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
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
Computer Science, Interdisciplinary Applications
Samuel Olivier, Will Pazner, Terry S. Haut, Ben C. Yee
Summary: We propose a family of discretization methods for the Variable Eddington Factor (VEF) equations that have high-order accuracy and efficient preconditioned iterative solvers. These methods are combined with the Discontinuous Galerkin transport discretization to form effective high-order, linear transport methods. The VEF discretizations are derived by extending the unified analysis of Discontinuous Galerkin methods for elliptic problems to the VEF equations. Numerical results demonstrate the accuracy, preservation of diffusion limit, and effectiveness of the VEF discretizations on curved meshes.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Jan Nordstrom, Andrew R. Winters
Summary: This study proves the stability of the most common filtering procedure for nodal discontinuous Galerkin methods, utilizing polynomial basis functions and accurate quadrature methods. Theoretical discussions recontextualize stable filtering results from finite difference methods to the DG framework, with numerical tests verifying the theoretical findings.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Jeonghun J. Lee, Omar Ghattas
Summary: In this paper, we propose a variant of enriched Galerkin methods for second order elliptic equations with over-penalization of interior jump terms and prove that optimal a priori error estimates with the standard discrete energy norm can be obtained by combining a priori and a posteriori error analysis techniques. We also show that the interior over-penalization is advantageous for constructing preconditioners robust to mesh refinement by analyzing the spectral equivalence of bilinear forms.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Zhaonan Dong, Emmanuil H. Georgoulis
Summary: A new variant of the IPDG method, called robust IPDG (RIPDG), is proposed, which involves weighted averages of the gradient of the approximate solution to enhance its robustness. Numerical experiments show that the RIPDG method performs better than the standard IPDG method in terms of error behavior and conditioning in scenarios with strong local variation.
JOURNAL OF SCIENTIFIC COMPUTING
(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
Computer Science, Interdisciplinary Applications
A. C. W. Creech, A. Jackson
Summary: This paper introduces a hybrid approach for explicitly-filtered Large Eddy Simulation using a Discontinous Galerkin discretisation for velocity, which incorporates information from a Continuous Galerkin version of the velocity field to improve computational performance while maintaining stability and accuracy.
COMPUTER PHYSICS COMMUNICATIONS
(2021)
Article
Mathematics, Applied
Siavash Hedayati Nasab, Carlos A. Pereira, Brian C. Vermeire
Summary: This paper presents optimized Runge-Kutta stability polynomials for multidimensional discontinuous Galerkin methods using the flux reconstruction approach. The stability polynomials significantly increase time-step sizes for various elements, with up to a speedup factor of 1.97 compared to classical methods. The optimization also yields modest performance benefits for certain elements and maintains the designed accuracy levels.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
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
Computer Science, Interdisciplinary Applications
Xiaofeng He, Kun Wang, Tiegang Liu, Yiwei Feng, Bin Zhang, Weixiong Yuan, Xiaojun Wang
Summary: This paper presents HODG, an open-source component-based development framework based on high order Discontinuous Galerkin methods. It is written in pure C++11 and supports various mesh types for solving compressible Euler and Navier-Stokes equations in 2D and 3D.
COMPUTER PHYSICS COMMUNICATIONS
(2023)
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
Zhaonan Dong, Emmanuil H. Georgoulis, Thomas Kappas
Summary: This paper introduces discontinuous Galerkin methods on polygonal/polyhedral meshes and parallel algorithms to address the computational complexity in the matrix-assembly step. The results demonstrate that using CUDA-enabled graphics cards for parallel implementation can greatly enhance the efficiency of these methods.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Article
Engineering, Multidisciplinary
Guosheng Fu, Zhiliang Xu
Summary: We introduce a novel class of high-order space-time finite element schemes for solving the Poisson-Nernst-Planck (PNP) equations. Our schemes achieve mass conservation, positivity preservation, and unconditional energy stability for any order of approximation. This is accomplished by employing the entropy variable formulation and a discontinuous Galerkin discretization in time.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(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
G. U. O. S. H. E. N. G. FU, W. E. N. Z. H. E. N. G. KUANG
Summary: We propose a novel monolithic divergence-conforming HDG scheme for a linear fluid-structure interaction problem with a thick structure. The semidiscrete scheme obtains a pressure-robust optimal energy-norm estimate. When combined with a Crank-Nicolson time discretization, our fully discrete scheme is energy stable and generates an exactly divergence-free fluid velocity approximation. The resulting linear system, which is symmetric and indefinite, is solved using a preconditioned MinRes method with a robust block algebraic multigrid preconditioner.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(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
Yanfang Yang, Shubin Fu, Eric T. Chung
Summary: In this paper, a efficient and robust two-grid preconditioner is proposed for solving the linear elasticity equation with high contrasts. The challenges imposed by multiple scales and high-contrast are addressed by constructing a coarse space within the framework of GMsFEM and controlling its dimension adaptively. The paper also introduces a parameter-independent efficient preconditioner for dealing with linear elasticity problems with stochastic coefficients.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Guosheng Fu
Summary: This study presents a new discontinuous Galerkin method for the nonlinear shallow water equation, which is able to maintain entropy stability and conservation properties on unstructured meshes. It also proposes a special treatment to handle dry areas where the water height is close to zero. One-dimensional and two-dimensional numerical experiments demonstrate the performance of the method.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Computer Science, Interdisciplinary Applications
Lina Zhao, Eric Chung
Summary: This paper introduces a novel residual-type a posteriori error estimator for Darcy flows in fractured porous media, using staggered DG methods on general polygonal meshes. The method is capable of handling fairly general meshes and incorporating hanging nodes for adaptive mesh refinement, demonstrating reliability and efficiency in error estimation.
