In arXiv:2307.03503 [math.NA] we commenced to study a variant of the Raviart-Thomas mixed finite element method for triangles, to solve second order elliptic equations in a curved domain with Neumann or mixed boundary conditions. It is well known that in such a case the normal component of the flux variable should 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 work by the first author et al. An order-preserving technique was studied therein, based on a parametric version of these elements with curved simplexes. Our variant is an alternative to the approach advocated in those articles, allowing to achieve the same effect with straight-edged triangles. 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. In this paper we first recall the description of this method, together with underlying uniform stability results given in arXiv:2307.03503 [math.NA]. Then we show that it gives rise to optimal-order interpolation in the space H(div). Accordingly a priori error estimates are obtained for the Poisson equation taken as a model.

翻译：在arXiv:2307.03503 [math.NA]中，我们开始研究三角形Raviart-Thomas混合有限元方法的一个变体，用于求解带有Neumann或混合边界条件的弯曲区域中的二阶椭圆方程。众所周知，在此情形下，流变量的法向分量不应在逼近多面体边界上沿相应法线方向偏移的节点处取值。这是因为方法精度会降低——第一作者等人在此前的研究工作中已指出该问题。他们在该研究中提出了一种基于曲边单形参数化版本的保阶技术。我们的变体是对这些文献所倡导方法的替代方案，允许使用直边三角网格实现相同效果。该方法的关键在于混合问题的Petrov-Galerkin格式，其中测试流空间与形状流空间略有不同。本文首先回顾该方法的描述及其在arXiv:2307.03503 [math.NA]中给出的相关一致稳定性结果，随后证明该方法可在H(div)空间中产生最优阶插值。据此，以Poisson方程为例，我们得到了先验误差估计。