes are instrumental to derive optimal cardinality of ADFEM. We show that ADFEM (and AFEM on nonconforming meshes) yields a decay rate of energy error plus oscillation in terms of number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.
The direct discontinuous Galerkin (ddg) methods for diffusion problems
 SIAM J. Numer. Anal
"... Abstract. Based on a novel numerical flux involving jumps of even order derivatives of the numerical solution, a direct discontinuous Galerkin (DDG) method for diffusion problems was introduced in [H. Liu and J. Yan, SIAM J. Numer. Anal. 47(1) (2009), 475698]. In this work, we show that higher orde ..."
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Cited by 20 (5 self)
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Abstract. Based on a novel numerical flux involving jumps of even order derivatives of the numerical solution, a direct discontinuous Galerkin (DDG) method for diffusion problems was introduced in [H. Liu and J. Yan, SIAM J. Numer. Anal. 47(1) (2009), 475698]. In this work, we show that higher order (k≥4) derivatives in the numerical flux can be avoided if some interface corrections are included in the weak formulation of the DDG method; still the jump of 2nd order derivatives is shown to be important for the method to be efficient with a fixed penalty parameter for all pk elements. The refined DDG method with such numerical fluxes enjoys the optimal (k+1)th order of accuracy. The developed method is also extended to solve convection diffusion problems in both one and twodimensional settings. A series of numerical tests are presented to demonstrate the high order accuracy of the method.
A Central Discontinuous Galerkin Method for HamiltonJacobi Equations
 J SCI COMPUT
, 2010
"... In this paper, a central discontinuous Galerkin method is proposed to solve for the viscosity solutions of HamiltonJacobi equations. Central discontinuous Galerkin methods were originally introduced for hyperbolic conservation laws. They combine the central scheme and the discontinuous Galerkin m ..."
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Cited by 15 (1 self)
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In this paper, a central discontinuous Galerkin method is proposed to solve for the viscosity solutions of HamiltonJacobi equations. Central discontinuous Galerkin methods were originally introduced for hyperbolic conservation laws. They combine the central scheme and the discontinuous Galerkin method and therefore carry many features of both methods. Since HamiltonJacobi equations in general are not in the divergence form, it is not straightforward to design a discontinuous Galerkin method to directly solve such equations. By recognizing and following a “weightedresidual” or “stabilizationbased” formulation of central discontinuous Galerkin methods when applied to hyperbolic conservation laws, we design a high order numerical method for HamiltonJacobi equations. The L2 stability and the error estimate are established for the proposed method when the Hamiltonians are linear. The overall performance of the method in approximating the viscosity solutions of general HamiltonJacobi equations are demonstrated through extensive numerical experiments, which involve linear, nonlinear, smooth, nonsmooth, convex, or nonconvex Hamiltonians.
Lockingfree ReissnerMindlin elements without reduced integration
 COMPUT. METHODS APPL. MECH. ENGRG
, 2006
"... In a recent paper of Arnold, Brezzi, and Marini [4], the ideas of discontinuous Galerkin methods were used to obtain and analyze two new families of locking free finite element methods for the approximation of the Reissner–Mindlin plate problem. By following their basic approach, but making differ ..."
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Cited by 14 (1 self)
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In a recent paper of Arnold, Brezzi, and Marini [4], the ideas of discontinuous Galerkin methods were used to obtain and analyze two new families of locking free finite element methods for the approximation of the Reissner–Mindlin plate problem. By following their basic approach, but making different choices of finite element spaces, we develop and analyze other families of locking free finite elements that eliminate the need for the introduction of a reduction operator, which has been a central feature of many lockingfree methods. For k ≥ 2, all the methods use piecewise polynomials of degree k to approximate the transverse displacement and (possibly subsets) of piecewise polynomials of degree k − 1 to approximate both the rotation and shear stress vectors. The approximation spaces for the rotation and the shear stress are always identical. The methods vary in the amount of interelement continuity required. In terms of smallest number of degrees of freedom, the simplest method approximates the transverse displacement with continuous, piecewise quadratics and both the rotation and shear stress with rotated linear BrezziDouglasMarini elements.
MULTILEVEL PRECONDITIONERS FOR DISCONTINUOUS GALERKIN APPROXIMATIONS OF ELLIPTIC PROBLEMS WITH JUMP COEFFICIENTS
, 2010
"... In this article we develop and analyze twolevel and multilevel methods for the family of Interior Penalty (IP) discontinuous Galerkin (DG) discretizations of second order elliptic problems with rough coefficients (exhibiting large jumps across interfaces in the domain). These methods are based on ..."
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Cited by 13 (5 self)
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In this article we develop and analyze twolevel and multilevel methods for the family of Interior Penalty (IP) discontinuous Galerkin (DG) discretizations of second order elliptic problems with rough coefficients (exhibiting large jumps across interfaces in the domain). These methods are based on a decomposition of the DG finite element space that inherently hinges on the diffusion coefficient of the problem. Our analysis of the proposed preconditioners is presented for both symmetric and nonsymmetric IP schemes, and we establish both robustness with respect to the jump in the coefficient and nearoptimality with respect to the mesh size. Following the analysis, we present a sequence of detailed numerical results which verify the
FUNCTIONAL A POSTERIORI ERROR ESTIMATES FOR DISCONTINUOUS GALERKIN APPROXIMATIONS OF ELLIPTIC PROBLEMS
"... Abstract. In this paper, we develop functional a posteriori error estimates for DG approximations of elliptic boundaryvalue problems. These estimates are based on a certain projection of DG approximations to the respective energy space and functional a posteriori estimates for conforming approximat ..."
