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

A $C^{0}$ finite element approximation of planar oblique derivative problems in non-divergence form