Statusvortrag im Promotionsver­fahren von Herrn Philipp Suhrcke

Titel: "An adaptive-resolution algorithm for meshfree collocation methods in complex geometries“

Abstract: Well-conditioned spatial discretization of computational domains is essential when numerically solving partial differential equations (PDEs). The smallest structures of the solution field dictate the minimum resolution required. If these structures are only localized to one part of the domain, however, having a highly resolved discretization everywhere is computationally inefficient. Adaptive-resolution methods avoid this by dynamically adapting the spatial discretization as required by the evolving solution and the stability of the numerical method. Adaptive-resolution schemes have been proposed for many numerical methods, including mesh-based and meshfree ones. 

Of these, meshfree methods offer a particular advantage in complex-shaped domains. They are not constrained to a lattice or mesh that needs to be of a certain regularity to ensure consistency and convergence. Instead, they work on irregular point clouds, which allows adaptation to arbitrary geometric shapes. So far, however, adaptive-resolution meshfree methods were only available for Lagrangian transport problems, and they have not been developed for solving long-range field equations. 

In this talk, I present an overview of adaptive-resolution algorithms for different numerical schemes, focusing on the particular challenges arising in meshfree collocation methods. I then propose a new method in which the meshfree collocation points (particles) interact via a pairwise potential, defining a pseudo-force for resolution adaptation. The sum of all forces within a given interaction radius leads to a displacement of the particles toward a locally regular but globally adaptive distribution. I present the details of the algorithm and rationalize my design choices with benchmarks. Finally, I demonstrate the ability of the proposed new algorithm to solve the challenging PDEs governing active hydrodynamics in living biological systems, such as cells and tissues.