  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.
