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
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
Shukai Du, Francisco-Javier Sayas
Summary: This study proposes a simple way of constructing HDG+ projections on polyhedral elements, enabling concise analysis of Lehrenfeld-Schoberl HDG methods and the reuse of analysis techniques from standard HDG methods. The novelty lies in an alternative method of constructing the projections without using M-decompositions. This extends previous results in elliptic problems and elasticity to polyhedral meshes.
MATHEMATICS OF COMPUTATION
(2021)
Article
Mathematics, Applied
Xiaobin Liu, Dazhi Zhang, Xiong Meng, Boying Wu
Summary: This paper investigates the superconvergence properties of the local discontinuous Galerkin methods with generalized alternating fluxes for one-dimensional linear convection-diffusion equations. By constructing special correction functions, the (2k + 1)th order superconvergence for cell averages and numerical traces in the discrete L-2 norm is proven. Additionally, superconvergence of orders k + 2 and k + 1 is obtained for errors and derivatives at generalized Radau points, all of which are confirmed by numerical experiments.
SCIENCE CHINA-MATHEMATICS
(2021)
Article
Mathematics, Applied
Dongwook Shin, Youngmok Jeon, Eun-Jae Park
Summary: This article introduces and analyzes arbitrary-order, locally conservative hybrid discontinuous Galkerin methods for linearized Navier-Stokes equations. The study proves the injectivity of the lifting operator associated with trace variables for any polynomial degree, and obtains optimal error estimates in the energy norm by introducing nonstandard projection operators for the hybrid DG method. The numerical results presented in the article validate the theory and demonstrate the performance of the algorithm.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Florian Kummer, Jens Weber, Martin Smuda
Summary: The software package BoSSS discretizes partial differential equations with discontinuous coefficients and/or time-dependent domains using an eXtended Discontinuous Galerkin (XDG) method. This work introduces the XDG method, develops a formal notation capturing important numerical details, and presents iterative solvers for extended DG systems.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Computer Science, Interdisciplinary Applications
Hendrik Ranocha
Summary: Nishikawa (2007) proposed a reformulation of the classical Poisson equation as a steady state problem for a linear hyperbolic system, which provides optimal error estimates for the solution of the elliptic equation and its gradient. However, it hinders the use of well-known solvers for elliptic problems. We establish connections to a discontinuous Galerkin (DG) method studied by Cockburn, Guzman, and Wang (2009) that is generally difficult to implement. Additionally, we demonstrate the efficient implementation of this method using summation by parts (SBP) operators, particularly in the context of SBP DG methods like the DG spectral element method (DGSEM). The resulting scheme combines desirable properties from both the hyperbolic and the elliptic perspective, offering a higher order of convergence for the gradients than what is typically expected from DG methods for elliptic problems.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Yingxia Xi, Xia Ji
Summary: This paper presents a new proof of the C(0)IPG method for the biharmonic eigenvalue problem. Instead of following the proof structure of the discontinuous Galerkin method, the problem is rewritten as the eigenvalue problem of a holomorphic Fredholm operator function. The convergence of C(0)IPG is proved using the abstract approximation theory for holomorphic operator functions. The spectral indicator method is employed for easy computation of eigenvalues, and numerical examples are provided for validation.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Engineering, Multidisciplinary
Tale Bakken Ulfsby, Andre Massing, Simon Sticko
Summary: We propose a novel cut discontinuous Galerkin (CutDG) method for solving stationary advection-reaction problems on surfaces embedded in Rd. The approach involves embedding the surface into a full-dimensional background mesh and using discontinuous piecewise polynomials as test and trial functions. By introducing a suitable stabilization technique, we are able to establish inf-sup stability, a priori error estimates, and condition number estimates using an augmented streamline-diffusion norm. Numerical examples validate our theoretical findings.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Mathematics, Applied
Minqiang Xu, Chungang Shi
Summary: In this paper, a new finite difference method for biharmonic equations is studied by combining Hessian recovery techniques and the ghost points method. Numerical results validate the optimal convergence orders of the proposed method in the L2 norm and H1 seminorm. Moreover, superconvergence properties of the recovered gradient and Hessian have been observed.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Mathematics, Applied
Simon Becher, Gunar Matthies
Summary: This paper presents a unified analysis for a family of variational time discretization methods applied to non-stiff initial value problems. The analysis includes discontinuous Galerkin methods and continuous Galerkin-Petrov methods, with a focus on global error and superconvergence properties under weak abstract assumptions. Numerical experiments support the theoretical results.
NUMERICAL ALGORITHMS
(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
Mathematics, Applied
Peng Zhu, Shenglan Xie
Summary: This paper presents a stabilizer-free weak Galerkin finite element scheme for non-self adjoint and indefinite elliptic equations, deriving supercloseness convergence of the method. After suitable postprocessing, the resulting solution converges with higher order than optimal in both H-1 semi-norm and L-2 norm on triangular meshes. Theoretical findings are verified through numerical experiments.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Hecong Gao, Hui Liang
Summary: This paper investigates the integral-algebraic equation (IAE) of index 1 and proposes a discontinuous Galerkin (DG) method to solve it. The numerical accuracy is improved by using the iterated DG method, and the global superconvergence of the iterated DG solution is derived. However, there is no local superconvergence for the mixed IAE system of first-kind and second-kind VIEs due to the lack of local superconvergence of the DG residual for first-kind VIEs.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
(2023)
Article
Mathematics, Applied
Zhiqiang Cai, Jing Yang
Summary: In this paper, a class of discontinuous Galerkin finite element methods for advection-diffusion-reaction problems is presented, and a priori error estimates are established when the solution is only in H1+s (Omega) space with s is an element of(0, 1/2].
