Analysis of a reduced-order HDG method for the Stokes equations

TL;DR

This paper analyzes a reduced-stabilization hybridized discontinuous Galerkin method for the Stokes equations, demonstrating improved efficiency, optimal error estimates, and connections to nonconforming finite element methods.

Contribution

It introduces a reduced-stabilization HDG method that decreases facet unknowns and links it to nonconforming finite element methods, with proven convergence properties.

Findings

Optimal error estimates in energy and L2 norms.

Convergence of the reduced method to nonconforming Gauss-Legendre solutions as stabilization parameter increases.

Reduced method closely related to Crouzeix-Raviart finite element method.

Abstract

In this paper, we analyze a hybridized discontinuous Galerkin(HDG) method with reduced stabilization for the Stokes equations. The reduced stabilization enables us to reduce the number of facet unknowns and improve the computational efficiency of the method. We provide optimal error estimates in an energy and $L^2$ norms. It is shown that the reduced method with the lowest-order approximation is closely related to the nonconforming Crouzeix-Raviart finite element method. We also prove that the solution of the reduced method converges to the nonconforming Gauss-Legendre finite element solution as a stabilization parameter $\tau$ tends to infinity and that the convergence rate is $O(\tau^{-1})$.