A superconvergent HDG method for Stokes flow with strongly enforced symmetry of the stress tensor

TL;DR

This paper introduces a superconvergent HDG method for Stokes flow that strongly enforces stress tensor symmetry, improving accuracy and efficiency over classical methods, validated through numerical experiments in 2D and 3D.

Contribution

The paper presents a novel HDG formulation that strongly enforces stress symmetry, achieves superconvergence, and enhances computational efficiency without additional discrete space enrichment.

Findings

Achieves optimal convergence for mixed variables.

Provides superconvergent velocity through post-processing.

Validates effectiveness in 2D and 3D simulations.

Abstract

This work proposes a superconvergent hybridizable discontinuous Galerkin (HDG) method for the approximation of the Cauchy formulation of the Stokes equation using same degree of polynomials for the primal and mixed variables. The novel formulation relies on the well-known Voigt notation to strongly enforce the symmetry of the stress tensor. The proposed strategy introduces several advantages with respect to the existing HDG formulations. First, it remedies the suboptimal behavior experienced by the classical HDG method for formulations involving the symmetric part of the gradient of the primal variable. The optimal convergence of the mixed variable is retrieved and an element-by-element post-process procedure leads to a superconvergent velocity field, even for low-order approximations. Second, no additional enrichment of the discrete spaces is required and a gain in computational efficiency follows from reducing the quantity of stored information and the size of the local problems. Eventually, the novel formulation naturally imposes physical tractions on the Neumann boundary. Numerical validation of the optimality of the method and its superconvergent properties is performed in 2D and 3D using meshes of different element types.