  We propose a numerical method to approximate the solution of second order
elliptic problems in nonvariational form. The method is of Galerkin type using
conforming finite elements and applied directly to the nonvariational
(nondivergence) form of a second order linear elliptic problem. The key tools
are an appropriate concept of a 'finite element Hessian' and a Schur complement
approach to solving the resulting linear algebra problem. The method is
illustrated with computational experiments on three linear and one quasilinear
PDE, all in nonvariational form.
