Uniform preconditioners for problems of negative order

TL;DR

This paper introduces a new class of uniform preconditioners for negative order operators that are efficient, mesh-independent, and easier to implement than previous methods, applicable to various polynomial discretizations.

Contribution

The paper constructs a novel preconditioner for negative order operators using an invertible operator of opposite order, avoiding complex matrix inversions and mesh refinements.

Findings

Preconditioner scales linearly with mesh size

Does not require inverse of non-diagonal matrices

Operates without mesh grading or barycentric refinement

Abstract

Uniform preconditioners for operators of negative order discretized by (dis)continuous piecewise polynomials of any order are constructed from a boundedly invertible operator of opposite order discretized by continuous piecewise linears. Besides the cost of the application of the latter discretized operator, the other cost of the preconditioner scales linearly with the number of mesh cells. Compared to earlier proposals, the preconditioner has the following advantages: It does not require the inverse of a non-diagonal matrix; it applies without any mildly grading assumption on the mesh; and it does not require a barycentric refinement of the mesh underlying the trial space.