标题： 任意阶各向异性三角形和四面体上的Brezzi-Douglas-Marini插值 显示英文标题

标题： Brezzi-Douglas-Marini interpolation of any order on anisotropic triangles and tetrahedra

摘要： 最近，针对二阶椭圆问题的$\vec{H}(\operatorname{div})$有限元族越来越受到关注，因为由于杂交化和随后计算效率的提高，它们的使用不再主要是理论上的。 它们在混合问题中能够产生精确的散度为零的解的特性，使其在各种应用中变得有趣，包括不可压缩流体。 在这个领域中，存在边界层和内部层，这需要使用各向异性单元。 虽然对于各向异性四面体上的任意阶 Raviart-Thomas 插值，已知最优误差估计，但本贡献将这些结果扩展到了 Brezzi-Douglas-Marini 有限元。 在对单元的两种不同正则性条件下证明了最优插值误差估计，这两种条件都放松了标准的最小角条件。 此外，还给出了一个关于 Stokes 方程的数值应用，以说明研究结果。 显示英文摘要

摘要： Recently, the $\vec{H}(\operatorname{div})$-conforming finite element families for second order elliptic problems have come more into focus, since due to hybridization and subsequent advances in computational efficiency their use is no longer mainly theoretical. Their property of yielding exactly divergence-free solutions for mixed problems makes them interesting for a variety of applications, including incompressible fluids. In this area, boundary and interior layers are present, which demand the use of anisotropic elements. While for the Raviart-Thomas interpolation of any order on anisotropic tetrahedra optimal error estimates are known, this contribution extends these results to the Brezzi-Douglas-Marini finite elements. Optimal interpolation error estimates are proved under two different regularity conditions on the elements, which both relax the standard minimal angle condition. Additionally a numerical application on the Stokes equations is presented to illustrate the findings.