Operator preconditioning: the simplest case

TL;DR

This paper presents a straightforward approach to operator preconditioning for elliptic operators, utilizing Calderón preconditioning with a composition of an opposite order operator and diagonal scalings, ensuring uniform preconditioning.

Contribution

It introduces a simple, effective preconditioning method for elliptic operators using Calderón framework with novel composition of operators and scalings.

Findings

Preconditioners are uniform across discretizations.

The method simplifies existing Calderón preconditioning techniques.

Effective for continuous finite element discretizations.

Abstract

Using the framework of operator or Calder\'on preconditioning, uniform preconditioners are constructed for elliptic operators discretized with continuous finite (or boundary) elements. The preconditioners are constructed as the composition of an opposite order operator, discretized on the same ansatz space, and two diagonal scaling operators.