# Uniform preconditioners for problems of positive order

## TL;DR

This paper develops uniform preconditioners for elliptic operators of various orders, enabling efficient solutions without constructing dual meshes, by leveraging operator preconditioning techniques.

## Contribution

It introduces a novel framework for uniform preconditioning of elliptic operators of order between 0 and 2, avoiding dual mesh construction and maintaining linear complexity.

## Key findings

- Preconditioners are effective across a range of elliptic operator orders.
- The approach avoids the need for dual mesh construction.
- Preconditioner application costs are linear in problem size.

## Abstract

Using the framework of operator or Cald\'{e}ron preconditioning, uniform preconditioners are constructed for elliptic operators of order $2s \in [0,2]$ discretized with continuous finite (or boundary) elements. The cost of the preconditioner is the cost of the application an elliptic opposite order operator discretized with discontinuous or continuous finite elements on the same mesh, plus minor cost of linear complexity. Herewith the construction of a so-called dual mesh is avoided.