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Bernardo Cockburn, Guosheng Fu
IMA JOURNAL OF NUMERICAL ANALYSIS
(2018)
Article
Mathematics, Applied
Bernardo Cockburn, John R. Singler, Yangwen Zhang
JOURNAL OF SCIENTIFIC COMPUTING
(2019)
Article
Mathematics, Applied
Gang Chen, Bernardo Cockburn, John Singler, Yangwen Zhang
JOURNAL OF SCIENTIFIC COMPUTING
(2019)
Article
Mathematics, Applied
Marta Benitez, Bernardo Cockburn
Summary: This paper analyzes a technique to improve the spatial accuracy of pure Lagrange-Galerkin (PLG) methods applied to convection-diffusion equations with time-dependent domains. The study shows that applying a local post-processing to the numerical solution of PLG schemes significantly enhances the spatial accuracy. Numerical tests confirm the theoretical results.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2022)
Article
Mathematics, Applied
Bernardo Cockburn, Shiqiang Xia
Summary: This study presents the first a priori error analysis of a new method for computing adjoint-based, super-convergent Galerkin approximations of linear functionals. Compared to existing methods, this new approach provides faster and more computationally efficient approximations, which have been validated through numerical experiments.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2022)
Article
Engineering, Multidisciplinary
Manuel A. Sanchez, Bernardo Cockburn, Ngoc-Cuong Nguyen, Jaime Peraire
Summary: This paper presents a class of high-order finite element methods that conserve linear and angular momenta as well as energy for equations of linear elastodynamics by exploiting and preserving the Hamiltonian structure. Experimental results confirm optimal convergence and conservation properties of these methods.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Engineering, Multidisciplinary
Stein K. F. Stoter, Bernardo Cockburn, Thomas J. R. Hughes, Dominik Schillinger
Summary: This article presents a theoretical framework for integrating discontinuous Galerkin methods in the variational multiscale paradigm, showing that existing formulations can be derived from a specific choice of multiscale projector. The fine-scale closure functions corresponding to discontinuous Galerkin methods exhibit more compact support and smaller amplitudes compared to classical finite element methods, providing a new perspective on the natural stability of discontinuous Galerkin methods for hyperbolic problems.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Computer Science, Interdisciplinary Applications
Bernardo Cockburn, Shiqiang Xia
Summary: The paper introduces a new method for computing high-order accurate approximations of eigenvalues defined in terms of Galerkin approximations, showing faster convergence rates compared to traditional finite element methods. The method is demonstrated on a second-order elliptic eigenvalue problem, with numerical results indicating superior performance in terms of convergence speed.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Bernardo Cockburn, Shiqiang Xia
Summary: This paper presents a novel, fully computable error approximation and mesh adaptation approach for functionals defined by second-order elliptic equations. The method approximates the functionals using the hybridizable discontinuous Galerkin method and obtains error approximation through adjoint-based method and a local post-processing technique. Compared to existing adjoint-based error estimations, this method does not require an auxiliary finer mesh or higher order approximation spaces, reducing computational cost and easing implementation. The local post-processing technique can be carried out in parallel, further improving the speed of the method.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Engineering, Multidisciplinary
Manuel A. Sanchez, Shukai Du, Bernardo Cockburn, Ngoc-Cuong Nguyen, Jaime Peraire
Summary: In this paper, several high-order accurate finite element methods for the Maxwell's equations are presented, which provide time-invariant, non-drifting approximations to the total electric and magnetic charges, and to the total energy. These methods are devised by taking advantage of the Hamiltonian structures of the Maxwell's equations and using spatial and temporal discretization techniques to ensure the conservation properties and convergence of the methods.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Mathematics, Applied
Bernardo Cockburn, Shukai Du, Manuel A. Sanchez
Summary: This paper presents a new class of discontinuous Galerkin methods for space discretization of the time-dependent Maxwell equations. The main feature of these methods is the use of time derivatives and/or integrals in the stabilization part of their numerical traces. It is shown that these methods conserve a discrete version of the energy, and fully discrete schemes also conserve the discrete energy when using the mid-point rule to march in time. The paper also proposes a three-step technique to devise fully discrete schemes of arbitrary order of accuracy which conserve the energy in time.
COMPUTATIONAL METHODS IN APPLIED MATHEMATICS
(2022)
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
Gang Chen, Bernardo Cockburn, John R. Singler, Yangwen Zhang
Summary: The research focuses on a superconvergent hybridizable discontinuous Galerkin (HDG) method defined on simplicial meshes for scalar reaction-diffusion equations, utilizing an interpolatory version to maintain convergence properties and reduce computational costs. By establishing a link between HDG and hybrid high-order (HHO) methods, the study extends the idea to HHO-inspired HDG methods on meshes consisting of general polyhedral elements. It is proven that for shape-regular polyhedral element meshes equivalent to a finite number of reference elements, the resulting interpolatory HDG methods converge at the same rate as for linear elliptic problems, thus retaining superconvergence properties. Numerical results are presented to illustrate the convergence theory.
COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics, Applied
Bernardo Cockburn, Jiguang Shen
RESULTS IN APPLIED MATHEMATICS
(2019)
Article
Mathematics, Applied
Bernardo Cockburn, Guosheng Fo, Weifeng Qiu
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2018)