A <i>C</i> ⁰ linear finite element method for a second‐order elliptic equation in non‐divergence form with Cordes coefficients

In this paper, we develop a gradient recovery based linear (GRBL) finite element method (FEM) and Hessian FEM for second-order elliptic equations in non-divergence form. The equation is casted into symmetric weak formulation, which derivatives of the unknown function are involved. We use operators to calculate approximations. Although, thanks low degrees freedom elements, implementation proposed schemes easy straightforward, performances methods competitive. unique solvability H 2 $$ {H}^2 seminorm error estimate GRBL scheme rigorously proved. Optimal estimates both L {L}^2 norm 1 {H}^1 have been proved when coefficient diagonal, confirmed by numerical experiments. Superconvergence errors has also observed. Moreover, our can handle computational domains with curved boundaries without loss accuracy from approximation boundaries. Finally, successfully applied solve fully nonlinear Monge–Ampère equations.