  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.

Numerical Analysis