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Journal articles on the topic 'Hybridizable discontinuous galerkin (HDG)'

Author: Grafiati

Published: 2 September 2023

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1

Chen, Gang, and Jintao Cui. "On the error estimates of a hybridizable discontinuous Galerkin method for second-order elliptic problem with discontinuous coefficients." IMA Journal of Numerical Analysis 40, no. 2 (February 6, 2019): 1577–600. http://dx.doi.org/10.1093/imanum/drz003.

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Abstract Hybridizable discontinuous Galerkin (HDG) methods retain the main advantages of standard discontinuous Galerkin (DG) methods, including their flexibility in meshing, ease of design and implementation, ease of use within an $hp$-adaptive strategy and preservation of local conservation of physical quantities. Moreover, HDG methods can significantly reduce the number of degrees of freedom, resulting in a substantial reduction of computational cost. In this paper, we study an HDG method for the second-order elliptic problem with discontinuous coefficients. The numerical scheme is proposed on general polygonal and polyhedral meshes with specially designed stabilization parameters. Robust a priori and a posteriori error estimates are derived without a full elliptic regularity assumption. The proposed a posteriori error estimators are proved to be efficient and reliable without a quasi-monotonicity assumption on the diffusion coefficient.

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2

Li, Meng, and Xianbing Luo. "Convergence Analysis and Cost Estimate of an MLMC-HDG Method for Elliptic PDEs with Random Coefficients." Mathematics 9, no. 9 (May 10, 2021): 1072. http://dx.doi.org/10.3390/math9091072.

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We considered an hybridizable discontinuous Galerkin (HDG) method for discrete elliptic PDEs with random coefficients. By an approach of projection, we obtained the error analysis under the assumption that a(ω,x) is uniformly bounded. Together with the HDG method, we applied a multilevel Monte Carlo (MLMC) method (MLMC-HDG method) to simulate the random elliptic PDEs. We derived the overall convergence rate and total computation cost estimate. Finally, some numerical experiments are presented to confirm the theoretical results.

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3

Kong, Yan, and Mei Yan. "A Hybridizable Discontinuous Galerkin and Boundary Element coupling method for electromagnetic simulations." Journal of Physics: Conference Series 2287, no. 1 (June 1, 2022): 012027. http://dx.doi.org/10.1088/1742-6596/2287/1/012027.

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Abstract In this paper, the hybridizable discontinuous Galerkin (HDG) method is combined with the electric field integral equation (EFIE) for the numerical simulation of time-harmonic Maxwell’s equation. The key feature of HDG-EFIE is that the HDG method and the boundary element method (BEM) can be coupled naturally through the extra hybrid variable on the faces of elements, the combined method produces a linear system in term of the degrees of freedom of the extra hybrid variable only. A numerical solution was compared to the analytical solution and found to be in excellent agreement.

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4

Wang, Bo, and B. C. Khoo. "Hybridizable discontinuous Galerkin method (HDG) for Stokes interface flow." Journal of Computational Physics 247 (August 2013): 262–78. http://dx.doi.org/10.1016/j.jcp.2013.03.064.

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5

Chen, Gang, Jintao Cui, and Liwei Xu. "Analysis of a hybridizable discontinuous Galerkin method for the Maxwell operator." ESAIM: Mathematical Modelling and Numerical Analysis 53, no. 1 (January 2019): 301–24. http://dx.doi.org/10.1051/m2an/2019007.

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In this paper, we study a hybridizable discontinuous Galerkin (HDG) method for the Maxwell operator. The only global unknowns are defined on the inter-element boundaries, and the numerical solutions are obtained by using discontinuous polynomial approximations. The error analysis is based on a mixed curl-curl formulation for the Maxwell equations. Theoretical results are obtained under a more general regularity requirement. In particular for the low regularity case, special treatment is applied to approximate data on the boundary. The HDG method is shown to be stable and convergence in an optimal order for both high and low regularity cases. Numerical experiments with both smooth and singular analytical solutions are performed to verify the theoretical results.

