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 a uniformly stable and optimally convergent method in the natural norm, taking the Poisson equation as a model problem.

翻译：对于由二阶椭圆方程建模的若干物理问题，采用七十年代提出的N-单纯形Raviart-Thomas族RTk混合有限元可进行高效求解。当诺伊曼条件施加于曲线边界时，通量变量的法向分量不宜取位于近似多面体边界沿法向偏移节点处的值。这是因为该方法精度会降低，第一作者等人在先前论文中已证明此现象。该研究基于采用曲线单纯形的参数化单元版本，探讨了一种保阶技术。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表述，其中测试通量空间与形函数通量空间略有不同。在描述RTk的此变体后，我们以泊松方程作为模型问题，证明该方法在自然范数下具有一致稳定性和最优收敛性。