Discontinuous Galerkin approximations in computational mechanics: hybridization, exact geometry and degree adaptivity

Abstract

Discontinuous Galerkin (DG) discretizations with exact representation of the geometry and local polynomial degree adaptivity are revisited. Hybridization techniques are employed to reduce the computational cost of DG approximations and devise the hybridizable discontinuous Galerkin (HDG) method. Exact geometry described by non-uniform rational B-splines (NURBS) is integrated into HDG using the framework of the NURBS-enhanced finite element method (NEFEM). Moreover, optimal convergence and superconvergence properties of HDG-Voigt formulation in presence of symmetric second-order tensors are exploited to construct inexpensive error indicators and drive degree adaptive procedures. Applications involving the numerical simulation of problems in electrostatics, linear elasticity and incompressible viscous flows are presented. Moreover, this is done for both high-order HDG approximations and the lowest-order framework of face-centered finite volumes (FCFV).

Hybridisable Discontinuous Galerkin Formulation of Compressible Flows

“…In addition, it permits an efficient exploitation of parallel computing architectures [52,111] and an easy implementation of adaptive strategies for non-uniform degree approximations [?, 3,10,59,65,77]. However, the duplication of nodes at the interface of neighbouring elements has limited its application mostly to academic problems, see the discussion in [57] and references therein.…”

HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB

“…Unified presentations of hybrid discretisation techniques and their relationship with other known numerical methods are available in [37,89,114]. Interested readers are also referred to the review papers [75,149] and to the recent monograph [112]. In the following subsections, an overview of the contributions on hybrid discretisation methods according to the authors' vision is presented.…”

“…It is worth noting that all the techniques mentioned above introduce geometric errors due to the polynomial approximation of the boundaries. In order to exploit the exact CAD representation of the boundaries, the NURBS-enhanced finite element method (NEFEM) [241,242] is employed in [149,239,247] to devise HDG formulations with exact geometry for Stokes, linear elastic and electrostatics problems.…”

“…In the context of adaptivity, on the one hand, the analysis of HDG approximations based on non-uniform polynomial degrees [60,61] and the superconvergence property of the postprocessed solution [77,89] prompted the development of degree adaptive procedures based on superparametric HDG methods [152,153] and on isoparametric HDG-NEFEM approaches [149,239,247]. Degree adaptivity is also applied to the simulation of cardiac electrophysiology in [159].…”

Coupling of hybridisable discontinuous Galerkin and finite volumes for transient compressible flows

“…The flexibility of uneven degree afforded by the discontinuous framework facilitates the tailoring of solution accuracy in the regions of interest. This facility has been previously used to devise degree adaptive algorithms, see [53,54,140,50].…”