HDGlab : An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB
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ORIGINAL PAPER
HDGlab: An Open‑Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB Matteo Giacomini1,2 · Ruben Sevilla3 · Antonio Huerta1,2 Received: 10 September 2020 / Accepted: 24 September 2020 © The Author(s) 2020
Abstract This paper presents HDGlab, 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 HDGlab. Ultimately, this is expected to make this relatively new advanced discretisation method more accessible to the computational engineering community. HDGlab 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 Gmsh is provided to facilitate its application to practical engineering problems. Keywords Hybridizable discontinuous Galerkin · High-order · Elliptic problems · MATLAB · Open-source Mathematics Subject Classification 35-04 · 35M32 · 65N30 · 68-04 · 97N80
1 Introduction In recent years, hybrid discretisation methods have received increasing attention by the applied mathematics and computational engineering community. The main interest in these methodologies is due to their reduced computational cost with respect to classical discontinuous Galerkin (DG) methods, see [135, 163, 170, 267], from which they inherit appealing stability and convergence properties as well as the flexibility to devise high-order, non-uniform degree and adaptive discretisations and the capability to efficiently
* Ruben Sevilla [email protected] 1
Laboratori de Calcul Numeric (LaCàN), ETS de Ingenieros de Caminos, Canales y Puertos, Universitat Politècnica de Catalunya, Barcelona, Spain
2
International Centre for Numerical Methods in Engineering (CIMNE), Barcelona, Spain
3
Zienkiewicz Centre for Computational Engineering, College of Engineering, Swansea University, Bay Campus, Swansea SA1 8EN, Wales, UK
exploit parallel computing architectures [42, 91, 108, 158, 226]. The purpose of the present contribution is two-fold: to present a review on the state-of-the-art of hybrid discretisation methods including both fundamental and applied contributions; to provide an educational implementation of the hybridisable discontinuous Galerkin (HDG) method in MATLAB, the so-called HDGlab library, and describe its structure, capabilities and functioning. HDGlab is an open-source library released under GNU GPL licence and designed for rapid prototyping and testing. It supports simplicial meshes and it provides a seamless 2D
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