Coupling of Raviart--Thomas and Hybridizable Discontinuous Galerkin Methods with BEM

Cockburn, Bernardo; Guzmán, Johnny; Sayas, Francisco–Javier

doi:10.1137/100818339

Cited by 13 publications

(11 citation statements)

References 12 publications

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“…During his Zaragoza period, Javier had been interested in the coupling of LDG and BE methods [2,45,46]. During his Minneapolis period, he continued to develop this interest in three papers published in 2012, namely, on the coupling of the RT or HDG with BE methods [23], on general symmetric couplings of DG and BE methods [29], and on coupling at a distance HDG and BE methods [31]. During his Delaware period, he worked on nonsymmetric couplings of HDG and BE methods in the 2017 paper [44].…”

Section: Coupling Hdg and Boundary Element Methodsmentioning

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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Section: Coupling Hdg and Boundary Element Methodsmentioning

confidence: 99%

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

Cockburn

2021

SeMA

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“…The classical bending-dominated problem of Cook's membrane 78 is revisited to take into account two materials with different mechanical properties as proposed in. 77 The tapered plate Ω defined by the points (0, 0), (48,44), (48,60), (0, 44) is decomposed in two nonoverlapping subdomains such that Ω CG is the convex region identified by the points (12, 20.25), (36, 38.75), (36, 50.25), (12, 38.75) and Ω HDG =Ω \ Ω CG . The first three levels of mesh refinement are displayed in Figure 9, where the subdomain Ω CG is represented in red, Ω HDG in blue and the interface Γ I in black.…”

Section: Bimaterials Cook's Membrane Problemmentioning

confidence: 99%

Hybrid coupling of CG and HDG discretizations based on Nitsche’s method

2019

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A strategy to couple continuous Galerkin (CG) and hybridizable discontinuous Galerkin (HDG) discretizations based only on the HDG hybrid variable is presented for linear thermal and elastic problems. The hybrid CG-HDG coupling exploits the definition of the numerical flux and the trace of the solution on the mesh faces to impose the transmission conditions between the CG and HDG subdomains. The continuity of the solution is imposed in the CG problem via Nitsche's method, whereas the equilibrium of the flux at the interface is naturally enforced as a Neumann condition in the HDG global problem. The proposed strategy does not affect the core structure of CG and HDG discretizations. In fact, the resulting formulation leads to a minimally-intrusive coupling, suitable to be integrated in existing CG and HDG libraries. Numerical experiments in two and three dimensions show optimal global convergence of the stress and superconvergence of the displacement field, locking-free approximation, as well as the potential to treat structural problems of engineering interest featuring multiple materials with compressible and nearly incompressible behaviors.

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“…HDG methods were proposed by Cockburn et al in [11] as an improvement of traditional discontinuous Galerkin (DG) methods and have many applications; see, e.g., [8,9,12,13,15,16,32]. HDG methods are based on mixed formulations and introduce a new variable to approximate the trace of the scalar variable along the element boundary.…”

Section: A Class Of Embedded Dg Formulationsmentioning

confidence: 99%

A Class of Embedded DG Methods for Dirichlet Boundary Control of Convection Diffusion PDEs

Chen

Singler

et al. 2019

J Sci Comput

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We investigated an hybridizable discontinuous Galerkin (HDG) method for a convection diffusion Dirichlet boundary control problem in our earlier work [SIAM J. Numer. Anal. 56 (2018) 2262-2287] and obtained an optimal convergence rate for the control under some assumptions on the desired state and the domain. In this work, we obtain the same convergence rate for the control using a class of embedded DG methods proposed by Nguyen, Peraire and Cockburn [J. Comput. Phys. vol. 302 (2015), pp. 674-692] for simulating fluid flows. Since the global system for embedded DG methods uses continuous elements, the number of degrees of freedom for the embedded DG methods are smaller than the HDG method, which uses discontinuous elements for the global system. Moreover, we introduce a new simpler numerical analysis technique to handle low regularity solutions of the boundary control problem. We present some numerical experiments to confirm our theoretical results. Recently, researchers have investigated discontinuous Galerkin (DG) methods for Dirichlet boundary control problems. We used a hybridizable discontinuous Galerkin (HDG) method for the Poisson equation in [25], and obtained a superlinear convergence rate for the control without using a special mesh or a higher order element. More recently, convection diffusion Dirichlet boundary control problems have gained more and more attention. Benner et al. in [3] used a local discontinuous Galerkin (LDG) method to obtain a sublinear convergence rate for the control. We considered an HDG method and proved optimal superlinear convergence rate for the control in [24] if the regularity of the solution is high, i.e., y ∈ H 1+s (Ω) with s ≥ 1/2. To overcome the difficulty for the low regularity case (0 ≤ s < 1/2), we utilized a special projection operator to get an optimal superlinear convergence rate in [18]; the numerical analysis was more complicated than in [24]. Furthermore, in contrast to [25], we obtained an optimal superliner convergence rate for the control by using a discontinuous higher order (quadratic) element.Although the degrees of freedom of the HDG method are significantly reduced compared to standard mixed methods, DG methods and LDG methods, they are still larger than the degrees of freedom of the CG method. In this work, we use embedded DG (EDG) and interior EDG (IEDG) methods to approximate the solution of the Dirichlet boundary control problem. The EDG and IEDG methods are obtained from the HDG methods, and the global systems both use the same continuous elements; this reduces the number of degrees of freedom considerably. To approximate the control, we use continuous element in the EDG method, and discontinuous elements in the IEDG method. Although the degrees of freedom of IEDG is slightly larger than the EDG method, the IEDG method provides greater flexibility for boundary control problems: we can use different finite element spaces for the control and the state. One possible benefit of the greater flexibility of the IEDG method is that discontinuous elements ...

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Coupling of Raviart--Thomas and Hybridizable Discontinuous Galerkin Methods with BEM

Cited by 13 publications

References 12 publications

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

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

Hybrid coupling of CG and HDG discretizations based on Nitsche’s method

A Class of Embedded DG Methods for Dirichlet Boundary Control of Convection Diffusion PDEs

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