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6

Moon, Minam, Hyung Kyu Jun, and Tay Suh. "Error Estimates on Hybridizable Discontinuous Galerkin Methods for Parabolic Equations with Nonlinear Coefficients." Advances in Mathematical Physics 2017 (2017): 1–11. http://dx.doi.org/10.1155/2017/9736818.

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HDG method has been widely used as an effective numerical technique to obtain physically relevant solutions for PDE. In a practical setting, PDE comes with nonlinear coefficients. Hence, it is inevitable to consider how to obtain an approximate solution for PDE with nonlinear coefficients. Research on using HDG method for PDE with nonlinear coefficients has been conducted along with results obtained from computer simulations. However, error analysis on HDG method for such settings has been limited. In this research, we give error estimations of the hybridizable discontinuous Galerkin (HDG) method for parabolic equations with nonlinear coefficients. We first review the classical HDG method and define notions that will be used throughout the paper. Then, we will give bounds for our estimates when nonlinear coefficients obey “Lipschitz” condition. We will then prove our main result that the errors for our estimations are bounded.

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7

Bastidas, Manuela, Bibiana Lopez-Rodríguez, and Mauricio Osorio. "A High-Order HDG Method with Dubiner Basis for Elliptic Flow Problems." Ingeniería y Ciencia 16, no. 32 (November 2020): 33–54. http://dx.doi.org/10.17230/ingciencia.16.32.2.

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We propose a standard hybridizable discontinuous Galerkin (HDG) method to solve a classic problem in fluids mechanics: Darcy’s law. This model describes the behavior of a fluid trough a porous medium and it is relevant to the flow patterns on the macro scale. Here we present the theoretical results of existence and uniqueness of the weak and discontinuous solution of the second order elliptic equation, as well as the predicted convergence order of the HDG method. We highlight the use and implementation of Dubiner polynomial basis functions that allow us to develop a general and efficient high order numerical approximation. We also show some numerical examples that validate the theoretical results.

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8

Cockburn, Bernardo. "The pursuit of a dream, Francisco Javier Sayas and the HDG methods." SeMA Journal 79, no. 1 (October 16, 2021): 37–56. http://dx.doi.org/10.1007/s40324-021-00273-y.

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AbstractFranciso 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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9

Feng, Xiaobing, Peipei Lu, and Xuejun Xu. "A Hybridizable Discontinuous Galerkin Method for the Time-Harmonic Maxwell Equations with High Wave Number." Computational Methods in Applied Mathematics 16, no. 3 (July 1, 2016): 429–45. http://dx.doi.org/10.1515/cmam-2016-0021.

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AbstractThis paper proposes and analyzes a hybridizable discontinuous Galerkin (HDG) method for the three-dimensional time-harmonic Maxwell equations coupled with the impedance boundary condition in the case of high wave number. It is proved that the HDG method is absolutely stable for all wave numbers ${\kappa>0}$ in the sense that no mesh constraint is required for the stability. A wave-number-explicit stability constant is also obtained. This is done by choosing a specific penalty parameter and using a PDE duality argument. Utilizing the stability estimate and a non-standard technique, the error estimates in both the energy-norm and the ${\mathbf{L}^{2}}$-norm are obtained for the HDG method. Numerical experiments are provided to validate the theoretical results and to gauge the performance of the proposed HDG method.

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10

Li, Binjie, Xiaoping Xie, and Shiquan Zhang. "Analysis of a Two-Level Algorithm for HDG Methods for Diffusion Problems." Communications in Computational Physics 19, no. 5 (May 2016): 1435–60. http://dx.doi.org/10.4208/cicp.scpde14.19s.

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AbstractThis paper analyzes an abstract two-level algorithm for hybridizable discontinuous Galerkin (HDG) methods in a unified fashion. We use an extended version of the Xu-Zikatanov (X-Z) identity to derive a sharp estimate of the convergence rate of the algorithm, and show that the theoretical results also are applied to weak Galerkin (WG) methods. The main features of our analysis are twofold: one is that we only need the minimal regularity of the model problem; the other is that we do not require the triangulations to be quasi-uniform. Numerical experiments are provided to confirm the theoretical results.

