Numerical Homogenization via Approximation of the Solution Operator

Abstract

The paper describes techniques for constructing simplified models for problems governed by elliptic partial differential equations involving heterogeneous media. Examples of problems under consideration include electro-statics and linear elasticity in composite materials, and flows in porous media. A common approach to such problems is to either up-scale the governing differential equation and then discretize the up-scaled equation, or to construct a discrete problem whose solution approximates the solution to the original problem under some constraints on the permissible loads. In contrast, the current paper suggests that it is in many situations advantageous to directly approximate the solution operatorto the original differential equation. Such an approach has become feasible due to recent advances in numerical analysis, and can in a natural way handle situations that are challenging to existing techniques, such as those involving, e.g.concentrated loads, boundary effects, and irregular micro-structures. The capabilities of the proposed methodology are illustrated by numerical examples involving domains that are loaded on the boundary only, in which case the solution operator is a boundary integral operator such as, e.g., a Neumann-to–Dirichlet operator.