HDG Methods for Stokes Equation Based on Strong Symmetric Stress Formulations
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HDG Methods for Stokes Equation Based on Strong Symmetric Stress Formulations Shukai Du1 Received: 8 March 2020 / Revised: 21 August 2020 / Accepted: 5 September 2020 / Published online: 23 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We propose a hybridizable discontinuous Galerkin (HDG) method for Stokes equation based on strong symmetric stress formulations. Stress-based method stands out compared to gradient or vorticity-based methods by its simple and natural way of enforcing Neumann type boundary conditions. Our method uses the polynomial spaces of orders k − (k + 1) − k for the stress–velocity–pressure triplet, and orders k − (k + 1) for the tangential and the normal components of the numerical trace space. By sending the normal stabilization to infinity, we obtain another method that provides exactly divergence-free solution and is pressure-robust. We prove that both methods are optimal for all variables and achieve super-convergence for the numerical trace. In addition, we build a quantitative connection between the normal stabilization and the pressure-robustness. Numerical experiments are also presented to validate our theoretical discoveries. Keywords Discontinuous Galerkin · Hybridization · Stokes equation · Strong symmetric stress · Super-convergence · Pressure-robust Mathematics Subject Classification 65N15 · 65N30
1 Introduction Hybridizable discontinuous Galerkin (HDG) methods [9] are developed based on discontinuous Galerkin (DG) methods and are also closely related to mixed finite element methods [14]. Since HDG methods naturally support static condensation, they can be efficiently implemented [22,32]. On the other hand, the fact that many DG, mixed and continuous Galerkin finite element methods can be related to HDG methods with suitable choices of stabilization functions [9] make HDG methods interesting from analysis perspectives.
This work is partially supported by the NSF Grant DMS-1818867.
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Shukai Du [email protected] University of Delaware, Newark, USA
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Journal of Scientific Computing (2020) 85:8
In this paper, we consider HDG methods for solving Stokes equations in the following stress-velocity-pressure formulation (stress-based): σ − ε(u) = 0
in Ω,
(1a)
−ν∇ · σ + ∇ p = f
in Ω,
(1b)
∇ ·u=0
in Ω,
(1c)
u=g
on Γ D ,
(1d)
νσ n − pn = h p = 0.
on Γ N ,
(1e)
Ω
(1f)
In the above equations, Ω ⊂ Rd is a Lipschitz polyhedral domain, Γ D and Γ N are its Dirichlet and Neumann boundaries and we assume they have non-vanishing (d − 1)measure. We assume the forcing term f ∈ L 2 (Ω; Rd ), the Dirichlet data g ∈ L 2 (Γ D ; Rd ), the Neumann data h ∈ L 2 (Γ N ; Rd ), and the viscosity 0 < ν ≤ 1 for convenience. The other two formulations are the gradient-velocity-pressure (gradient-based) and the vorticityvelocity-pressure (vorticity-based) formulations, which respectively replace Eq. (1a) by L − ∇u = 0 (gradient based), w + ∇ × u = 0 (vorticity based). Due to the divergence-free condition, the above three form
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