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Engineering, Multidisciplinary
Andrea La Spina, Jacob Fish
Summary: The work proposes a hybridizable discontinuous Galerkin (HDG) method for weakly compressible magnetohydrodynamic (MHD) problems, demonstrating its superior properties and superconvergence characteristics. Different MHD formulations are discussed, and the convergence properties of the proposed methods under various conditions are extensively examined through numerical examples.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Mathematics, Applied
Xiaobin Liu, Dazhi Zhang, Xiong Meng, Boying Wu
Summary: This paper investigates the superconvergence properties of the local discontinuous Galerkin methods for solving nonlinear convection-diffusion equations in one-dimensional space. The study introduces a new projection and proves superconvergence of order (2k + 1) for cell averages and numerical flux based on elaborate error estimates. Additionally, improvements in convergence orders for errors at Radau points are discussed, and the theoretical findings are confirmed through numerical experiments.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Computer Science, Interdisciplinary Applications
Carlos A. Pereira, Brian C. Vermeire
Summary: This paper presents a unified framework for hybridizing flux reconstruction (FR) schemes and analyzes the performance and accuracy of hybridized flux reconstruction (HFR) and interior-embedded flux reconstruction (EFR) methods on quadrilateral elements. It is found that EFR methods generally introduce additional numerical error but behave closer to FR methods for higher orders. The choice of correction function has an impact on the performance of the implicit solver, with EFR methods showing potential for higher accuracy and reduced computational cost.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Gautam Singh, Srinivasan Natesan
Summary: The parabolic convection-diffusion-reaction problem is discretized using the NIPG method in space and the DG method in time. Piecewise Lagrange interpolation at Gauss points is used to improve the order of convergence, and the error bound in the discrete energy norm is estimated. The study demonstrates superconvergence properties of the DG method with (k+1)-order convergence in space and (l+1)-order convergence in time, with numerical results confirming the theoretical findings.
NUMERICAL ALGORITHMS
(2022)
Article
Computer Science, Interdisciplinary Applications
Ruben Sevilla
Summary: The hybridisable discontinuous Galerkin (HDG) method, proposed by Cockburn and co-workers, is popular for reducing the global number of coupled degrees of freedom compared to other DG methods. This work introduces a dual time stepping (DTS) approach to solve the global system of equations in the HDG formulation of convection-diffusion problems, presenting a proof of the existence and uniqueness of the steady state solution. The stability limit and optimal choice for the dual time step are derived, and different time marching approaches are compared for convection-dominated problems.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Stephen Metcalfe, Siva Nadarajah
Summary: In this work, a new quasi-optimal test norm for discontinuous Petrov-Galerkin (DPG) discretization of the ultra-weak formulation of the convection-diffusion equation is proposed. The theoretical analysis shows that the proposed test norm leads to favorable scalings between the target norm and the energy norm. Numerical experiments are conducted to confirm the theoretical results.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Bernardo Cockburn
Summary: In the framework of steady-state diffusion problems, this paper describes the development history of hybridizable discontinuous Galerkin (HDG) methods since their introduction in 2009. It explains the parallel development of hybridized mixed (HM) methods and how the introduction of M-decompositions led to the creation of superconvergent HM and HDG methods for elements of general shapes. It also reveals a new connection between HM and HDG methods, stating that any HM method can be rewritten as an HDG method by transforming a subspace of the approximate fluxes of the HM method into a stabilization function. The paper concludes by presenting several open problems resulting from this discovery.
JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Limin Ma
Summary: In this paper, a unified analysis of superconvergence property for a large class of mixed discontinuous Galerkin methods is presented. This analysis is applicable to the Poisson equation and linear elasticity problems with symmetric stress formulations. Numerical experiments validate the effectiveness of locally postprocess schemes in improving displacement accuracy.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Jingjun Zhao, Wenjiao Zhao, Yang Xu
Summary: This work focuses on the numerical solution of initial and boundary value problems for space-time fractional advection-diffusion equations. The well-posedness of weak solutions is proven using the Lax-Milgram lemma. Two fully discrete methods are established, combining a hybridizable discontinuous Galerkin approach in the spatial direction and two finite difference schemes in the temporal direction: L1 formula and the weighted and shifted Grunwald-Letnikov formula. The stability and convergence analyses of these methods are derived in detail, and several numerical experiments are provided to illustrate the theoretical results.
APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Mathematics, Applied
A. A. Sivas, B. S. Southworth, S. Rhebergen
Summary: This paper investigates the efficiency, robustness, and scalability of AIR algebraic multigrid as a preconditioner in the all-at-once solution of a space-time hybridizable discontinuous Galerkin discretization of advection-dominated flows. Numerical examples demonstrate the effectiveness of AIR as a preconditioner for advection-diffusion problems on fixed and time-dependent domains, using both slab-by-slab and all-at-once space-time discretizations. The geometric coarsening structure in AIR explains why it can provide robust, scalable, space-time convergence on advective and hyperbolic problems.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Review
Computer Science, Interdisciplinary Applications
Matteo Giacomini, Ruben Sevilla, Antonio Huerta
Summary: This paper introduces HDGlab, an open source MATLAB implementation of the hybridisable discontinuous Galerkin (HDG) method, with the goal of providing a detailed description of the HDG method and its implementation in HDGlab. HDGlab offers unique features not found in other implementations, such as high-order polynomial shape functions and support for curved isoparametric simplicial elements, making it a valuable tool for the computational engineering community.
ARCHIVES OF COMPUTATIONAL METHODS IN ENGINEERING
(2021)
Article
Mathematics, Applied
Aycil Cesmelioglu, Jeonghun J. Lee, Sander Rhebergen
Summary: We introduce and analyze a hybridizable discontinuous Galerkin finite element method for the coupled Stokes-Biot problem. The method has the property that the discrete velocities and displacements satisfy the compressibility equations pointwise on the elements. We prove well-posedness of the discretization and provide a priori error estimates that demonstrate the method is free of volumetric locking. Numerical examples further demonstrate optimal rates of convergence for all unknowns and locking-free discretization.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Hamid Safdari, Majid Rajabzadeh, Moein Khalighi
Summary: A local discontinuous Galerkin method is proposed for solving a nonlinear convection-diffusion equation with a fractional diffusion, nonlinear diffusion, and nonlinear convection term, achieving higher accuracy using Spline interpolations. Compared to the direct Galerkin method, this proposed method is demonstrated to be suitable for general fractional convection-diffusion problems, significantly improving stability and providing a convergence order of O(h(k+1)), where k represents the degree of polynomials.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Engineering, Multidisciplinary
Helene Barucq, Julien Diaz, Rose-Cloe Meyer, Ha Pham
Summary: The study utilizes the HDG method to numerically solve the two-dimensional anisotropic poroelastic wave equations, inheriting the advantages of the discontinuous Galerkin method without drastic increase in degrees of freedom. Through comparisons with analytical solutions and sensitivity analysis of stabilization parameters, the method's accuracy and practicality are confirmed.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
(2021)
Article
Mathematics, Applied
Y. A. O. Cheng, S. H. A. N. Jiang, M. A. R. T. I. N. Stynes
Summary: This paper numerically solves a convection-diffusion problem on the unit square in Double-struck capital R2 with exponential boundary layers using the local discontinuous Galerkin (LDG) method. They establish the superconvergence property of the LDG solution on three types of layer-adapted meshes, which leads to an optimal bound for the L2 error. Numerical experiments confirm their theoretical results.
MATHEMATICS OF COMPUTATION
(2023)
Article
Mathematics, Applied
Guosheng Fu, Weifeng Qiu, Wujun Zhang
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(2016)
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Bernardo Cockburn, Guosheng Fu, Weifeng Qiu
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Weifeng Qiu, Ke Shi
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Weifeng Qiu, Minglei Wang, Jiahao Zhang
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Huangxin Chen, Weifeng Qiu, Ke Shi, Manuel Solano
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Weifeng Qiu, Manuel Solano, Patrick Vega
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Peipei Lu, Huangxin Chen, Weifeng Qiu
MATHEMATICS OF COMPUTATION
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Aycil Cesmelioglu, Bernardo Cockburn, Weifeng Qiu
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(2017)
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Huangxin Chen, Weifeng Qiu
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Weifeng Qiu, Jiguang Shen, Ke Shi
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(2018)
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Eric T. Chung, Weifeng Qiu
SIAM JOURNAL ON NUMERICAL ANALYSIS
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Huangxin Chen, Weifeng Qiu, Ke Shi
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IMA JOURNAL OF NUMERICAL ANALYSIS
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