$p$-Multilevel preconditioners for HHO discretizations of the Stokes equations with static condensation

TL;DR

This paper introduces a $p$-multilevel preconditioner for HHO discretizations of the Stokes equations, analyzing its efficiency and comparing it with classical methods through extensive numerical experiments.

Contribution

It develops a novel $p$-multilevel preconditioner for HHO methods, incorporating static condensation and efficient implementation strategies, validated on 2D and 3D problems.

Findings

The preconditioner improves convergence rates for HHO discretizations.

Static condensation influences the size and sparsity of the linear systems.

The proposed implementation is efficient and scalable for complex problems.

Abstract

We propose a $p$-multilevel preconditioner for Hybrid High-Order discretizations (HHO) of the Stokes equation, numerically assess its performance on two variants of the method, and compare with a classical Discontinuous Galerkin scheme. We specifically investigate how the combination of $p$-coarsening and static condensation influences the performance of the $V$-cycle iteration for HHO. Two different static condensation procedures are considered, resulting in global linear systems with a different number of unknowns and non-zero elements. An efficient implementation is proposed where coarse level operators are inherited using $L^2$-orthogonal projections defined over mesh faces and the restriction of the fine grid operators is performed recursively and matrix-free. The various resolution strategies are thoroughly validated on two- and three-dimensional problems.