We consider a mixed dimensional elliptic partial differential equation posed in a bulk domain with a large number of embedded interfaces. In particular, we study well-posedness of the problem and regularity of the solution. We also propose a fitted finite element approximation and prove an a priori error bound. For the solution of the arising linear system we propose and analyze an iterative method based on subspace decomposition. Finally, we present numerical experiments and achieve rapid convergence using the proposed preconditioner, confirming our theoretical findings.

翻译：我们考虑的是,在有大量嵌入界面的大片域中,形成一个混合的二维椭圆部分差异方程式。特别是,我们研究了问题的正确性和解决办法的规律性。我们还提出了一个合适的有限元素近似值,并证明存在先验错误。为了解决正在形成的线性系统,我们提出并分析基于子空间分解的迭接方法。最后,我们提出数字实验,并利用拟议的先决条件实现快速趋同,证实了我们的理论结论。