  We analyze the stability and functional superconvergence of discretizations
of diffusion problems with the narrow-stencil second-derivative generalized
summation-by-parts (SBP) operators coupled with simultaneous approximation
terms (SATs). Provided that the primal and adjoint solutions are sufficiently
smooth and the SBP-SAT discretization is primal and adjoint consistent, we show
that linear functionals associated with the steady diffusion problem
superconverge at a rate of $ 2p $ when a degree $ p+1 $ narrow-stencil or a
degree $ p $ wide-stencil generalized SBP operator is used for the spatial
discretization. Sufficient conditions for stability of adjoint consistent
discretizations with the narrow-stencil generalized SBP operators are
presented. The stability analysis assumes nullspace consistency of the
second-derivative operator and the invertibility of the matrix approximating
the first derivative at the element boundaries. The theoretical results are
verified by numerical experiments with the one-dimensional Poisson problem.
