Finite element network approximation of conductivity in particle composites

A new finite element method computes conductivity in some unstructured particle-reinforced composite material. The 2-phase material under consideration is composed of a poorly conducting matrix material filled by highly conducting circular inclusions which are randomly dispersed. The mathematical model is a Poisson-type problem with discontinuous coefficients. The discontinuities are huge in contrast and quantity. The proposed method generalizes classical continuous piecewise affine finite elements to special computational meshes which encode the particles in a network structure. Important geometric parameters such as the volume fraction are preserved exactly. The computational complexity of the resulting method is proportional to the number of inclusions and this is optimal in comparison to data complexity. The discretization error is proportional to the particle distance and does not depend on the conductivity contrast in the medium.