Tutorial on Hybridizable Discontinuous Galerkin (HDG) Formulation for Incompressible Flow Problems

TL;DR

This paper presents a comprehensive tutorial on the HDG method for solving linearized incompressible Navier-Stokes equations, emphasizing stability, optimal convergence, and superconvergence through Voigt notation and polynomial approximations.

Contribution

It introduces a stable HDG formulation with optimal and superconvergent properties for incompressible flow, including implementation details and a postprocessing strategy.

Findings

Achieves optimal convergence order k+1 for velocity, pressure, and strain-rate tensor.

Develops a superconvergent velocity approximation of order k+2.

Provides a detailed tutorial for implementing HDG in incompressible flow simulations.

Abstract

A hybridizable discontinuous Galerkin (HDG) formulation of the linearized incompressible Navier-Stokes equations, known as Oseen equations, is presented. The Cauchy stress formulation is considered and the symmetry of the stress tensor and the mixed variable, namely the scaled strain-rate tensor, is enforced pointwise via Voigt notation. Using equal-order polynomial approximations of degree k for all variables, HDG provides a stable discretization. Moreover, owing to Voigt notation, optimal convergence of order k+1 is obtained for velocity, pressure and strain-rate tensor and a local postprocessing strategy is devised to construct an approximation of the velocity superconverging with order k+2, even for low-order polynomial approximations. A tutorial for the numerical solution of incompressible flow problems using HDG is presented, with special emphasis on the technical details required for its implementation.