The Compact Discontinuous Galerkin (CDG) Method for Elliptic Problems

By Jaime Peraire, Per-Olof Persson. SIAM Journal on Scientific Computing, 2007

418 citations recorded by the source corpus.

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. Compared to the BR2 [F. Bassi and S. Rebay, J. Comput. Phys., 131 (1997), pp. 267-279] and IP [J. Douglas, Jr., and T. Dupont, in Computing Methods in Applied Sciences (Second Internat. Sympos., Versailles, 1975), Lecture Notes in Phys. 58, Springer, Berlin, 1976, pp. 207-216] methods, which are known to be compact, the present method produces fewer nonzero elements in the matrix and is computationally more efficient.

Subfields: Computational Mechanics, Electrical and Electronic Engineering

View Paper: https://www.semanticscholar.org/paper/01a9ecab606fd06db94afc511663f01c55332e6a

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Citations in the corpus

Hybridizable discontinuous Galerkin p‐adaptivity for wave propagation problems (2013, methodology): “Note that the computational overhead for CG and CDG is non-negligible and it is not introduced in the comparisons.”
High‐order continuous and discontinuous Galerkin methods for wave problems (2013, methodology): “Following the rationale detailed in [34], the CDG weak form associated to (6) requires to find uh ∈ V̂h such that â(uh, v) = l̂(v) ∀v ∈ V̂h, where the bilinear and linear forms, â(·, ·) and l̂(·) are given by”
Explicit Discontinuous Galerkin methods for unsteady problems (2012, background): “The extension to problems of viscous gas dynamics was initiated in [1, 3] and this again has led to several related formulations [24, 31, 21] for the compressible Navier-Stokes equations.”
A Reconstruction Approach to High -Order Schemes Including Discontinuous Galerkin for Diffusion (2009, methodology): “J. Peraire AND P.-O. Persson, “The compact discontinuous Galerkin (CDG) method for elliptic problems”, SIAM J. Sci. Comput., Vol. 30, No. 4 (2008) pp. 1806-1824.”
Efficiency of high‐order elements for continuous and discontinuous Galerkin methods (2013, background): “And finally, it is also important to note that different Galerkin approaches are considered: CG, with static condensation, compact DG (CDG) as proposed in [19] because it presents the smallest stencils, and HDG.”
A Hybridizable Discontinuous Galerkin Method for the Compressible Euler and Navier-Stokes Equations (2010, neutral): “[19] has also been applied to compressible viscous ﬂows 20 and turbulent ﬂows.”
Differential Formulation of Discontinuous Galerkin and Related Methods for the Navier-Stokes Equations (2013, methodology): “The four schemes of compact stencil employed are BR2 (Bassi and Rebay) [4], compact DG or CDG [20], interior penalty [9,12], and I-continuous (the value and derivative are continuous across the interface) [14].”
An Embedded Discontinuous Galerkin Method for the Compressible Euler and Navier-Stokes Equations (2011, neutral): “In this section, we compare the EDG method to two other classes of DG methods, the HDG method 12,13 and other DG methods such as the CDG method 22 or the method of Bassi and Reaby.”
An implicit high-order hybridizable discontinuous Galerkin method for linear convection-diffusion equations (2009, neutral): “The Compact DG scheme introduced in [17] is not an HDG method; it is a modiﬁcation of an LDG method.)”
Implicit Large Eddy Simulation of transition to turbulence at low Reynolds numbers using a Discontinuous Galerkin method (2011, methodology): “DOI: 10.1002/nme.3036 Implicit Large Eddy Simulation of transition to turbulence at low Reynolds numbers using a Discontinuous Galerkin method A. Uranga1, P.-O. Persson2, M. Drela1 and J. Peraire1,∗,† 1Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA, ”
A review of flux reconstruction or correction procedure via reconstruction method for the Navier-Stokes equations (2016, methodology): “(2.19) For 𝜅𝜅 = 0 or 𝜅𝜅 = 1, we have the one-sided formula used in the local DG or LDG (Cockburn and Shu 1998) as well as the compact DG or CDG methods (Peraire and Persson 2008).”