Conditions for superconvergence of HDG methods for second-order elliptic problems

Abstract

Abstract. We provide a projection-based analysis of a large class of finite element methods for second order elliptic problems. It includes the hybridized version of the main mixed and hybridizable discontinuous Galerkin methods. The main feature of this unifying approach is that it reduces the main difficulty of the analysis to the verification of some properties of an auxiliary, locally defined projection and of the local spaces defining the methods. Sufficient conditions for the optimal convergence of the approximate flux and the superconvergence of an element-by-element postprocessing of the scalar variable are obtained. New mixed and hybridizable discontinuous Galerkin methods with these properties are devised which are defined on squares, cubes and prisms.

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

“…The conditions for superconvergence imposed upon the finite element space depending on the element shape are discussed in [18]. Numerical results demonstrating the L 2 errors before and after postprocessing for hexahedral and tetrahedral elements (see Section 5) indicate that the technique works as intended for both types of elements.…”

“…Remark 2 In [18], the space for the numerical flux q that ensures superconvergence of the HDG method on a hexahedral element is slightly larger than the standard tensor-product space Q k (for details, see Table 6 in [18]). In our implementation, we use the Q k space to represent the flux variable and still observe superconvergence properties on hexahedral elements numerically.…”

“…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 superconvergent HDG method for the incompressible Navier–Stokes equations on general polyhedral meshes

“…Notice that as a built-in feature of HDG methods, see [11], the degrees of freedom of the globally-coupled unknown comes from the numerical trace of the velocity on the mesh skeleton. From the point of view of the global degrees of freedom, the method provides optimal convergent approximations to the velocity, velocity gradient and pressure in L 2 -norm for k ≥ 0, superconvergent approximation to the velocity in the discrete H 1 -norm without postprocessing for k ≥ 1, and superconvergent approximation to the velocity in L 2 norm without postprocessing for k ≥ 1.…”

A hybridizable discontinuous Galerkin method for fractional diffusion problems

“…The choice of the HDG methods for the problem under consideration can be easily justified. Indeed, the HDG methods are a relatively new class of DG methods introduced in [6] in the framework of steady-sate diffusion which share with the classical (hybridized version of the) mixed finite element methods their remarkable convergence properties, [7,8,9], as well as the way in which they can be efficiently implemented, [18]. They provide approximations that are more accurate than the ones given by any other DG method for second-order elliptic problems [36].…”