A new perspective on the fictitious space lemma

TL;DR

This paper offers a novel proof of the fictitious space lemma by explicitly expressing the inverse of additive Schwarz preconditioners using Moore-Penrose pseudo-inverses, enhancing theoretical understanding.

Contribution

It introduces a new proof of the fictitious space lemma with an explicit inverse expression for additive Schwarz preconditioners.

Findings

Explicit inverse expression for additive Schwarz preconditioners

New proof of the fictitious space lemma

Enhanced theoretical insight into domain decomposition methods

Abstract

In the present contribution we propose a new proof of the so-called fictitious space lemma. For the proof, we exhibit an explicit expression for the inverse of additive Schwarz preconditioners in terms of Moore-Penrose pseudo inverse of the map associated to the decomposition over the subdomain partition.