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

Abstract

This paper presents , an open source MATLAB implementation of the hybridisable discontinuous Galerkin (HDG) method. The main goal is to provide a detailed description of both the HDG method for elliptic problems and its implementation available in . Ultimately, this is expected to make this relatively new advanced discretisation method more accessible to the computational engineering community. presents some features not available in other implementations of the HDG method that can be found in the free domain. First, it implements high-order polynomial shape functions up to degree nine, with both equally-spaced and Fekete nodal distributions. Second, it supports curved isoparametric simplicial elements in two and three dimensions. Third, it supports non-uniform degree polynomial approximations and it provides a flexible structure to devise degree adaptivity strategies. Finally, an interface with the open-source high-order mesh generator is provided to facilitate its application to practical engineering problems.

A scalable elliptic solver with task-based parallelism for the SpECTRE numerical relativity code

“…However, the larger number of elements with shared boundaries also means that we need more grid points than SpEC to reach the same accuracy for a black-hole binary initial-data problem. Variations of the DG scheme, such as a hybridizable DG method, can provide a possible resolution to this effect [64][65][66]. Even without changing the DG scheme, we expect that optimizations of our binary compact-object domain can significantly reduce the number of grid points required to reach a certain accuracy.…”

A posteriori goal-oriented bounds for the Poisson problem using potential and equilibrated flux reconstructions: application to the hybridizable discontinuous Galerkin method

“…Let ũh and ξh be two potential reconstructions of u and ξ respectively and consider qh = q = −ν∇u and ζh = ζ = −ν∇ξ. Noting that in this case s∓ h = s ∓ h , so that s∓ h can be rewritten as shown in equation (B.2), the exact representation for the quantity of interest (19) yields after some rearrangements to…”

“…Also, goal-oriented error indicators are provided to enhance the convergence of adaptive remeshing for non-smooth problems. HDG methods have gained popularity in the last decade due to their reduced computational cost with respect to classical discontinuous Galerkin methods while retaining superconvergence properties [19]. Also, a very attractive feature is that a simple post-process of the solution yields equilibrated H(div; Ω) approximations of the fluxes.…”

A Coupled HDG-FV Method for Unsteady Compressible Flows

“…In some studies, the reduction in terms of computing time has also been demonstrated [1]. Out of the many possible high-order methods available, the hybridisable discontinuous Galerkin (HDG) [2,3] is an attractive option due to the reduced number of degrees of freedom, compared to other DG methods, and the ability to introduce the stabilisation required in convection dominated flows [4].…”