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45
Energy norm a posteriori error estimation for mixed discontinuous Galerkin approximations of the Maxwell operator
, 2003
"... Abstract. In this paper, we develop the a posteriori error estimation of mixed discontinuous Galerkin finite element approximations of the Stokes problem. In particular, we derive computable upper bounds on the error, measured in terms of a natural (mesh–dependent) energy norm. This is done by rewri ..."
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Cited by 53 (8 self)
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Abstract. In this paper, we develop the a posteriori error estimation of mixed discontinuous Galerkin finite element approximations of the Stokes problem. In particular, we derive computable upper bounds on the error, measured in terms of a natural (mesh–dependent) energy norm. This is done by rewriting the underlying method in a nonconsistent form using appropriate lifting operators, and by employing a decomposition result for the discontinuous spaces. A series of numerical experiments highlighting the performance of the proposed a posteriori error estimator on adaptively refined meshes are presented.
Spacetime discontinuous Galerkin method for nonlinear water waves, preprint for
 J. Comput. Phys.,
, 2006
"... Summary. A spacetime discontinuous Galerkin (DG) finite element method for nonlinear water waves in an inviscid and incompressible fluid is presented. The spacetime DG method results in a conservative numerical discretization on time dependent deforming meshes which follow the free surface evolut ..."
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Cited by 44 (13 self)
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Summary. A spacetime discontinuous Galerkin (DG) finite element method for nonlinear water waves in an inviscid and incompressible fluid is presented. The spacetime DG method results in a conservative numerical discretization on time dependent deforming meshes which follow the free surface evolution. The dispersion and dissipation errors of the scheme are investigated and the algorithm is demonstrated with the simulation of waves generated by a wave maker.
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 locally conservative LDG method for the incompressible NavierStokes equations
 Math. Comp
"... Abstract. In this paper a new local discontinuous Galerkin method for the incompressible stationary NavierStokes equations is proposed and analyzed. Four important features render this method unique: its stability, its local conservativity, its highorder accuracy, and the exact satisfaction of the ..."
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Cited by 38 (13 self)
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Abstract. In this paper a new local discontinuous Galerkin method for the incompressible stationary NavierStokes equations is proposed and analyzed. Four important features render this method unique: its stability, its local conservativity, its highorder accuracy, and the exact satisfaction of the incompressibility constraint. Although the method uses completely discontinuous approximations, a globally divergencefree approximate velocity in H(div; Ω) is obtained by simple, elementbyelement postprocessing. Optimal error estimates are proven and an iterative procedure used to compute the approximate solution is shown to converge. This procedure is nothing but a discrete version of the classical fixed point iteration used to obtain existence and uniqueness of solutions to the incompressible NavierStokes equations by solving a sequence of Oseen problems. Numerical results are shown which verify the theoretical rates of convergence. They also confirm the independence of the number of fixed point iterations with respect to the discretization parameters. Finally, they show that the method works well for a wide range of Reynolds numbers. 1.
THE DERIVATION OF HYBRIDIZABLE DISCONTINUOUS GALERKIN METHODS FOR STOKES FLOW
"... Abstract. In this paper, we introduce a new class of discontinuous Galerkin methods for the Stokes equations. The main feature of these methods is that they can be implemented in an efficient way through a hybridization procedure which reduces the globally coupled unknowns to certain approximations ..."
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Cited by 36 (5 self)
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Abstract. In this paper, we introduce a new class of discontinuous Galerkin methods for the Stokes equations. The main feature of these methods is that they can be implemented in an efficient way through a hybridization procedure which reduces the globally coupled unknowns to certain approximations on the element boundaries. We present four ways of hybridizing the methods, which differ by the choice of the globally coupled unknowns. Classical methods for the Stokes equations can be thought of as limiting cases of these new methods.
Stabilized interior penalty methods for the timeharmonic Maxwell equations
 ComputerMethods in AppliedMechanics and Engineering
, 2002
"... We propose stabilized interior penalty discontinuous Galerkin methods for the indefinite time–harmonic Maxwell system. The methods are based on a mixed formulation of the boundary value problem chosen to provide control on the divergence of the electric field. We prove optimal error estimates for th ..."
