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：多个由二阶椭圆方程建模的物理问题可借助二十世纪七十年代提出的N-单纯形Raviart-Thomas族RTk混合有限元高效求解。当在曲边界上施加纽曼条件时，通量变量的法向分量应避免沿法线方向移位至逼近多面体边界节点处取值，原因在于此操作会降低方法精度（第一作者及其合作者此前已论证）。该研究提出了一种基于曲边单纯形参数化单元的保阶技术。本文针对二维问题提出一种采用直边三角形的替代方案。该方法的关键在于构建混合问题的彼得罗夫-伽辽金格式，其中测试通量空间与形状通量空间略有差异。在描述该RTk变体基本原理后，我们以泊松方程为模型问题证明其可生成一致稳定的方法。