Decoupling integration schemes of higher order for poroelastic networks

多孔弹性多重网络模型在地球科学和生物医学等领域有着广泛的应用。例如，对脑水肿的研究考虑了不同的血液循环和脑脊液，从而导致总共四个液体室。相应的系统是椭圆和抛物型的耦合偏微分方程（PDE），特别是在3维情况下，如果应用标准方法，则计算上具有挑战性或甚至不可行。这个项目的目的是建立一个新的高效率的集成计划的高阶联合收割机的简单性与巨大的速度加快的迭代方法，解耦的问题。由于耦合偏微分方程的空间离散化的结果在一个微分代数方程（DAE），收敛性分析呈现出一个具有挑战性的任务。该项目的第一个主要贡献是申请人最近引入的想法扩展到非线性情况。收敛性分析是基于这样一个事实，即半显式的计划可以被解释为一个相关的延迟方程的隐式计划，其中的时间延迟等于步长。其次，通过寻找合适的延迟方程构造高阶半显式格式，使得延迟方程与原模型之间的误差达到所需的阶数。自适应步长选择机制的实现和分析进一步提高了该方法的效率。由于收敛性分析依赖于所谓的弱耦合条件，我们，第三，证明耦合条件，它是直接相关的相关延迟方程的渐近稳定性。利用这种联系，我们将建立半显式时间步长方法的收敛性与中立型延迟方程和延迟DAE的渐近稳定性之间的等价结果，从而连接不同的研究社区。第四，在弱的数据正则性假设下，我们分析了相关时滞方程解的正则性。该项目是一个协同合作的两个申请人的互补专业知识的时间离散计划耦合偏微分方程和分析的时间延迟系统。

Poroelastic multiple-network models arise in a variety of different application domains, including geoscience and biomedicine. The investigation of cerebral edema, for instance, accounts for different blood cycles and a cerebrospinal fluid, resulting in a total of four fluid comparments. The corresponding system is a coupled partial differential equation (PDE) of elliptic and parabolic type that, in particular in the 3-dimensional case, is computationally challenging or even unfeasible if standard methods are applied. This project aims to construct a novel class of highly efficient integration schemes of higher order that combine the simplicity of monolithic approaches with the tremendous speed-ups of iterative methods that decouple the problem. Since the spatial discretization of the coupled PDE results in a differential-algebraic equation (DAE), the convergence analysis renders a challenging task. The first main contribution of the project is the extension of recently introduced ideas of the applicants to the nonlinear case. The convergence analysis is based on the fact that the semi-explicit scheme can be interpreted as an implicit scheme for a related delay equation, where the time-delay equals the step size. Second, higher-order semi-explicit schemes are constructed by finding suitable delay equations, such that the error between the delay equation and the original model is of the desired order. Implementation and analysis of an adaptive step size selection mechanism further improve the method's efficiency. Since the convergence analysis relies on a so-called weak coupling condition, we, third, justify the coupling condition by showing that it is directly related to the asymptotic stability of the related delay equation. Exploiting this connection, we will establish an equivalence result between the convergence of semi-explicit time-stepping methods and the asymptotic stability of neutral delay equations and delay DAEs, thus, connecting the different research communities. Fourth, we analyze the regularity of solutions of the related delay equations under weak assumptions on the regularity of the data. The project is a synergistic collaboration of the two applicants' complementary expertise on time discretization schemes for coupled PDEs and analysis of time-delay systems.