$C^0$ finite element approximations of linear elliptic equations in non-divergence form and Hamilton-Jacobi-Bellman equations with Cordes coefficients

This paper is concerned with $C^0$ (non-Lagrange) finite element approximations of the linear elliptic equations in non-divergence form and the Hamilton-Jacobi-Bellman (HJB) equations with Cordes coefficients. Motivated by the Miranda-Talenti estimate, a discrete analog is proved once the finite element space is $C^0$ on the $(n-1)$-dimensional subsimplex (face) and $C^1$ on $(n-2)$-dimensional subsimplex. The main novelty of the non-standard finite element methods is to introduce an interior penalty term to argument the PDE-induced variational form of the linear elliptic equations in non-divergence form or the HJB equations. As a distinctive feature of the proposed methods, no penalization or stabilization parameter is involved in the variational forms. As a consequence, the coercivity constant (resp. monotonicity constant) for the linear elliptic equations in non-divergence form (resp. the HJB equations) at discrete level is exactly the same as that from PDE theory. The quasi-optimal order error estimates as well as the convergence of the semismooth Newton method are established. Numerical experiments are provided to validate the convergence theory and to illustrate the accuracy and computational efficiency of the proposed methods.