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Cited by 11 (2 self)
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Abstract. In this paper, we develop functional a posteriori error estimates for DG approximations of elliptic boundaryvalue problems. These estimates are based on a certain projection of DG approximations to the respective energy space and functional a posteriori estimates for conforming approximations (see [30, 31]). On these grounds we derive twosided guaranteed and computable bounds for the errors in ”broken ” energy norms. A series of numerical examples presented confirm the efficiency of the estimates. 1.
Low order discontinuous Galerkin methods for second order elliptic problems
 SIAM J. Numer. Anal
"... Abstract. We consider DGmethods for 2nd order scalar elliptic problems using piecewise affine approximation in two or three space dimensions. We prove that both the symmetric and the nonsymmetric version of the DGmethod are wellposed also without penalization of the interelement solution jumps pr ..."
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Cited by 10 (3 self)
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Abstract. We consider DGmethods for 2nd order scalar elliptic problems using piecewise affine approximation in two or three space dimensions. We prove that both the symmetric and the nonsymmetric version of the DGmethod are wellposed also without penalization of the interelement solution jumps provided boundary conditions are imposed weakly. Optimal convergence is proved for sufficiently regular meshes and data. We then propose a discontinuous Galerkin method using piecewise affine functions enriched with quadratic bubbles. Using this space we prove optimal convergence in the energy norm for both a symmetric and nonsymmetric DGmethod without stabilization. All these proposed methods share the feature that they conserve mass locally independent of the penalty parameter. Key words. Discontinuous Galerkin; elliptic equation; CrouzeixRaviart approximation; interior penalty; local mass conservation. AMS subject classifications. 65M160, 65M15 1. Introduction. The discontinuous Galerkin method (DG) for (2n)thorder elliptic problems was introduced and analysed by Baker [2] with special focus on the fourth order case. In this work both the jumps of the solution and its gradient were
A Multilevel Method for Discontinuous Galerkin Approximation of Threedimensional Elliptic Problems
"... Summary. We construct optimal order multilevel preconditioners for interiorpenalty discontinuous Galerkin (DG) finite element discretizations of 3D elliptic boundaryvalue problems. A specific assembling process is proposed which allows us to characterize the hierarchical splitting locally. This is ..."
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Cited by 8 (1 self)
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Summary. We construct optimal order multilevel preconditioners for interiorpenalty discontinuous Galerkin (DG) finite element discretizations of 3D elliptic boundaryvalue problems. A specific assembling process is proposed which allows us to characterize the hierarchical splitting locally. This is also the key for a local analysis of the angle between the resulting subspaces. Applying the corresponding twolevel basis transformation recursively, a sequence of algebraic problems is generated. These discrete problems can be associated with coarse versions of DG approximations (of the solution to the original variational problem) on a hierarchy of geometrically nested meshes. The presented numerical results demonstrate the potential of this approach. 1
LOCAL ERROR ANALYSIS OF DISCONTINUOUS GALERKIN METHODS FOR ADVECTIONDOMINATED ELLIPTIC LINEARQUADRATIC OPTIMAL CONTROL PROBLEMS ∗
"... Abstract. This paper analyzes the local properties of the symmetric interior penalty upwind discontinuous Galerkin method (SIPG) for the numerical solution of optimal control problems governed by linear reactionadvectiondiffusion equations with distributed controls. The theoretical and numerical r ..."
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Abstract. This paper analyzes the local properties of the symmetric interior penalty upwind discontinuous Galerkin method (SIPG) for the numerical solution of optimal control problems governed by linear reactionadvectiondiffusion equations with distributed controls. The theoretical and numerical results presented in this paper show that for advectiondominated problems the convergence properties of the SIPG discretization can be superior to the convergence properties of stabilized finite element discretizations such as the streamline upwind Petrov Galerkin (SUPG) method. For example, we show that for a small diffusion parameter the SIPG method is optimal in the interior of the domain. This is in sharp contrast to SUPG discretizations, for which it is known that the existence of boundary layers can pollute the numerical solution of optimal control problems everywhere even into domains where the solution is smooth and, as a consequence, in general reduces the convergence rates to only first order. In order to prove the nice convergence properties of the SIPG discretization for optimal control problems, we first improve local error estimates of the SIPG discretization for single advectiondominated equations by showing that the size of the numerical boundary layer is controlled not by the mesh size but rather by the size of the diffusion parameter. As a result, for small diffusion, the boundary layers are too ”weak ” to pollute the SIPG solution into domains of smoothness in optimal control problems. This favorable property of the SIPG method is due to the weak treatment of boundary conditions which is natural for discontinuous Galerkin methods, while for SUPG methods strong imposition of boundary conditions is more conventional. The importance of the weak treatment of boundary conditions for the solution of advection dominated optimal control problems with distributed controls is also supported by our numerical results. Key words. Optimal control, advectiondiffusion equations, discontinuous Galerkin methods, discretization, local error estimates, distributed control.