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11

Chen, Gang, and Xiaoping Xie. "A Robust Weak Galerkin Finite Element Method for Linear Elasticity with Strong Symmetric Stresses." Computational Methods in Applied Mathematics 16, no. 3 (July 1, 2016): 389–408. http://dx.doi.org/10.1515/cmam-2016-0012.

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AbstractThis paper proposes and analyzes a weak Galerkin (WG) finite element method with strong symmetric stresses for two- and three-dimensional linear elasticity problems on conforming or nonconforming polygon/polyhedral meshes. The WG method uses piecewise-polynomial approximations of degreesk(${\geq 1}$) for the stress,${k+1}$for the displacement, andkfor the displacement trace on the inter-element boundaries. It is shown to be equivalent to a hybridizable discontinuous Galerkin (HDG) finite element scheme. We show that the WG methods are robust in the sense that the derived a priori error estimates are optimal and uniform with respect to the Lamé constant λ. Numerical experiments confirm the theoretical results.

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12

Chen, Yanlai, Bo Dong, and Jiahua Jiang. "Optimally convergent hybridizable discontinuous Galerkin method for fifth-order Korteweg-de Vries type equations." ESAIM: Mathematical Modelling and Numerical Analysis 52, no. 6 (November 2018): 2283–306. http://dx.doi.org/10.1051/m2an/2018037.

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We develop and analyze the first hybridizable discontinuous Galerkin (HDG) method for solving fifth-order Korteweg-de Vries (KdV) type equations. We show that the semi-discrete scheme is stable with proper choices of the stabilization functions in the numerical traces. For the linearized fifth-order equations, we prove that the approximations to the exact solution and its four spatial derivatives as well as its time derivative all have optimal convergence rates. The numerical experiments, demonstrating optimal convergence rates for both the linear and nonlinear equations, validate our theoretical findings.

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13

Gong, Wei, Weiwei Hu, Mariano Mateos, John R. Singler, and Yangwen Zhang. "Analysis of a hybridizable discontinuous Galerkin scheme for the tangential control of the Stokes system." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 6 (November 2020): 2229–64. http://dx.doi.org/10.1051/m2an/2020015.

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We consider an unconstrained tangential Dirichlet boundary control problem for the Stokes equations with an L2 penalty on the boundary control. The contribution of this paper is twofold. First, we obtain well-posedness and regularity results for the tangential Dirichlet control problem on a convex polygonal domain. The analysis contains new features not found in similar Dirichlet control problems for the Poisson equation; an interesting result is that the optimal control has higher local regularity on the individual edges of the domain compared to the global regularity on the entire boundary. Second, we propose and analyze a hybridizable discontinuous Galerkin (HDG) method to approximate the solution. For convex polygonal domains, our theoretical convergence rate for the control is optimal with respect to the global regularity on the entire boundary. We present numerical experiments to demonstrate the performance of the HDG method.

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14

Chen, Jia-Fen, Xian-Ming Gu, Liang Li, and Ping Zhou. "An Optimized Schwarz Method for the Optical Response Model Discretized by HDG Method." Entropy 25, no. 4 (April 19, 2023): 693. http://dx.doi.org/10.3390/e25040693.

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An optimized Schwarz domain decomposition method (DDM) for solving the local optical response model (LORM) is proposed in this paper. We introduce a hybridizable discontinuous Galerkin (HDG) scheme for the discretization of such a model problem based on a triangular mesh of the computational domain. The discretized linear system of the HDG method on each subdomain is solved by a sparse direct solver. The solution of the interface linear system in the domain decomposition framework is accelerated by a Krylov subspace method. We study the spectral radius of the iteration matrix of the Schwarz method for the LORM problems, and thus propose an optimized parameter for the transmission condition, which is different from that for the classical electromagnetic problems. The numerical results show that the proposed method is effective.

