Simulation-based model reduction for partial differential equations on networks

More about the book

This thesis addresses model reduction for parameter-dependent parabolic PDEs on networks with variable composition. The Reduced Basis Element Method (RBEM), developed by Maday and Rønquist, is employed since a solution across the entire domain is unnecessary. This method constructs a reduced basis for each component and couples them using a mortar-like approach. However, this decomposition can be challenging, particularly for networks with many edges, as the variable composition complicates predictions at interfaces, potentially resulting in inadequate basis functions and poor global solution approximations. We propose an extension of the RBEM that enhances basis representation for individual edges by utilizing a spline-based boundary parametrization in local basis construction. To validate the approximation properties, we develop an error estimate for local basis construction using Proper Orthogonal Decomposition (POD) or POD-Greedy. We also establish existence, uniqueness, and regularity results for parabolic PDEs on one-dimensional networks, crucial for error analysis. Our method is illustrated through three examples: the first showcases two networks of one-dimensional heat equations with varying thermal conductivity, while the second and third demonstrate the method's applicability to component-based domains in two dimensions and nonlinear PDEs. This research is part of a project funded by the German Federal Ministry of Ed