Conditions for superconvergence of HDG methods for Stokes flow

Abstract

We provide an a priori error analysis of a wide class of finite element methods for the Stokes equations. The methods are based on the velocity gradient-velocity-pressure formulation of the equations and include new and old mixed and hybridizable discontinuous Galerkin methods. We show how to reduce the error analysis to the verification of some properties of an elementwise-defined projection and of the local spaces defining the methods. We also show that the projection of the errors only depends on the approximation properties of the projection. We then provide sufficient conditions for the superconvergence of the projection of the error in the approximate velocity. We give many examples of these methods and show how to systematically construct them from similar methods for the diffusion equation.

Analysis of an HDG method for linear elasticity

“…Moreover, the methods provide approximate stresses and displacements converging with order kC1 when the exact solution is smooth enough. This has been observed experimentally for the method proposed in [1,22], and theoretically proven in [28,29] for the method in [24,25], and in [26] for the method proposed therein. So, in terms of convergence properties, the only main difference between the methods seems to be the superconvergence properties of the displacement.…”

To CG or to HDG: A Comparative Study in 3D

“…The last step is to express v e from Equation (14) (the local problem) and plug it into Equation (19) above. This will give us our global system for the trace unknowns:…”

“…The HDG method has proven to be a popular method and has, in recent years, been applied in the context of steady-state di↵usion [9,10,16,18], Maxwell's equations [32,33,36], convection-di↵usion problems [7,13,34,35], linear elasticity [45], Timoshenko beam model [4,5], elastodynamics [37], Stokes equations [12,17,19,34], compressible [26,49] and incompressible Navier-Stokes, and Oseen equations [6,37,39,40]. We note, however, that the majority of these works focus either on the theoretical aspects of the method such as formulation and analysis for a specific equation type, or the specific benefits such as accurately captured solution features that the HDG method can o↵er.…”

A class of embedded discontinuous Galerkin methods for computational fluid dynamics

“…This is the case for HDG methods for linear convection-diffusion problems [51,12,8,9] and nonlinear convection-diffusion problems [12,52,65] in the diffusion-dominated regime, for HDG methods for the Stokes system of incompressible flow [13,15,47,53,26], for HDG methods for the incompressible Navier-Stokes equations [49,50,54] in the diffusion-dominated regime, and for HDG methods for linear elasticity [26,27]. In particular, a unique feature of the HDG method for incompressible fluid flow is that the approximate velocity, pressure and velocity gradient converge with the optimal order k + 1 in the L 2 -norm for diffusion-dominated flows for any k ≥ 0.…”