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15

Xiong, Chunguang, Fusheng Luo, and Xiuling Ma. "Uniform in time error analysis of HDG approximation for Schrödinger equation based on HDG projection." ESAIM: Mathematical Modelling and Numerical Analysis 52, no. 2 (March 2018): 751–72. http://dx.doi.org/10.1051/m2an/2017058.

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This paper presents error analysis of hybridizable discontinuous Galerkin (HDG) time-domain method for solving time dependent Schrödinger equations. The numerical trace and numerical flux are constructed to preserve the conservative property for the density of the particle described. We prove that there exist the superconvergence properties of the HDG method, which do hold for second-order elliptic problems, uniformly in time for the semidiscretization by the same method of Schrödinger equations provided that enough regularity is satisfied. Thus, if the approximations are piecewise polynomials of degree r, the approximations to the wave function and the flux converge with order r + 1. The suitably chosen projection of the wave function into a space of lower polynomial degree superconverges with order r + 2 for r ≥ 1 uniformly in time. The application of element-by-element postprocessing of the approximate solution which provides an approximation of the potential convergence with order r + 2 for r ≥ 1 in L2 is also uniformly in time.

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16

Marche, Fabien. "Combined Hybridizable Discontinuous Galerkin (HDG) and Runge-Kutta Discontinuous Galerkin (RK-DG) formulations for Green-Naghdi equations on unstructured meshes." Journal of Computational Physics 418 (October 2020): 109637. http://dx.doi.org/10.1016/j.jcp.2020.109637.

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17

Araya, Rodolfo, Manuel Solano, and Patrick Vega. "Analysis of an adaptive HDG method for the Brinkman problem." IMA Journal of Numerical Analysis 39, no. 3 (June 7, 2018): 1502–28. http://dx.doi.org/10.1093/imanum/dry031.

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Abstract We introduce and analyze a hybridizable discontinuous Galerkin method for the gradient-velocity-pressure formulation of the Brinkman problem. We present an a priori error analysis of the method, showing optimal order of convergence of the error. We also introduce an a posteriori error estimator, of the residual type, which helps us to improve the quality of the numerical solution. We establish reliability and local efficiency of our estimator for the $L^{2} $-error of the velocity gradient and the pressure and the $ H^{1} $-error of the velocity, with constants written explicitly in terms of the physical parameters and independent of the size of the mesh. In particular, our results are also valid for the Stokes problem. Finally, we provide numerical experiments showing the quality of our adaptive scheme.

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18

Qiu, Weifeng, and Ke Shi. "A mixed DG method and an HDG method for incompressible magnetohydrodynamics." IMA Journal of Numerical Analysis 40, no. 2 (January 15, 2019): 1356–89. http://dx.doi.org/10.1093/imanum/dry095.

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Abstract In this paper we propose and analyze a mixed discontinuous Galerkin (DG) method and an hybridizable DG (HDG) method for the stationary magnetohydrodynamics (MHD) equations with two types of boundary (or constraint) conditions. The mixed DG method is based on a recent work proposed by Houston et al. (2009, A mixed DG method for linearized incompressible magnetohydrodynamics. J. Sci. Comput., 40, 281–314) for the linearized MHD. With two novel discrete Sobolev embedding type estimates for the discontinuous polynomials, we provide a priori error estimates for the method on the nonlinear MHD equations. In the smooth case we have optimal convergence rate for the velocity, magnetic field and pressure in the energy norm; the Lagrange multiplier only has suboptimal convergence order. With the minimal regularity assumption on the exact solution, the approximation is optimal for all unknowns. To the best of our knowledge, these are the first a priori error estimates for DG methods for the nonlinear MHD equations. In addition, we also propose and analyze the first divergence-free HDG method for the problem with several unique features comparing with the mixed DG method.

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19

Gürkan, Ceren, Esther Sala-Lardies, Martin Kronbichler, and Sonia Fernández-Méndez. "eXtended Hybridizable Discontinous Galerkin (X-HDG) for Void Problems." Journal of Scientific Computing 66, no. 3 (July 6, 2015): 1313–33. http://dx.doi.org/10.1007/s10915-015-0066-8.

