Multilevel methods for spatial network models

Spatial network models are used as simplified discrete representations of more complex geometric structures.

Blood vessels may e.g. be modelled as one dimensional tubes, represented by a network of nodes and edges; a porous medium may be represented by a pore network model of throats and pore cavities; paper may be represented by connected one dimensional beams, forming a spatial network.

These simplifications reduces the complexity from a full three dimensional geometric description to a discrete model that can be utilized for computer simulations.Still the complex underlying geometric structure of the network is a major challenge when simulating e.g. elastic deformation. The heterogeneity causes the discrete model to be poorly conditioned and, hence, difficult to solve numerically.

Successful numerical algorithms, for solving vast discrete models, like multigrid and multilevel Monte Carlo, use representations of the underlying model on multiple spatial scales. For discretized PDEs this can be accomplished using nested meshes. In this setting the convergence analysis is well developed.

For spatial networks it is less clear how to introduce coarse scales and new mathematical tools are needed for the convergence analysis. The main objective of this proposal is to develop and analyze robust solvers for to spatial network models. We will consider both deterministic and random networks.

The target application is simulation of mechanical properties of paper.