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：摘要：在arXiv:2307.03503 [math.NA]中，我们开始研究一种基于三角形的Raviart-Thomas混合有限元方法变体，用于求解弯曲区域中具有Neumann或混合边界条件的二阶椭圆方程。众所周知，在这种情况下，通量变量的法向分量不应取节点移至逼近多面体边界对应法线方向上的数值，因为该方法精度会降低——这已在第一作者等人的前期工作中证明。此前研究提出了一种基于弯曲单形参数化版本的保阶技术。我们的变体是这些文献中方法的替代方案，允许使用直边三角形实现相同效果。该方法的关键在于混合问题的Petrov-Galerkin格式，其中测试通量空间与形状通量空间略有不同。本文首先回顾了该方法及其在arXiv:2307.03503 [math.NA]中给出的底层一致稳定性结果，随后证明该格式在H(div)空间中产生最优阶插值。据此，以Poisson方程为模型获得了先验误差估计。