Decoupling integration schemes of higher order for poroelastic networks

• Applicants: Professor Dr. Robert Altmann; Dr. Benjamin Unger
• Subject Area: Mathematics
• Term: since 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 467107679

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.

• DFG Programme: Research Grants