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20

McLachlan, Robert I., and Ari Stern. "Multisymplecticity of Hybridizable Discontinuous Galerkin Methods." Foundations of Computational Mathematics 20, no. 1 (April 22, 2019): 35–69. http://dx.doi.org/10.1007/s10208-019-09415-1.

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21

Celiker, Fatih, Bernardo Cockburn, and Ke Shi. "Hybridizable Discontinuous Galerkin Methods for Timoshenko Beams." Journal of Scientific Computing 44, no. 1 (March 7, 2010): 1–37. http://dx.doi.org/10.1007/s10915-010-9357-2.

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22

Huang, Jianguo, and Xuehai Huang. "A Hybridizable Discontinuous Galerkin Method for Kirchhoff Plates." Journal of Scientific Computing 78, no. 1 (July 7, 2018): 290–320. http://dx.doi.org/10.1007/s10915-018-0780-0.

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23

Cockburn, Bernardo, Ricardo H. Nochetto, and Wujun Zhang. "Contraction property of adaptive hybridizable discontinuous Galerkin methods." Mathematics of Computation 85, no. 299 (August 17, 2015): 1113–41. http://dx.doi.org/10.1090/mcom/3014.

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24

Nguyen, N. C., J. Peraire, and B. Cockburn. "A hybridizable discontinuous Galerkin method for Stokes flow." Computer Methods in Applied Mechanics and Engineering 199, no. 9-12 (January 2010): 582–97. http://dx.doi.org/10.1016/j.cma.2009.10.007.

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25

Soon, S. C., B. Cockburn, and Henryk K. Stolarski. "A hybridizable discontinuous Galerkin method for linear elasticity." International Journal for Numerical Methods in Engineering 80, no. 8 (June 17, 2009): 1058–92. http://dx.doi.org/10.1002/nme.2646.

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26

Cockburn, Bernardo, and Jiguang Shen. "A Hybridizable Discontinuous Galerkin Method for the $p$-Laplacian." SIAM Journal on Scientific Computing 38, no. 1 (January 2016): A545—A566. http://dx.doi.org/10.1137/15m1008014.

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27

Cockburn, Bernardo, and Kassem Mustapha. "A hybridizable discontinuous Galerkin method for fractional diffusion problems." Numerische Mathematik 130, no. 2 (October 9, 2014): 293–314. http://dx.doi.org/10.1007/s00211-014-0661-x.

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28

Kabaria, Hardik, Adrian J. Lew, and Bernardo Cockburn. "A hybridizable discontinuous Galerkin formulation for non-linear elasticity." Computer Methods in Applied Mechanics and Engineering 283 (January 2015): 303–29. http://dx.doi.org/10.1016/j.cma.2014.08.012.

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29

Kauffman, Justin A., Jason P. Sheldon, and Scott T. Miller. "Overset meshing coupled with hybridizable discontinuous Galerkin finite elements." International Journal for Numerical Methods in Engineering 112, no. 5 (March 1, 2017): 403–33. http://dx.doi.org/10.1002/nme.5512.

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30

Hoermann, Julia M., Cristóbal Bertoglio, Martin Kronbichler, Martin R. Pfaller, Radomir Chabiniok, and Wolfgang A. Wall. "An adaptive hybridizable discontinuous Galerkin approach for cardiac electrophysiology." International Journal for Numerical Methods in Biomedical Engineering 34, no. 5 (February 12, 2018): e2959. http://dx.doi.org/10.1002/cnm.2959.

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31

Zhang, Xiao, Xiaoping Xie, and Shiquan Zhang. "An Optimal Embedded Discontinuous Galerkin Method for Second-Order Elliptic Problems." Computational Methods in Applied Mathematics 19, no. 4 (October 1, 2019): 849–61. http://dx.doi.org/10.1515/cmam-2018-0007.

