Several physical problems modeled by second-order partial differential equations can be efficiently solved using mixed finite elements of the Raviart-Thomas family for N-simplexes, introduced in the seventies. In case Neumann conditions are prescribed on a curvilinear boundary, the normal component of the flux variable should preferably not take up values at nodes shifted to the boundary of the approximating polytope in the corresponding normal direction. This is because the method's accuracy downgrades, which was shown in \cite{FBRT}. In that work an order-preserving technique was studied, based on a parametric version of these elements with curved simplexes. In this paper an alternative with straight-edged triangles for two-dimensional problems is proposed. The key point of this method is a Petrov-Galerkin formulation of the mixed problem, in which the test-flux space is a little different from the shape-flux space. After carrying out a well-posedness and stability analysis, error estimates of optimal order are proven.

翻译：许多由二阶偏微分方程建模的物理问题可通过20世纪70年代提出的Raviart-Thomas族N-单纯形混合有限元高效求解。当纽曼边界条件定义于曲边界时，通量变量的法向分量不宜取在近似多面体边界沿对应法向平移后的节点上，因为这会降低方法精度（见参考文献 \cite{FBRT}）。该文献研究了一种基于曲边单纯形参数化版本且保持阶数的技术。本文针对二维问题提出了一种采用直边三角形替代方案。该方法的关键在于对混合问题采用Petrov-Galerkin变分形式，其中检验通量空间与形状通量空间略有差异。通过适定性与稳定性分析后，证明了最优阶误差估计。