Optimal geometric multigrid preconditioners for HDG-P0 schemes for the reaction-diffusion equation and the generalized Stokes equations

TL;DR

This paper introduces optimal HDG-P0 schemes with numerical integration for reaction-diffusion and Stokes equations, providing error analysis and multigrid preconditioners based on equivalence with nonconforming discretizations, verified by numerical tests.

Contribution

It presents the first optimal a priori error analysis for HDG-P0 schemes with numerical integration and develops geometric multigrid preconditioners leveraging their equivalence to nonconforming methods.

Findings

Optimal error estimates for HDG-P0 schemes established.

Effective multigrid preconditioners demonstrated for the linear systems.

Numerical results confirm theoretical predictions.

Abstract

We present the lowest-order hybridizable discontinuous Galerkin schemes with numerical integration (quadrature), denoted as HDG-P0, for the reaction-diffusion equation and the generalized Stokes equations on conforming simplicial meshes in two- and three-dimensions. Here by lowest order, we mean that the (hybrid) finite element space for the global HDG facet degrees of freedom (DOFs) is the space of piecewise constants on the mesh skeleton. A discontinuous piecewise linear space is used for the approximation of the local primal unknowns. We give the optimal a priori error analysis of the proposed {\sf HDG-P0} schemes, which hasn't appeared in the literature yet for HDG discretizations as far as numerical integration is concerned. Moreover, we propose optimal geometric multigrid preconditioners for the statically condensed HDG-P0 linear systems on conforming simplicial meshes. In both cases, we first establish the equivalence of the statically condensed HDG system with a (slightly modified) nonconforming Crouzeix-Raviart (CR) discretization, where the global (piecewise-constant) HDG finite element space on the mesh skeleton has a natural one-to-one correspondence to the nonconforming CR (piecewise-linear) finite element space that live on the whole mesh. This equivalence then allows us to use the well-established nonconforming geometry multigrid theory to precondition the condensed HDG system. Numerical results in two- and three-dimensions are presented to verify our theoretical findings.