Article
Computer Science, Interdisciplinary Applications
Michael Neunteufel, Joachim Schoberl
Summary: This study presents a novel application of the (high-order) H(div)-conforming Hybrid Discontinuous Galerkin finite element method for monolithic fluid-structure interaction (FSI), which yields exact divergence free fluid velocity solutions by introducing the Piola transformation. With the use of hp-refinement strategies, singularities and boundary layers are overcome leading to optimal spatial convergence rates. Copyright (C) 2020 Elsevier Ltd.
COMPUTERS & STRUCTURES
(2021)
Article
Mathematics, Applied
Linshuang He, Minfu Feng, Jun Guo
Summary: In this paper, we propose and analyze an H(div) conforming CDG method for the three-field Biot's consolidation model with displacement reconstruction technique. The method utilizes kth-order Brezzi-Douglas-Marini element for discretizing the displacement, and k-1th-order Raviart-Thomas-Nedelec element pairs for approximating the fluid flux and pore pressure. The H(div) CDG method is derived from the H(div) conforming formulation by replacing the gradient operator with a weak gradient operator, and optimal a-priori error estimates are proven for both semi-discrete scheme and fully-discrete scheme with backward Euler discretization in time.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Guosheng Fu, Christoph Lehrenfeld, Alexander Linke, Timo Streckenbach
Summary: This study discusses the issues of volume-locking and gradient robustness in linear elasticity, proposing novel Hybrid Discontinuous Galerkin methods for discretization. By utilizing divergence-conforming discretization and additional optimization measures, effective control over gradient fields and spurious displacements is achieved.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Computer Science, Interdisciplinary Applications
Walter Boscheri, Giacomo Dimarco, Lorenzo Pareschi
Summary: We propose a novel Structure-Preserving Discontinuous Galerkin (SPDG) operator that recovers the algebraic property related to the div-curl problem at the discrete level. A staggered Cartesian grid is adopted in 3D, and a high order DG divergence operator is built upon integration by parts. The novel SPDG schemes are capable of obtaining a zero div-curl identity with high accuracy and can be applied to solving the incompressible Navier-Stokes equations.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Qian Zhang, Zhimin Zhang
Summary: Several smooth finite element de Rham complexes are constructed in three-dimensional space, which lead to three families of grad div-conforming finite elements. The simplest element has only 8 degrees of freedom (DOFs) for a tetrahedron and 14 DOFs for a 3-rectangle. These elements are shown to provide conforming and convergent approximations to quad-div problems, while also generating some grad div-nonconforming elements. Numerical experiments validate the correctness and efficiency of the nonconforming elements for solving the quad-div problem.
NUMERISCHE MATHEMATIK
(2022)
Article
Computer Science, Interdisciplinary Applications
Xiu Ye, Shangyou Zhang
Summary: A new discontinuous Galerkin finite element method for the Stokes equations has been developed, which utilizes discontinuous polynomials on general polygonal/polyhedral meshes without stabilizers for velocity and pressure. The method shows optimal error estimates for numerical approximations in various norms, and has been tested for low and high order elements up to degree four in both 2D and 3D spaces.
INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS
(2021)
Article
Mathematics, Applied
Yongbin Han, Yanren Hou
Summary: The paper focuses on the velocity error analysis of H(div)-conforming DG method for semi-discrete time-dependent Navier-Stokes equations. It proves the optimality and pressure-robustness of velocity errors, but dependency on viscosity inversely affects semi-robustness. The study introduces Raviart-Thomas interpolation operator for pressure-robust and semi-robust velocity error analysis at high Reynolds numbers, showing quasi-optimality under certain conditions.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Raman Kumar, Bhupen Deka
Summary: In this paper, a fitted weak Galerkin (WG) finite element scheme is demonstrated for solving H(div)-elliptic equation with discontinuous coefficients and interface. Error estimates of optimal orders in both L2 norm and H1 norm are discussed for the H(div)-elliptic interface problems. High-order convergence rates are achieved by using suitable WG approximation spaces of higher degrees. Numerical tests confirm the theoretical findings of the proposed WG algorithm and show its capability to handle geometrically complicated and very irregular interfaces with sharp edges, cusps, and tips.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mechanics
Xi Chen, Corina Drapaca
Summary: This paper presents a systematic construction of H(div)-conforming numerical dissipation for time-dependent incompressible Euler and Navier-Stokes equations to improve the performance of the central flux scheme. The method generalizes the upwind flux scheme from a dissipation point of view, utilizing discontinuity of numerical quantities across interior edges. Experimental results demonstrate the effectiveness of the added dissipation in reducing errors and preserving physics.
