  This paper proposes a $C^{0}$ (non-Lagrange) primal finite element
approximation of the linear elliptic equations in non-divergence form with
oblique boundary conditions in planar, curved domains. As an extension of
[Calcolo, 58 (2022), No. 9], the Miranda-Talenti estimate for oblique boundary
conditions at a discrete level is established by enhancing the regularity on
the vertices. Consequently, the coercivity constant for the proposed scheme is
exactly the same as that from PDE theory. The quasi-optimal order error
estimates are established by carefully studying the approximation property of
the finite element spaces. Numerical experiments are provided to verify the
convergence theory and to demonstrate the accuracy and efficiency of the
proposed methods.
