Iterative solution of spatial network models by subspace decomposition

TL;DR

This paper introduces a preconditioned conjugate gradient method for efficiently solving spatial network models, with a convergence analysis based on finite element theory, applicable to diffusion and structural mechanics simulations.

Contribution

The work provides the first convergence analysis of a subspace decomposition-based PCG method tailored for spatial network problems, extending finite element techniques to network models.

Findings

Convergence rate depends on network bounds and constants.

Method effectively solves diffusion and mechanics simulations.

Numerical experiments confirm theoretical predictions.

Abstract

We present and analyze a preconditioned conjugate gradient method (PCG) for solving spatial network problems. Primarily, we consider diffusion and structural mechanics simulations for fiber based materials, but the methodology can be applied to a wide range of models, fulfilling a set of abstract assumptions. The proposed method builds on a classical subspace decomposition into a coarse subspace, realized as the restriction of a finite element space to the nodes of the spatial network, and localized subspaces with support on mesh stars. The main contribution of this work is the convergence analysis of the proposed method. The analysis translates results from finite element theory, including interpolation bounds, to the spatial network setting. A convergence rate of the PCG algorithm, only depending on global bounds of the operator and homogeneity, connectivity and locality constants of the network, is established. The theoretical results are confirmed by several numerical experiments.