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Finite element methods for the Stokes problem on complicated domains

机译：复杂域Stokes问题的有限元方法

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摘要

It is a standard assumption in the error analysis of finite element methods that the underlying finite element mesh has to resolve the physical domain of the modeled process. In case of complicated domains which appear in many applications such as ground water flows this requirement sometimes becomes a bottleneck. The resolution condition links the computational complexity to the number (and size) of geometric details although the accuracy requirements, possibly, are moderate and would allow a (locally) coarse mesh width. Therefore even the coarsest available discretization can lead to a huge number of unknowns. The composite mini element is a remedy to this dilemma because the degrees of freedom are not linked to the number of geometric details. The basic concept for the Stokes problem with uniform no-slip boundary conditions has been introduced and analyzed in [D. Peterseim, S. Sauter, The composite mini element - coarse mesh computation of Stokes flows on complicated domains, SINUM, 46(6) (2008) 3181-3206]. Here, we generalize the composite mini element to slip, leak and Neumann boundary conditions so that it becomes applicable to this much larger and more important problem class. The main results are (a) the algorithmic concept remains unchanged and the new boundary conditions can be implemented as a weighted quadrature rule, (b) the stability and convergence can be proved under very mild assumption on the domain geometries, (c) the analysis is far from trivial and requires the development of substantially new tools compared to the simple case of uniform no-slip boundary conditions.
机译：在有限元方法的误差分析中，一个标准的假设是基础有限元网格必须解析建模过程的物理域。在诸如地下水流等许多应用中出现复杂区域的情况下，此要求有时会成为瓶颈。分辨率条件将计算复杂性与几何细节的数量（和大小）联系在一起，尽管精度要求可能适中，并且允许（局部）粗网格宽度。因此，即使是最粗略的离散化也可能导致大量未知数。微型复合材料元件可以解决这个难题，因为自由度与几何细节的数量没有关系。 ［D］中介绍并分析了具有统一防滑边界条件的斯托克斯问题的基本概念。 Peterseim，S. Sauter，《复合微型元素-复杂域上Stokes流的粗网格计算》，SINUM，46（6）（2008）3181-3206]。在这里，我们将复合微型元素推广到滑移，泄漏和诺伊曼边界条件，从而使其适用于更大，更重要的问题类别。主要结果是：（a）算法概念保持不变，并且新的边界条件可以实现为加权正交规则；（b）可以在非常温和的假设下对域几何形状证明稳定性和收敛性；（c）分析与统一的无滑移边界条件的简单情况相比，它绝非易事，需要开发实质上新的工具。

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