  We propose a least squares Galerkin based gradient recovery to approximate
Dirichlet problems for strong solutions of linear elliptic problems in
nondivergence form and corresponding apriori and aposteriori error bounds. This
approach is used to tackle fully nonlinear elliptic problems, e.g.,
Monge-Amp\`ere, Hamilton-Jacobi-Bellman, using the smooth (vanilla) and the
semismooth Newton linearization. We discuss numerical results, including
adaptive methods based on the aposteriori error indicators.
