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514
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 36 (7 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
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.
A Compact Discontinuous Galerkin (CDG) Method for Elliptic Problems,” submitted
 SIAM J. for Numerical Analaysis
, 2006
"... Abstract. We present a compact discontinuous Galerkin (CDG) method for an elliptic model problem. The problem is first cast as a system of first order equations by introducing the gradient of the primal unknown, or flux, as an additional variable. A standard discontinuous Galerkin (DG) method is the ..."
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Cited by 34 (14 self)
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Abstract. We present a compact discontinuous Galerkin (CDG) method for an elliptic model problem. The problem is first cast as a system of first order equations by introducing the gradient of the primal unknown, or flux, as an additional variable. A standard discontinuous Galerkin (DG) method is then applied to the resulting system of equations. The numerical interelement fluxes are such that the equations for the additional variable can be eliminated at the element level, thus resulting in a global system that involves only the original unknown variable. The proposed method is closely related to the local discontinuous Galerkin (LDG) method [B. Cockburn and C.W. Shu, SIAM J. Numer. Anal., 35 (1998), pp. 2440–2463], but, unlike the LDG method, the sparsity pattern of the CDG method involves only nearest neighbors. Also, unlike the LDG method, the CDG method works without stabilization for an arbitrary orientation of the element interfaces. The computation of the numerical interface fluxes for the CDG method is slightly more involved than for the LDG method, but this additional complication is clearly offset by increased compactness and flexibility.
Generalized Multiscale Finite Element Methods (GMsFEM)
, 2013
"... In this paper, we propose a general approach called Generalized Multiscale Finite Element Method (GMsFEM) for performing multiscale simulations for problems without scale separation over a complex input space. As in multiscale finite element methods (MsFEMs), the main idea of the proposed approach i ..."
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Cited by 32 (10 self)
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In this paper, we propose a general approach called Generalized Multiscale Finite Element Method (GMsFEM) for performing multiscale simulations for problems without scale separation over a complex input space. As in multiscale finite element methods (MsFEMs), the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space. In the proposed approach, we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage. The enrichment in the online stage is performed based on a spectral decomposition of the offline space. In the online stage, for any input parameter, a multiscale space is constructed to solve the global problem on a coarse grid. The online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues. The computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be reused for solving the forward problem with any forcing and boundary condition. Compared with the other approaches where global snapshots are used, the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarsegrid level. We present various examples in the paper and some numerical results to demonstrate the effectiveness of our method. 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.
Discontinuous Galerkin method for Helmholtz equation with large wave numbers
 SIAM J. Numer. Anal
"... Abstract. This paper develops and analyzes some interior penalty discontinuous Galerkin methods using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary condition in the two and three dimensions. It is proved that the proposed discontinuous Galerkin metho ..."
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Cited by 32 (6 self)
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Abstract. This paper develops and analyzes some interior penalty discontinuous Galerkin methods using piecewise linear polynomials for the Helmholtz equation with the first order absorbing boundary condition in the two and three dimensions. It is proved that the proposed discontinuous Galerkin methods are stable (hence wellposed) without any mesh constraint. For each fixed wave number k, optimal order (with respect to h) error estimate in the broken H1norm and suboptimal order estimate in the L2norm are derived without any mesh constraint. The latter estimate improves to optimal order when the mesh size h is restricted to the preasymptotic regime (i.e., k2h � 1). Numerical experiments are also presented to gauge the theoretical result and to numerically examine the pollution effect (with respect to k) in the error bounds. The novelties of the proposed interior penalty discontinuous Galerkin methods include: first, the methods penalize not only the jumps of the function values across the element edges but also the jumps of the normal and tangential derivatives; second, the penalty parameters are taken as complex numbers of positive imaginary parts so essentially and practically no constraint is imposed on the penalty parameters. Since the Helmholtz problem is a nonHermitian and indefinite linear problem, as expected, the crucial and the most difficult part of the whole analysis is to establish the stability estimates (i.e., a priori estimates) for the numerical solutions. To the end, the cruxes of our analysis are to establish and to make use of a local version of the Rellich identity (for the Laplacian) and to mimic the stability analysis for the PDE solutions given in [23, 24, 35].
Interior penalty method for the indefinite timeharmonic Maxwell equations
 IN THE SECOND MODEL, THE
, 2003
"... ..."
SUPERCONVERGENT DISCONTINUOUS GALERKIN METHODS FOR SECONDORDER ELLIPTIC PROBLEMS
"... Abstract. We identify discontinuous Galerkin methods for secondorder elliptic problems in several space dimensions having superconvergence properties similar to those of the RaviartThomas and the BrezziDouglasMarini mixed methods. These methods use polynomials of degree k ≥ 0 for both the potent ..."
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Cited by 30 (9 self)
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Abstract. We identify discontinuous Galerkin methods for secondorder elliptic problems in several space dimensions having superconvergence properties similar to those of the RaviartThomas and the BrezziDouglasMarini mixed methods. These methods use polynomials of degree k ≥ 0 for both the potential as well as the flux. We show that the approximate flux converges in L 2 with the optimal order of k + 1, and that the approximate potential and its numerical trace superconverge, in L 2like norms, to suitably chosen projections of the potential, with order k + 2. We also apply elementbyelement postprocessing of the approximate solution to obtain new approximations of the flux and the potential. The new approximate flux is proven to have normal components continuous across interelement boundaries, to converge in L 2 with order k + 1, and to have a divergence converging in L 2 also with order k+1. The new approximate potential is proven to converge with order k+2in L 2. Numerical experiments validating these theoretical results are presented. 1.
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 30 (7 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