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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 and Suli [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.

Variational Formulation and Numerical Analysis of Linear Elliptic Equations in Nondivergence form with Cordes Coefficients