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

Abstract

Since the inception of discontinuous Galerkin (DG) methods for elliptic problems, there has existed a question of whether DG methods can be made more computationally e cient than continuous Galerkin (CG) methods. Fewer degrees of freedom, approximation properties for elliptic problems together with the number of optimization techniques, such as static condensation, available within CG framework make it challenging for DG methods to be competitive until recently. However, with the introduction of a static-condensation-amenable DG method -the hybridizable discontinuous Galerkin (HDG) method -it has become possible to perform a realistic comparison of CG and HDG methods when applied to elliptic problems. In this work, we extend upon an earlier 2D comparative study, providing numerical results and discussion of the CG and HDG method performance in three dimensions. The comparison categories covered include steady-state elliptic and time-dependent parabolic problems, various element types and serial and parallel performance. The postprocessing technique, which allows for superconvergence in the HDG case, is also discussed. Depending on the linear system solver used and the type of the problem (steady-state vs time-dependent) in question the HDG method either outperforms or demonstrates a comparable performance when compared with the CG method. The HDG method however falls behind performance-wise when the iterative solver is used, which indicates the need for an e↵ective preconditioning strategy for the method.

On continuous, discontinuous, mixed, and primal hybrid finite element methods for second‐order elliptic problems

“…In this context, the continuous and discontinuous methods demonstrate better performances than the hybrid method. This conclusion is in contrast with the previous efficiency results in the works of Giorgiani et al, and Forti et al, and the given justification is the use of better performance matrix‐free implementations.…”

“…For instance, efficiency comparisons were presented in the work of Kirby et al for two‐dimensional (2D) elliptic problems discretized by hybrid DG (HDG) methods based on triangular and quadrilateral meshes. In the work of Yakovlev et al, the work is extended for three‐dimensional (3D) steady‐state elliptic and time‐dependent parabolic problems, using serial execution and direct solvers. In this work, the authors have found that the hybrid approaches can outperform the H 1 ‐conforming method.…”

Efficient construction of unified continuous and discontinuous Galerkin formulations for the 3D Euler equations

“…Hybridizable Discontinuous Galerkin (HDG) [7,23,32,31,40] is a method that aims to make DG computationally competitive with CG using the static condensation model reduction technique discussed in the previous section. Hybridization was introduced in [41] for efficient solution of finite element approximations of linear elasticity problems.…”

“…Hybridization was introduced in [41] for efficient solution of finite element approximations of linear elasticity problems. Both HDG and CG with static condensation follow the same pipeline [31,40], namely construction of local problems, global formulation and post-processing. The local problem is the procedure to find the solution on an element based on the solution at the boundary assuming fixed values.…”

eXtended hybridizable discontinuous Galerkin for incompressible flow problems with unfitted meshes and interfaces

“…Thus, the number of DOFs for the velocity is slightly larger for HDG compared to CG, due to the duplication of the vertex nodes (and edge nodes in 3D), but HDG has fewer DOFs for the pressure. Moreover, the block structure of the sparsity pattern of the HDG matrices has been shown to be convenient for linear solvers, concluding that HDG is very competitive in front of CG (see other works for efficiency comparisons for heat and wave problems). A comparison of CPU time for the solution of 2D incompressible flow problems in the work of Paipuri et al has found that HDG requires less CPU time for the direct linear solver than CG for the same level of accuracy.…”