A hybridized discontinuous Galerkin method with reduced stabilization

TL;DR

This paper introduces a hybridized discontinuous Galerkin method with reduced stabilization for the Poisson equation, allowing lower-degree polynomial approximations and providing error estimates and numerical validation.

Contribution

It presents a novel HDG method with reduced stabilization that uses different polynomial degrees for element and inter-element unknowns, improving efficiency.

Findings

Error estimates in energy and L2 norms are established.

The method is related to the Crouzeix-Raviart nonconforming finite element for k=1.

Numerical results confirm the method's validity.

Abstract

In this paper, we propose a hybridized discontinuous Galerkin(HDG) method with reduced stabilization for the Poisson equation. The reduce stabilization proposed here enables us to use piecewise polynomials of degree $k$ and $k-1$ for the approximations of element and inter-element unknowns, respectively, unlike the standard HDG methods. We provide the error estimates in the energy and $L^2$ norms under the chunkiness condition. In the case of $k=1$, it can be shown that the proposed method is closely related to the Crouzeix-Raviart nonconforming finite element method. Numerical results are presented to verify the validity of the proposed method.