  Several types of simultaneous approximation term (SAT) for diffusion problems
discretized with multidimensional summation-by-parts (SBP) operators are
analyzed based on a common framework. Conditions under which the SBP-SAT
discretizations are consistent, conservative, adjoint consistent, and energy
stable are presented. For SATs leading to primal and adjoint consistent
discretizations, the error in output functionals is shown to be of order
$h^{2p}$ when a degree $p$ multidimensional SBP operator is used to discretize
the spatial derivatives. SAT penalty coefficients corresponding to various
discontinuous Galerkin fluxes developed for elliptic partial differential
equations are identified. We demonstrate that the original method of Bassi and
Rebay, the modified method of Bassi and Rebay, and the symmetric interior
penalty method are equivalent when implemented with SBP diagonal-E operators
that have diagonal norm matrix, e.g., the Legendre-Gauss-Lobatto SBP operator
in one space dimension. Similarly, the local discontinuous Galerkin and the
compact discontinuous Galerkin schemes are equivalent for this family of
operators. The analysis remains valid on curvilinear grids if a degree $\le
p+1$ bijective polynomial mapping from the reference to physical elements is
used. Numerical experiments with the two-dimensional Poisson problem support
the theoretical results.
