Effects of nonaffine elements on the accuracy of 3D H(div)‐conforming finite elements are studied. Instead of convergence order k+1 for the flux and the divergence of the flux obtained with Raviart‐Thomas or Nédélec spaces with normal traces of degree k, based on affine hexahedra or triangular prisms, reduced orders k for the flux and k−1 for the divergence of the flux may occur for distorted elements. To improve this scenario, a hierarchy of enriched flux approximations is considered, by adding internal shape functions up to a higher degree k+n, n>0, while keeping the original normal traces of degree k. The resulting enriched approximations, using multilinear transformations, keep the original flux accuracy (of order k+1 with affine elements or reduced order k otherwise), but enhanced divergence (of order k+n+1, in the affine case, or k+n−1 otherwise) can be reached. The reduced flux accuracy due to quadrilateral face distortions cannot be corrected by including higher order internal functions. The enriched spaces are applied to the mixed finite element formulation of Darcy's model. The computational cost of matrix assembly increases with n, but the condensed system to be solved has the same dimension and structure as the original scheme.

中文翻译：

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基于非仿射网格的H（div）有限元，用于流动问题的3D混合公式，具有任意高阶的通量散度精度

研究了非仿射元素对符合3D H（div）的有限元精度的影响。代替的收敛阶ķ对于通量和与Raviart -托马斯或Nédélec间隔所获得与度正常痕迹光通量的发散1 ķ，基于仿射六面体或三角棱镜，订单减少ķ对于通量和ķ -1对于变形的元件，可能会发生磁通量的发散。为了改善这种情况，考虑了富通量近似的层次结构，方法是将内部形状函数添加到更高的度数k + n，n > 0，同时保持度数k的原始法线轨迹。使用多线性变换得到的富集近似值保持原始通量精度（仿射元素为k +1阶，否则为k阶），但散度增强（仿射情况为k + n +1阶，或k +否则可以达到n -1）。由于四边形面畸变而导致的磁通精度降低，无法通过包含更高阶的内部函数来校正。丰富的空间应用于达西模型的混合有限元公式。矩阵装配的计算成本随着n的增加而增加，但要求解的浓缩系统具有与原始方案相同的尺寸和结构。