An Invitation to the Theory of the Hybridizable Discontinuous Galerkin Method

Du, Shukai; Sayas, Francisco–Javier

doi:10.1007/978-3-030-27230-2

Cited by 17 publications

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“…which proves that the sum of p γ n p Σ Ḱ and p γ p V Ḱ is orthogonal. Since p γ : p V Ḱ Ñ x M is injective (this result is known; see, for instance, [22] and [12]), the property (e) will follow from having proved that p γ n : p Σ Ḱ Ñ x M is injective. We first prove the following technical result: if σ P p Σ`satisfies div σ " 0 and p γ n σ " 0, then σ P p Σ, i.e., the component in p Σ fill vanishes.…”

Section: B)mentioning

confidence: 93%

New analytical tools for HDG in elasticity, with applications to elastodynamics

Sayas

2019

Math. Comp.

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We present some new analytical tools for the error analysis of hybridizable discontinuous Galerkin (HDG) method for linear elasticity. These tools allow us to analyze more variants of HDG method using the projection-based approach, which renders the error analysis simple and concise. The key result is a tailored projection for the Lehrenfeld-Schöberl type HDG (HDG+ for simplicity) methods. By using the projection we recover the error estimates of HDG+ for steady-state and time-harmonic elasticity in a simpler analysis. We also present a semi-discrete (in space) HDG+ method for transient elastic waves and prove it is uniformly-in-time optimal convergent by using the projection-based error analysis. Numerical experiments supporting our analysis are presented at the end.

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Section: B)mentioning

confidence: 93%

New analytical tools for HDG in elasticity, with applications to elastodynamics

Sayas

2019

Math. Comp.

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“…The book on the right, written with his then Ph.D. student Shukai Du, is about the socalled hybridizable discontinuous Galerkin (HDG) methods [38]. It was completed only a few weeks before Javier's death.…”

Section: Javier's Delaware Triptychmentioning

confidence: 99%

The pursuit of a dream, Francisco Javier Sayas and the HDG methods

Cockburn

2021

SeMA

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Franciso Javier Sayas, man of grit and determination, left his hometown of Zaragoza in 2007 in pursuit of a dream, to become a scholar in the USA. I hosted him in Minneapolis, where he spent three long years of an arduous transition before obtaining a permanent position at the University of Delaware. There, he enthusiastically worked on the unfolding of his dream until his life was tragically cut short by cancer, at only 50. In this paper, I try to bring to light the part of his academic life he shared with me. As we both worked on hybridizable discontinuous Galerkin methods, and he wrote a book on the subject, I will tell Javier’s life as it developed around this topic. First, I will show how the ideas of static condensation and hybridization, proposed back in the mid 60s, lead to the introduction of those methods. This background material will allow me to tell the story of the evolution of the hybridizable discontinuous Galerkin methods and describe Javier’s participation in it. Javier faced death with open eyes and poised dignity. I will end with a poem he liked.

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“…The Lehrenfeld-Schöberl HDG (HDG+) methods [16] have recently gained considerable interest since they superconverge on polyhedral meshes in addition to the easiness of implementation. In [12] (elliptic problems) and [13] (elasticity), we proposed mathematical tools to incorporate the analysis of the HDG+ methods into the projection-based error analysis setting [8]. In this way, we can reuse existing analysis techniques and avoid repeated or unnecessary arguments.…”

Section: Introductionmentioning

confidence: 99%

“…In this way, we can reuse existing analysis techniques and avoid repeated or unnecessary arguments. In [12] and [13], the projections were devised for simplicial elements. In this paper, we extend the results to polyhedral elements.…”

Section: Introductionmentioning

confidence: 99%

“…Following this idea, we have devised the HDG+ projections in [12] for ellptic problems and in [13] for elasticity with strong symmetric stress formulation. We have sucessfully used the projections, combined with some existing analysis techniques of the standard HDG methods, to derive the error estimates of the HDG+ methods for heat diffusion and acoustic waves in [12], and for time-harmonic and transient elastic waves in [13]. For simplicity, we have limited the discussions on simplicial meshes in [12,13].…”

Section: Introductionmentioning

confidence: 99%

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A note on devising HDG+ projections on polyhedral elements

Sayas

2020

Preprint

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In this note, we propose a simple way of constructing HDG+ projections on polyhedral elements. The projections enable us to analyze the Lehrenfeld-Schöberl HDG (HDG+) methods in a very concise manner, and make many existing analysis techniques of standard HDG methods reusable for HDG+. The novelty here is an alternative way of constructing the projections without using M -decomposition as a middle step. This extends our previous results [S. Du and F.-J. Sayas, SpringerBriefs in Mathematics ( 2019)] (elliptic problems) and [S. Du and F.-J. Sayas, arXiv:1903.11766] (elasticity) to polyhedral meshes.

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An Invitation to the Theory of the Hybridizable Discontinuous Galerkin Method

Cited by 17 publications

References 0 publications

New analytical tools for HDG in elasticity, with applications to elastodynamics

New analytical tools for HDG in elasticity, with applications to elastodynamics

The pursuit of a dream, Francisco Javier Sayas and the HDG methods

A note on devising HDG+ projections on polyhedral elements

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