Analysis of a finite element method for second order uniformly elliptic PDEs in non-divergence form

We propose one finite element method for both second order linear uniformly elliptic PDE in non-divergence form and the elliptic Hamilton-Jacobi-Bellman (HJB) equation. For the linear elliptic PDE in non-divergence form, we consider two scenarios of the matrix coefficient matrix $A$. One is $A$ is uniformly continuous. The other is $A$ is discontinuous but $\gamma A$ is dominated by $I_{d}$ where $\gamma$ is a positive weight function. We prove that optimal convergence in discrete $W^{2,p}$-norm of the numerical approximation to the strong solution for $1<p\leq 2$ on convex polyhedra in $\mathbb{R}^{d}$ ($d=2,3$). If the domain is a two dimensional non-convex polygon, $p$ is valid in a neighbourhood of $\frac{4}{3}$. We also prove the well-posedness of strong solution in $W^{2,p}(\Omega)$ for both linear elliptic PDE in non-divergence form and the HJB equation for $1< p \leq 2$ on convex polyhedra in $\mathbb{R}^{d}$ ($d=2,3$) and for $p$ in an open interval starting from $1$ and including $\frac{4}{3}$ on two dimensional non-convex polygon. Furthermore, we relax the assumptions on the continuity of coefficients of the HJB equation, which have been widely used in literature.