Optimal quasi-diagonal preconditioners for pseudodifferential operators of order minus two

TL;DR

This paper introduces quasi-diagonal preconditioners for discretized pseudodifferential operators of order minus two, demonstrating their asymptotic optimality across multiple dimensions through theoretical proofs and numerical experiments.

Contribution

It develops and proves the effectiveness of quasi-diagonal preconditioners for high-dimensional pseudodifferential operators, extending their applicability to various polynomial degrees and dimensions.

Findings

Preconditioners are asymptotically optimal in dimensions > 1.

Numerical experiments confirm theoretical condition number estimates.

Effective for piecewise constant and linear polynomials across multiple dimensions.

Abstract

We present quasi-diagonal preconditioners for piecewise polynomial discretizations of pseudodifferential operators of order minus two in any space dimension. Here, quasi-diagonal means diagonal up to a sparse transformation. Considering shape regular simplicial meshes and arbitrary fixed polynomial degrees, we prove, for dimensions larger than one, that our preconditioners are asymptotically optimal. Numerical experiments in two, three and four dimensions confirm our results. For each dimension, we report on condition numbers for piecewise constant and piecewise linear polynomials.