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

TL;DR

This paper introduces a novel $C^{0}$ finite element method for solving planar elliptic equations in non-divergence form with oblique boundary conditions, achieving optimal error estimates and verified by numerical experiments.

Contribution

It extends finite element approximation to non-divergence form equations with oblique boundary conditions, establishing a discrete Miranda-Talenti estimate and exact coercivity constants.

Findings

The scheme achieves quasi-optimal error estimates.

Numerical results confirm convergence and accuracy.

The method is efficient for curved domain problems.

Abstract

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.