Applications to Finite Element Matrices

Abstract

In §9.2.2 it was shown that the matrices arising from finite element discretisations (in what follows called the finite element matrices) are not only sparse but also belong to the $\mathcal{H}$-matrix set $\mathcal{H}(r, P)$ for all $r \in {_0}$, where P is the standard partition. This allows us to consider all finite element matrices as hierarchical matrices. In particular, no truncation is needed to use a finite element matrix as an input parameter for the inversion or for the LU algorithm.

In Section 11.1 we discuss the inverse of the mass matrix. Using tools from §9.5, we show that the inverse can be approximated by a hierarchical matrix. This result is required in the later analysis.

Section 11.2 is concerned with the continuous and discrete Green operator.

The analysis of the Green function in Section 11.3 yields the $\mathcal{H}$-matrix property of the inverse finite element matrix.

The results of this chapter have been improved by recent contributions mentioned in Section 11.4.