Optimal $hp$-error estimates and $p$-multigrid convergence for Hybrid High-Order discretizations of the Poisson equation

Abstract

This paper presents two new theoretical results for Hybrid High-Order (HHO) methods applied to elliptic problems. First, we establish $hp$-error estimates for the HHO discretization of the Poisson problem that achieve optimal approximation rates with respect to both the mesh size $h$ and the polynomial degree $k$. These results improve upon previous analyses of hybrid methods, whose convergence estimates were suboptimal in $k$. Second, building on these estimates, we develop and analyze a non-inherited $p$-multigrid solver for the statically condensed HHO system. We prove results that improve upon the corresponding theory available for other non-conforming methods and constitute, to the best of our knowledge, the first rigorous convergence analysis of a $p$-multigrid algorithm for HHO discretizations.