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Cited by 26 (6 self)
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We propose stabilized interior penalty discontinuous Galerkin methods for the indefinite time–harmonic Maxwell system. The methods are based on a mixed formulation of the boundary value problem chosen to provide control on the divergence of the electric field. We prove optimal error estimates for the methods in the special case of smooth coefficients and perfectly conducting boundary using a duality approach. Key words: Finite elements, discontinuous Galerkin methods, interior penalty methods, timeharmonic Maxwell’s equations 1
A note on discontinuous Galerkin divergencefree solutions of the NavierStokes equations
 J. Sci. Comput
"... We present a class of discontinuous Galerkin methods for the incompressible NavierStokes equations yielding exactly divergencefree solutions. Exact incompressibility is achieved by using divergenceconforming velocity spaces for the approximation of the velocities. The resulting methods are locall ..."
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Cited by 25 (5 self)
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We present a class of discontinuous Galerkin methods for the incompressible NavierStokes equations yielding exactly divergencefree solutions. Exact incompressibility is achieved by using divergenceconforming velocity spaces for the approximation of the velocities. The resulting methods are locally conservative, energystable, and optimally convergent. We present a set of numerical tests that confirm these properties. The results of this note naturally expand the work in [15].
The local discontinuous Galerkin method for the Oseen equations
 Math. Comp
, 2002
"... We introduce and analyze the local discontinuous Galerkin method for the Oseen equations of incompressible fluid flow. For a class of shaperegular meshes with hanging nodes, we derive optimal a priori estimates for the errors in the velocity and the pressure in L 2 and negativeorder norms. Nume ..."
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Cited by 17 (5 self)
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We introduce and analyze the local discontinuous Galerkin method for the Oseen equations of incompressible fluid flow. For a class of shaperegular meshes with hanging nodes, we derive optimal a priori estimates for the errors in the velocity and the pressure in L 2 and negativeorder norms. Numerical experiments are presented which verify these theoretical results and show that the method performs well for a wide range of Reynolds numbers.
The coupling of local discontinuous Galerkin and conforming finite element methods
 J. Sci. Comput
, 2000
"... The finite element formulation resulting from coupling the local discontinuous Galerkin method with a standard conforming finite element method for elliptic problems is analyzed. The transmission conditions across the interface separating the subdomains where the di erent formulations are applied ar ..."
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Cited by 14 (1 self)
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The finite element formulation resulting from coupling the local discontinuous Galerkin method with a standard conforming finite element method for elliptic problems is analyzed. The transmission conditions across the interface separating the subdomains where the di erent formulations are applied are taken into account by a suitable definition of the socalled numerical fluxes, and the resulting coupled method is shown to be stable. Optimal a priori error estimates are derived for arbitrary meshes with possible hanging nodes and elements of various shapes. Numerical experiments validating the theoretical results are also presented.
hpDiscontinuous Galerkin approximations for the Stokes problem
 Math. Models Meth. Appl. Sci
"... We propose and analyze a discontinuous Galerkin approximation for the Stokes problem. The finite element triangulation employed is not required to be conforming and we use discontinuous pressures and velocities. No additional unknown fields need to be introduced, but only suitable bilinear forms def ..."
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Cited by 14 (0 self)
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We propose and analyze a discontinuous Galerkin approximation for the Stokes problem. The finite element triangulation employed is not required to be conforming and we use discontinuous pressures and velocities. No additional unknown fields need to be introduced, but only suitable bilinear forms defined on the interfaces between the elements, involving the jumps of the velocity and the average of the pressure. We consider hp approximations using Qk ′Qk velocitypressure pairs with k ′ = k + 2, k + 1, k. Our methods show better stability properties than the corresponding conforming ones. We prove that our first two choices of velocity spaces ensure uniform divergence stability with the respect to the mesh size h. Numerical results show that they are uniformly stable with respect to the local polynomial degree k, a property that has no analog in the conforming case. An explicit bound in k which is not sharp is also proven. Numerical results show that if equal order approximation is chosen for the velocity and pressure, no spurious pressure modes are present but the method is not uniformly stable either with respect to h or k. We derive a priori error estimates generalizing the abstract theory of mixed methods. Optimal error estimates in h are proven. As for discontinuous Galerkin methods for scalar diffusive problems, half power of k is lost for p and hp approximations independently of the divergence stability.