Convergence analysis of a finite element method for second order non-variational elliptic problems

We introduce and analyze a family of finite element methods for elliptic partial differential equations in non-variational form with non-differentiable coefficients. The finite element method studied is a variant of the one recently proposed in [Lakkis & Pryer, SIAM J. Sci. Comput., 2011], where a finite element Hessian is introduced as an auxiliary unknown. We modify the definition of the finite element Hessian rendering the auxiliary variable completely local, thus resulting in a more efficient scheme. We show that the method is stable under general conditions on the coefficient matrix and derive error estimates in a discrete H-2-norm provided the discretization parameter is sufficiently small. Numerical experiments are presented which verify the theoretical results.