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, taking the Poisson equation as a model problem.

翻译：若干由二阶椭圆方程建模的物理问题，可通过七十年代提出的N-单纯形上的Raviart-Thomas族RTk混合有限元方法高效求解。当在曲线边界上施加Neumann条件时，通量变量的法向分量不宜取值为边界上沿对应法向方向平移至逼近多面体边界的节点值。这是因为该方法精度会下降——这一结论由第一作者等人在前期论文中给出。该工作提出了一种基于带曲边单纯形参数化版本的保阶技术。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。该方法的关键在于混合问题的Petrov-Galerkin公式，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，证明该方法具有一致稳定性和最优收敛性。