Several physical problems modeled by second-order elliptic equations can be efficiently solved using mixed finite elements of the Raviart-Thomas family RTk 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 previous papers by the first author et al. In that work an order-preserving technique was studied, based on a parametric version of these elements with curved simplexes. In this article 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 describing the underlying variant of RTk we show that it gives rise to uniformly stable methods taking the Poisson equation as a model problem.

翻译：许多由二阶椭圆方程建模的物理问题可以通过二十世纪七十年代提出的、用于N单形的Raviart-Thomas族RTk混合有限元高效求解。当在曲线边界上给定纽曼条件时，通量变量的法向分量最好不应取近似多面体边界节点沿相应法线方向偏移后的数值，因为这会降低方法的精度——首作者等人在先前论文中已指出这一点。该工作研究了基于曲单形参数化版本的一种保序技术。本文针对二维问题提出了一种采用直边三角形的替代方案。该方法的关键点在于混合问题的Petrov-Galerkin公式化，其中测试通量空间与形状通量空间略有不同。在描述RTk的底层变体后，我们以泊松方程为模型问题，证明该变体能导出均匀稳定的方法。