Multiscale models and simulations for diffusion and interaction in heterogeneous domains

Abstract: We investigate multiscale and multiphysics models for evolution systems in heterogeneous domains, with a focus on multiscale diffusions. Although diffusion is often studied in terms of continuum observables, it is a consequence of the motion of individual particles. Incorporating interactions between constituents and geometry often runs into complications, since interactions typically act on multiple length scales. We address this issue by studying different types of multiscale models and by applying them to a variety of scenarios known for their inherent complexity.Our contributions can be grouped in two parts. In the first part, we pose two-scale reaction-diffusion systems in domains with varying microstructures. We prove well-posedness and construct finite element schemes with desirable approximation properties that resolve the microscopic domain variations and support parallel execution. In the second part of the thesis, we investigate certain interacting particle systems and their links to families of partial differential equations. In this spirit, we analyze a model of interacting populations, admitting dual descriptions from a system of ordinary differential equations and a porous media-like equation. We construct a multiscale simulation to evaluate scenarios in population dynamics. Finally, we investigate non-equilibrium dynamics and phase transitions within an interacting particle system in an extension of the classical Ehrenfest model.Our overall focus is two-fold. On the one hand, we increase the theoretical understanding of multiscale models by providing modeling, analysis and simulation of specific two-scale couplings. On the other hand, we design computational frameworks and tailored implementations to improve the application of multiscale modeling to complex scenarios and large-scale systems. In this way, our contributions aim to expand the capacity of mathematical modeling to numerically approximate the rich and complex physical world.