COMPUTATIONAL GEOSCIENCES
(2022)
Article
Mathematics, Applied
Lina Zhao, Eric Chung, Eun-Jae Park
Summary: This paper proposes and analyzes a staggered discontinuous Galerkin method for a five-field formulation of the Biot system of poroelasticity on general polygonal meshes. The method is locking-free and can handle highly distorted grids, and a fixed stress splitting scheme is introduced to reduce the size of the global system.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2023)
Article
Computer Science, Interdisciplinary Applications
Tak Shing Au Yeung, Ka Chun Cheung, Eric T. Chung, Shubin Fu, Jianliang Qian
Summary: We propose a deep learning approach to extract ray directions at discrete locations by analyzing wave fields. A deep neural network is trained to predict ray directions based on local plane-wave fields. The resulting network is then applied to solve the Helmholtz equations at higher frequencies. The numerical results demonstrate the efficiency and accuracy of the proposed scheme.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Software Engineering
Changqing Ye, Eric T. Chung
Summary: This paper studies the convergences of several FFT-based discretization schemes in computational micromechanics, including Moulinec-Suquet's scheme, Willot's scheme, and the FEM scheme. It proves that the effective coefficients obtained by these schemes converge to the theoretical ones under reasonable assumptions. Convergence rate estimates are provided for the FEM scheme under additional regularity assumptions.
BIT NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Denis Spiridonov, Maria Vasilyeva, Min Wang, Eric T. Chung
Summary: In this paper, a class of Mixed Generalized Multiscale Finite Element Methods is proposed for solving elliptic problems in thin two-dimensional domains. The method utilizes multiscale basis functions and local snapshot space to construct a lower dimensional model and achieve multiscale approximation.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Uygulaana Kalachikova, Maria Vasilyeva, Isaac Harris, Eric T. Chung
Summary: This paper investigates the scattering problem in a heterogeneous domain using the Helmholtz equation and absorbing boundary conditions. A fine unstructured grid that resolves grid-level perforation is constructed for the finite element method solution. The large system of equations resulting from these approximations is reduced using the Generalized Multiscale Finite Element Method. The method constructs a multiscale space using the solution of local spectral problems on the snapshot space in each local domain, and two types of multiscale basis functions are presented and studied. Numerical results for the Helmholtz problem in a heterogeneous domain with obstacles of varying properties are provided, examining different wavenumbers and numbers of multiscale basis functions.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
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
Guosheng Fu, Wenzheng Kuang
Summary: In this paper, we present a lowest-order hybridizable discontinuous Galerkin (HDG) method, denoted as HDG-P0, for solving the reaction-diffusion equation and the generalized Stokes equations on conforming simplicial meshes in two- and three-dimensions. The proposed HDG-P0 schemes use piecewise constant finite element space for the global HDG facet degrees of freedom and discontinuous piecewise linear space for the local primal unknowns. We provide optimal a priori error analysis and propose optimal geometric multigrid preconditioners for the statically condensed HDG-P0 linear systems on conforming simplicial meshes.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Leonardo A. Poveda, Shubin Fu, Eric T. Chung, Lina Zhao
Summary: This paper presents a new Finite Element Method called CEM-GMsFEM for solving single-phase non-linear compressible flows in highly heterogeneous media. The method constructs basis functions by solving local spectral problems and local energy minimization problems. The convergence of the method is shown to only depend on the coarse grid size and the method is enhanced with an online enrichment guided by an a posteriori error estimator. Numerical experiments confirm the theoretical findings and demonstrate the efficiency and accuracy of the method.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
