  This paper studies formulations of second-order elliptic partial differential
equations in nondivergence form on convex domains as equivalent variational
problems. The first formulation is that of Smears \& S\"uli [SIAM J.\ Numer.\
Anal.\ 51(2013), pp.\ 2088--2106.], and the second one is a new symmetric
formulation based on a least-squares functional. These formulations enable the
use of standard finite element techniques for variational problems in subspaces
of $H^2$ as well as mixed finite element methods from the context of fluid
computations. Besides the immediate quasi-optimal a~priori error bounds, the
variational setting allows for a~posteriori error control with explicit
constants and adaptive mesh-refinement. The convergence of an adaptive
algorithm is proved. Numerical results on uniform and adaptive meshes are
included.