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AbstractThe embedded discontinuous Galerkin (EDG) method by Cockburn, Gopalakrishnan and Lazarov [B. Cockburn, J. Gopalakrishnan and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second-order elliptic problems, SIAM J. Numer. Anal. 47 2009, 2, 1319–1365] is obtained from the hybridizable discontinuous Galerkin method by changing the space of the Lagrangian multiplier from discontinuous functions to continuous ones, and adopts piecewise polynomials of equal degrees on simplex meshes for all variables. In this paper, we analyze a new EDG method for second-order elliptic problems on polygonal/polyhedral meshes. By using piecewise polynomials of degrees {k+1}, {k+1}, k ({k\geq 0}) to approximate the potential, numerical trace and flux, respectively, the new method is shown to yield optimal convergence rates for both the potential and flux approximations. Numerical experiments are provided to confirm the theoretical results.

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32

Fabien, Maurice S. "A GPU-Accelerated Hybridizable Discontinuous Galerkin Method for Linear Elasticity." Communications in Computational Physics 27, no. 2 (June 2020): 513–45. http://dx.doi.org/10.4208/cicp.oa-2018-0235.

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33

La Spina, Andrea, and Jacob Fish. "A superconvergent hybridizable discontinuous Galerkin method for weakly compressible magnetohydrodynamics." Computer Methods in Applied Mechanics and Engineering 388 (January 2022): 114278. http://dx.doi.org/10.1016/j.cma.2021.114278.

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34

Gander, Martin J., and Soheil Hajian. "Analysis of Schwarz Methods for a Hybridizable Discontinuous Galerkin Discretization." SIAM Journal on Numerical Analysis 53, no. 1 (January 2015): 573–97. http://dx.doi.org/10.1137/140961857.

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35

Cockburn, Bernardo, Daniele A. Di Pietro, and Alexandre Ern. "Bridging the hybrid high-order and hybridizable discontinuous Galerkin methods." ESAIM: Mathematical Modelling and Numerical Analysis 50, no. 3 (May 2016): 635–50. http://dx.doi.org/10.1051/m2an/2015051.

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36

Sheldon, Jason P., Scott T. Miller, and Jonathan S. Pitt. "A hybridizable discontinuous Galerkin method for modeling fluid–structure interaction." Journal of Computational Physics 326 (December 2016): 91–114. http://dx.doi.org/10.1016/j.jcp.2016.08.037.

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37

Cockburn, Bernardo, Bo Dong, and Johnny Guzmán. "A Hybridizable and Superconvergent Discontinuous Galerkin Method for Biharmonic Problems." Journal of Scientific Computing 40, no. 1-3 (February 24, 2009): 141–87. http://dx.doi.org/10.1007/s10915-009-9279-z.

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38

Cockburn, Bernardo, and Jayadeep Gopalakrishnan. "The Derivation of Hybridizable Discontinuous Galerkin Methods for Stokes Flow." SIAM Journal on Numerical Analysis 47, no. 2 (January 2009): 1092–125. http://dx.doi.org/10.1137/080726653.

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39

Nguyen, N. C., J. Peraire, and B. Cockburn. "Hybridizable discontinuous Galerkin methods for the time-harmonic Maxwell’s equations." Journal of Computational Physics 230, no. 19 (August 2011): 7151–75. http://dx.doi.org/10.1016/j.jcp.2011.05.018.

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40

Sánchez-Vizuet, Tonatiuh, and Manuel E. Solano. "A Hybridizable Discontinuous Galerkin solver for the Grad–Shafranov equation." Computer Physics Communications 235 (February 2019): 120–32. http://dx.doi.org/10.1016/j.cpc.2018.09.013.

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41

Cesmelioglu, Aycil, Jeonghun J. Lee, and Sander Rhebergen. "Hybridizable discontinuous Galerkin methods for the coupled Stokes–Biot problem." Computers & Mathematics with Applications 144 (August 2023): 12–33. http://dx.doi.org/10.1016/j.camwa.2023.05.024.

