An HDG method with orthogonal projections in facet integrals

TL;DR

This paper introduces a novel HDG method for second-order elliptic problems that incorporates $L^2$-orthogonal projections into facet integrals, achieving optimal convergence and superconvergence for scalar variables.

Contribution

The paper develops a new HDG formulation using orthogonal projections in facet integrals, enabling superconvergence without postprocessing.

Findings

Optimal convergence rates for all variables with polynomial degrees $k+l$, $k+1$, and $k$.

Superconvergence for scalar variables achieved without postprocessing.

Numerical results confirm theoretical convergence and superconvergence properties.

Abstract

We propose and analyze a new hybridizable discontinuous Galerkin (HDG) method for second-order elliptic problems. Our method is obtained by inserting the $L^2$-orthogonal projection onto the approximate space for a numerical trace into all facet integrals in the usual HDG formulation. The orders of convergence for all variables are optimal if we use polynomials of degree $k+l$, $k+1$ and $k$, where $k$ and $l$ are any non-negative integers, to approximate the vector, scalar and trace variables, which implies that our method can achieve superconvergence for the scalar variable without postprocessing. Numerical results are presented to verify the theoretical results.