Uniform auxiliary space preconditioning for HDG methods for elliptic operators with a parameter dependent low order term

TL;DR

This paper develops uniform auxiliary space preconditioners for HDG methods applied to various elliptic problems with low order terms, ensuring optimality and parameter independence.

Contribution

It introduces a unified ASP approach for multiple HDG schemes solving elliptic equations with low order terms, proving their optimality and uniformity.

Findings

Uniform preconditioners achieve optimal convergence rates.

Preconditioners are robust with respect to mesh size and low order parameter.

The approach applies to scalar, vectorial, and biharmonic elliptic problems.

Abstract

The auxiliary space preconditioning (ASP) technique is applied to the HDG schemes for three different elliptic problems with a parameter dependent low order term, namely, a symmetric interior penalty HDG scheme for the scalar reaction-diffusion equation, a divergence-conforming HDG scheme for a vectorial reaction-diffusion equation, and a C 0 -continuous interior penalty HDG scheme for the generalized biharmonic equation with a low order term. Uniform preconditioners are obtained for each case and the general ASP theory by J. Xu [21] is used to prove the optimality with respect to the mesh size and uniformity with respect to the low order parameter.