Discrete Geometric Homogenisation and Inverse Homogenisation of an Elliptic Operator

Abstract

We show how to parameterise a homogenised conductivity in $R^2$ by a scalar function $s(x)$, despite the fact that the conductivity parameter in the related up-scaled elliptic operator is typically tensor valued. Ellipticity of the operator is equivalent to strict convexity of $s(x)$, and with consideration to mesh connectivity, this equivalence extends to discrete parameterisations over triangulated domains. We apply the parameterisation in three contexts: (i) sampling $s(x)$ produces a family of stiffness matrices representing the elliptic operator over a hierarchy of scales; (ii) the curvature of $s(x)$ directs the construction of meshes well-adapted to the anisotropy of the operator, improving the conditioning of the stiffness matrix and interpolation properties of the mesh; and (iii) using electric impedance tomography to reconstruct $s(x)$ recovers the up-scaled conductivity, which while anisotropic, is unique. Extensions of the parameterisation to $R^3$ are introduced.