A Superconvergent HDG Method for Stokes Flow with Strongly Enforced Symmetry of the Stress Tensor

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 1 arXiv:1802.09394v1 [math.NA] 26 Feb 2018 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.

A face‐centred finite volume method for second‐order elliptic problems

“…The optimality is retrieved using simplicial elements or enriching the discrete spaces via the introduction of additional basis functions. Numerical experiments using high‐order HDG discretisations (cf the work of Giacomini et al) confirm that quadrilateral meshes experience a slight loss of accuracy in the approximation of pressure and gradient of velocity. Nevertheless, it is worth noting that the convergence order always deteriorates less than the predicted value 1/2.…”

A second-order face-centred finite volume method for elliptic problems

“…To simplify the presentation, this work considers the traditional velocity-pressure HDG formulation of the Stokes equation [32], where the vector t does not correspond to the boundary traction and it is usually called a pseudo-traction [13]. It is worth emphasising that the so-called Cauchy formulation could also be employed here, using the formulation proposed in [16].…”

A superconvergent hybridisable discontinuous Galerkin method for linear elasticity

“…An alternative physical interpretation of conditions and is given in the work of Giacomini et al, exploiting the definition of the vorticity of a fluid as the curl of its velocity field. For additional details on the postprocess procedures inspired by the velocity‐pressure‐vorticity formulation of the Stokes equation, the interested reader is referred to the works of Cockburn et al…”