Article
Mathematics, Applied
Huihui Cao, Yunqing Huang, Nianyu Yi
Summary: This paper investigates the adaptive direct discontinuous Galerkin method for second order elliptic equations in two dimensions, introducing a numerical flux with general weighted averages and proper weights for interface problems. In addition, it proposes a residual-type a posteriori error estimator and establishes global upper bounds and local lower bounds for errors in the DG norm. Several numerical examples are conducted to confirm the reliability and efficiency of the proposed error estimator and method.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Computer Science, Interdisciplinary Applications
Assyr Abdulle, Giacomo Rosilho de Souza
Summary: This paper introduces a local adaptive discontinuous Galerkin method for convection-diffusion-reaction equations. The method improves the accuracy of the solution by solving local elliptic problems in refined subdomains and provides an algorithm for the automatic identification of these subdomains. Numerical comparisons demonstrate the efficiency of the method.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Water Resources
Jean-Baptiste Clement, Frederic Golay, Mehmet Ersoy, Damien Sous
Summary: The paper proposes an adaptive strategy to deal with complex wetting fronts in both space and time, enhancing the reliability of nonlinear convergence with the use of backward difference formula and adaptive mesh refinement, combined with a weighted discontinuous Galerkin framework to better approximate wetting fronts.
ADVANCES IN WATER RESOURCES
(2021)
Article
Mathematics, Applied
Huihui Cao, Yunqing Huang, Nianyu Yi
Summary: In this paper, a gradient recovery method for the direct discontinuous Galerkin (DDG) method is proposed. The recovered gradient, which is obtained by fitting a quadratic polynomial to the gradient of the numerical solution, is defined on a piecewise continuous space and may be discontinuous on the whole domain. This recovered gradient is used to introduce a posteriori error estimator and validate its accuracy using benchmark test problems.
Article
Physics, Mathematical
Jiachuan Zhang, Ran Zhang, Xiaoshen Wang
Summary: Based on auxiliary subspace techniques, this paper presents a posteriori error estimator of nonconforming weak Galerkin finite element method (WGFEM) for the Stokes problem in two and three dimensions. Without the saturation assumption, it is proved that the WGFEM approximation error is bounded by the error estimator up to an oscillation term. The computational cost of the approximation and error problems is considered in terms of the size and sparsity of the system matrix. To reduce the computational cost of the error problem, an equivalent error problem is constructed using diagonalization techniques, which only requires solving two diagonal linear algebraic systems corresponding to the degree of freedom (d.o.f) to obtain the error estimator. Numerical experiments are provided to demonstrate the effectiveness and robustness of the a posteriori error estimator.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2023)
Article
Engineering, Mechanical
Zhen-Hua Jiang, Chao Yan, Jian Yu
Summary: A simple and efficient troubled-cell indicator based on a posteriori limiting paradigm is proposed for the discontinuous Galerkin (DG) method on the triangular grids. The developed methodology utilizes discrete solution from different time levels in the von Neumann neighborhood to maintain the compactness of the DG schemes. Different limitation can then be applied to the resulting troubled cells to achieve favorable numerical characteristic including positivity-preserving and oscillation-suppressing. The present indicator has been implemented and compared with other indicators for solving the two-dimensional Euler equations on unstructured grids, showing its effectiveness and robustness.
ACTA MECHANICA SINICA
(2023)
Article
Mathematics, Applied
Haitao Leng
Summary: In this paper, a hybridizable discontinuous Galerkin method with divergence-free and H(div)-conforming velocity field is proposed for the stationary incompressible Navier-Stokes equations. The pressure-robustness, which ensures that the a priori error estimates of the velocity are independent of the pressure error, is satisfied. Additionally, an efficient and reliable a posteriori error estimator is derived for the L-2 errors in the velocity gradient and pressure, under a smallness assumption. Numerical examples are provided to demonstrate the pressure-robustness and the performance of the obtained a posteriori error estimator.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
R. H. W. Hoppe
Summary: This paper presents a space-time adaptive C-0 Interior Penalty Discontinuous Galerkin (C(0)IPDG) approximation method for the dynamic quasi-static von Karman equations, including homogeneous Dirichlet boundary conditions and an equilibrated a posteriori error estimator. The backward Euler scheme is used for time discretization, and the C(0)IPDG method is derived from a six-field formulation of the finite element discretized von Karman equations. The equilibrated a posteriori error estimator provides an upper bound for the discretization error in terms of associated energy functionals. It requires the construction of equilibrated fluxes and moment tensors computed on local patches around interior nodal points.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
R. H. W. Hoppe
Summary: We consider an adaptive C0 Interior Penalty Discontinuous Galerkin (C0IPDG) approximation of the fourth order von Karman equations with homogeneous Dirichlet boundary conditions and an equilibrated a posteriori error estimator. The C0IPDG method can be derived from a six-field formulation of the finite element discretized von Karman equations. The equilibrated a posteriori error estimator provides an upper bound for the discretization error in the broken W 2,2 0 norm in terms of the associated primal and dual energy functionals. It requires the construction of equilibrated fluxes and equilibrated moment tensors which can be computed on local patches around interior nodal points of the triangulations. The relationship with a residual-type a posteriori error estimator is studied as well. Numerical results illustrate the performance of the suggested approach.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Mirjam Walloth, Winnifried Wollner
