Error estimation for second-order partial differential equations in nonvariational form

Second-order partial differential equations (PDEs) in nondivergence form are considered. Equations of this kind typically arise as subproblems for the solution of Hamilton-Jacobi-Bellman equations in the context of stochastic optimal control, or as the linearization of fully nonlinear second-order PDEs. The nondivergence form in these problems is natural. If the coefficients of the diffusion matrix are not differentiable, the problem cannot be transformed into the more convenient variational form. We investigate tailored nonconforming finite element approximations of second-order PDEs in nondivergence form, utilizing finite-element Hessian recovery strategies to approximate second derivatives in the equation. We study both approximations with continuous and discontinuous trial functions. Of particular interest are a priori and a posteriori error estimates as well as adaptive finite element methods. In numerical experiments our method is compared with other approaches known from the literature.