Chapter 3: Decomposition of Scales in Elliptic Problems
Axel Målqvist, Daniel Peterseim · Society for Industrial and Applied Mathematics eBooks · 2020
In this chapter we consider an elliptic model problem with a rough diffusion matrix, posed on a bounded domain in ℝd. We do not assume periodicity or scale separation in the diffusion matrix. In the spirit of Section 2.4 we construct a finite-dimensional function space that is ideal for numerical homogenization. A key component in the construction is the use of a quasi-interpolation operator. The kernel of that operator (corresponding to the space W in Chapter 2) defines the fine scales of the problem, and its a-orthogonal complement defines the ideal function space used for numerical homogenization (corresponding to in Chapter 2). There are no analytical expressions for the basis functions that span this space for d > 1. Instead, the basis has to be computed numerically.