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42

Yadav, Sangita, and Amiya K. Pani. "Superconvergent discontinuous Galerkin methods for nonlinear parabolic initial and boundary value problems." Journal of Numerical Mathematics 27, no. 3 (September 25, 2019): 183–202. http://dx.doi.org/10.1515/jnma-2018-0035.

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Abstract In this article, we discuss error estimates for nonlinear parabolic problems using discontinuous Galerkin methods which include HDG method in the spatial direction while keeping time variable continuous. When piecewise polynomials of degree k ⩾ 1 are used to approximate both the potential as well as the flux, it is shown that the error estimate for the semi-discrete flux in L∞(0, T; L2)-norm is of order k + 1. With the help of a suitable post-processing of the semi-discrete potential, it is proved that the resulting post-processed potential converges with order of convergence $\begin{array}{} \displaystyle O\big(\!\sqrt{{}\log(T/h^2)}\,h^{k+2}\big) \end{array}$ in L∞(0, T; L2)-norm. These results extend the HDG analysis of Chabaud and Cockburn [Math. Comp. 81 (2012), 107–129] for the heat equation to non-linear parabolic problems.

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43

Jaust, Alexander, Balthasar Reuter, Vadym Aizinger, Jochen Schütz, and Peter Knabner. "FESTUNG: A MATLAB/GNU Octave toolbox for the discontinuous Galerkin method. Part III: Hybridized discontinuous Galerkin (HDG) formulation." Computers & Mathematics with Applications 75, no. 12 (June 2018): 4505–33. http://dx.doi.org/10.1016/j.camwa.2018.03.045.

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44

Kang, Shinhoo, Francis X. Giraldo, and Tan Bui-Thanh. "IMEX HDG-DG: A coupled implicit hybridized discontinuous Galerkin and explicit discontinuous Galerkin approach for shallow water systems." Journal of Computational Physics 401 (January 2020): 109010. http://dx.doi.org/10.1016/j.jcp.2019.109010.

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45

Terrana, S., J. P. Vilotte, and L. Guillot. "A spectral hybridizable discontinuous Galerkin method for elastic–acoustic wave propagation." Geophysical Journal International 213, no. 1 (December 22, 2017): 574–602. http://dx.doi.org/10.1093/gji/ggx557.

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46

Cockburn, Bernardo, Johnny Guzmán, and Francisco-Javier Sayas. "Coupling of Raviart--Thomas and Hybridizable Discontinuous Galerkin Methods with BEM." SIAM Journal on Numerical Analysis 50, no. 5 (January 2012): 2778–801. http://dx.doi.org/10.1137/100818339.

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47

MOON, MINAM, and YANG HWAN LIM. "SUPERCONVERGENCE OF HYBRIDIZABLE DISCONTINUOUS GALERKIN METHOD FOR SECOND-ORDER ELLIPTIC EQUATIONS." Journal of the Korea Society for Industrial and Applied Mathematics 20, no. 4 (December 25, 2016): 295–308. http://dx.doi.org/10.12941/jksiam.2016.20.295.

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Cockburn, Bernardo, and Jiguang Shen. "An algorithm for stabilizing hybridizable discontinuous Galerkin methods for nonlinear elasticity." Results in Applied Mathematics 1 (June 2019): 100001. http://dx.doi.org/10.1016/j.rinam.2019.01.001.

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Gürkan, Ceren, Martin Kronbichler, and Sonia Fernández-Méndez. "eXtended Hybridizable Discontinuous Galerkin with Heaviside Enrichment for Heat Bimaterial Problems." Journal of Scientific Computing 72, no. 2 (January 28, 2017): 542–67. http://dx.doi.org/10.1007/s10915-017-0370-6.

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Stanglmeier, M., N. C. Nguyen, J. Peraire, and B. Cockburn. "An explicit hybridizable discontinuous Galerkin method for the acoustic wave equation." Computer Methods in Applied Mechanics and Engineering 300 (March 2016): 748–69. http://dx.doi.org/10.1016/j.cma.2015.12.003.

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