Summary: This article introduces a residual-type a posteriori error estimator for a time discrete quasi-static phase-field fracture model, focusing on the robustness of the error estimator for the variational inequality governing the phase-field evolution with respect to the phase-field regularization parameter ε. Numerical examples demonstrate the performance of the proposed a posteriori error estimators on three standard test cases.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Mirjam Walloth, Winnifried Wollner
Summary: This article develops a residual-type a posteriori error estimator for a time discrete quasi-static phase-field fracture model, focusing on the robustness of the error estimator with respect to the phase-field regularization parameter epsilon. Numerical examples are provided to demonstrate the performance of the proposed error estimators on three standard test cases.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Mahboub Baccouch
Summary: This paper investigates the superconvergence properties of the local discontinuous Galerkin method for linear second-order elliptic problems on Cartesian grids, and presents an efficient and reliable a posteriori error estimator. The proposed method converges to the true errors under mesh refinement, with a convergence order of p + 2.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Jing Wen, Jian Su, Yinnian He, Hongbin Chen
Summary: In this paper, semi-discrete and fully discrete schemes of the Stokes-Biot model are proposed and analyzed in detail. The existence and uniqueness of the semi-discrete scheme are proved, with a-priori error estimates derived. Numerical tests under matching and non-matching meshes validate the convergence analysis and support the theoretical results.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Engineering, Multidisciplinary
Robert E. Bird, Charles E. Augarde, William M. Coombs, Ravindra Duddu, Stefano Giani, Phuc T. Huynh, Bradley Sims
Summary: This paper presents a 2D hp-adaptive discontinuous Galerkin finite element method for phase field fracture that can reliably and efficiently solve phase field fracture problems with arbitrary initial meshes. The method uses a posteriori error estimators to drive mesh adaptivity based on both elasticity and phase field errors, and it is validated on several example problems.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Mathematics, Applied
Dongjie Liu, Le Zhou, Xiaoping Zhang
Summary: This paper explores the adaptive local discontinuous Galerkin (LDG) method for the p-Laplace problem in polygonal regions in R-2. A new, sharper a posteriori error estimate for the LDG approximation of the p-Laplacian is proposed in a new framework. Several examples are provided to confirm the reliability of the estimate.
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING
(2022)
Article
Mathematics, Applied
Lina Zhao, Dohyun Kim, Eun-Jae Park, Eric Chung
Summary: In this paper, a staggered discontinuous Galerkin method for Darcy flows in fractured porous media is presented and analyzed. The method uses a staggered discontinuous Galerkin method and a standard conforming finite element method with appropriate inclusion of interface conditions. The optimal convergence estimates for all the variables are proved, and the error estimates are shown to be fully robust with respect to the heterogeneity and anisotropy of the permeability coefficients.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Engineering, Multidisciplinary
Yiran Wang, Eric Chung, Shubin Fu
Summary: In this paper, a local-global multiscale method is proposed for highly heterogeneous stochastic groundwater flow problems. The method combines the reduced basis method and the generalized multiscale finite element method to achieve computational efficiency. The authors provide rigorous analysis and extensive numerical examples to demonstrate the accuracy and efficiency of the method.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Mathematics, Applied
Dmitry Ammosov, Maria Vasilyeva, Eric T. Chung
Summary: In this paper, the thermoporoelasticity problem in heterogeneous and fractured media is considered. The proposed multiscale method reduces the size of the discrete system and provides good accuracy by solving local spectral problems to compute the multiscale basis functions.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Eric T. Chung, Yalchin Efendiev, Wing Tat Leung, Petr N. Vabishchevich
Summary: This work proposes contrast-independent partially explicit time discretizations for wave equations in heterogeneous high-contrast media. The spatial space is split into contrast-dependent and contrast-independent components through multiscale space decomposition. The proposed splitting is unconditionally stable under suitable conditions and identifies local features for implicit treatment. The numerical results demonstrate that the proposed methods yield results similar to implicit methods with contrast-independent timestep.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Eric T. Chung, Uygulaana Kalachikova, Maria Vasilyeva, Valentin Alekseev
Summary: In this study, a Generalized Multiscale Discontinuous Galerkin Method (GMsDGM) is proposed for the convection-diffusion equation in perforated media. The method utilizes fine and coarse grid approximations and investigates various constructions of multiscale basis functions for numerical solutions.
MATHEMATICS AND COMPUTERS IN SIMULATION
(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
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
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)
Article
Mathematics, Applied
Jie Du, Eric Chung, Yang Yang
Summary: This paper studies the classical Allen-Cahn equations and investigates the maximum-principle-preserving (MPP) techniques. It discusses the application of the local discontinuous Galerkin (LDG) method and the use of conservative modified exponential Runge-Kutta methods. Numerical experiments are used to demonstrate the effectiveness of the MPP LDG scheme.
COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics, Applied
Lina Zhao, Ming Fai Lam, Eric Chung
Summary: This paper proposes a uniformly robust staggered DG method for the unsteady Darcy-Forchheimer-Brinkman problem, based on velocity gradient-velocity-pressure formulation for general polygonal meshes. The relaxation of tangential continuity for velocity is essential in achieving uniform robustness, and error analysis shows velocity error estimates are independent of pressure. Numerical experiments confirm the theoretical findings.
COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION
(2022)