Article
Engineering, Multidisciplinary
Ruo Li, Qicheng Liu, Fanyi Yang
Summary: In this paper, a high-order discontinuous Galerkin finite element method is proposed to solve the H(div)- and H(curl)-elliptic interface problems on unfitted meshes. The vector-valued approximation space is constructed using patch reconstruction with at most d degrees of freedom per element. The method allows for C2-smooth interfaces intersecting elements in a general setup. The patch reconstruction naturally provides stability near the interface without any additional stabilization strategy. The method is based on the symmetric interior penalty method and optimal and suboptimal convergence rates under the energy norm and L2 norm are derived. Numerical examples in both two and three dimensions are presented to demonstrate the accuracy of the method.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Mathematics, Applied
Yue Wang, Fuzheng Gao, Jintao Cui
Summary: A new conforming discontinuous Galerkin method is proposed for solving second order elliptic interface problems with discontinuous coefficient. Compared with known weak Galerkin algorithms, the method studied in this paper has no stabilizer and fewer unknowns. Error estimates in H-1 and L-2 norms are established, showing optimal order convergence.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Shashank Jaiswal
Summary: Adaptivity is crucial for addressing practical challenges, especially in computational fluid dynamics workflow. The mixed non-conforming discontinuous Galerkin discretization method is introduced for the full Boltzmann equation, providing optimal convergence for non-linear kinetic systems on non-orthogonal grids. The method allows for analysis of complex problems on massively parallel scales and is applicable to a wide range of rarefied flows. The computational overhead for solving kinetic equations on non-conforming structured/unstructured domains is negligible compared to conforming domains.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
J. Gopalakrishnan, L. Kogler, P. L. Lederer, J. Schoeberl
Summary: This paper introduces two new lowest order methods, a mixed method and a hybrid discontinuous Galerkin method, for approximating incompressible flows. Both methods use divergence-conforming linear Brezzi-Douglas-Marini space to approximate the velocity and the lowest order Raviart-Thomas space to approximate the vorticity. The methods provide exact divergence-free discrete velocity solutions, optimal error estimates and are based on the physically correct viscous stress tensor of the fluid.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Xu Li, Hongxing Rui
Summary: In this paper, a P-1(c) circle plus RT0 - P0 discretization method is proposed for solving the Stokes equations on general simplicial meshes in two/three dimensions. The method provides an exactly divergence-free and pressure-independent velocity approximation with optimal order. Additionally, the method can be easily transformed into a pressure-robust and stabilized discretization method, which has a much smaller number of degrees of freedom.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2022)
Article
Mathematics, Applied
Yue Wang, Fuzheng Gao, Jintao Cui
Summary: A new conforming discontinuous Galerkin method is studied for linear elasticity interface problems with discontinuous coefficients and displacement. This method, based on a new definition of weak gradient operator, does not require a stabilizer and employs a different weak divergence operator than the traditional weak Galerkin finite element method, resulting in reduced computational cost. Error estimates in discrete L-2 and H-1 norms are established with optimal order. Numerical examples validate the efficiency, accuracy, and locking-free property of the numerical method.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Xiaoxuan Yu, Yan Xu, Qiang Du
Summary: The study focuses on a quadrature-based finite difference discretization of one-dimensional scalar linear nonlocal conservation laws with spatially varying range of nonlocal interactions. It addresses the convergence of the discrete approximation in both nonlocal and local settings, presenting the first complete proof of numerical discretization convergence to feasible kernels in the nonlocal regime and local limit, establishing the asymptotic compatibility of the scheme. Numerical results are also provided to illustrate the impact of variable horizon on wave propagation in the nonlocal model.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Physics, Mathematical
Weijie Zhang, Yulong Xing, Yinhua Xia, Yan Xu
Summary: This paper proposes a high-order accurate DG method for the compressible Euler equations on unstructured meshes under gravitational fields, which preserves a general hydrostatic equilibrium state and guarantees the positivity of density and pressure. Through a special way to recover the equilibrium state and the design of novel interface variables, the scheme achieves well-balanced and positivity-preserving properties.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2022)
Article
Physics, Mathematical
Jianfang Lin, Yan Xu, Huiwen Xue, Xinghui Zhong
Summary: In this paper, two finite difference weighted essentially non-oscillatory (WENO) schemes with unequal-sized sub-stencils are developed for solving the Degasperis-Procesi (DP) and mu-Degasperis-Procesi (mu DP) equations. The proposed schemes are simple in stencil choice and allow for arbitrary positive linear weights. Numerical tests demonstrate the high order accuracy and non-oscillatory properties of the schemes.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
Xiaoxuan Yu, Yan Xu, Qiang Du
Summary: We study the propagation of singularities in solutions of linear convection equations with spatially heterogeneous nonlocal interactions. We are interested in understanding the impact on singularity propagation due to the heterogeneities of the nonlocal horizon and the local and nonlocal transition. We derive equations to characterize the propagation of different types of singularities for various forms of nonlocal horizon parameters in the nonlocal regime, and use numerical simulations to illustrate the propagation patterns in different scenarios.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Zhongjie Lu, Yan Xu
Summary: This paper introduces a method to compute the discrete Maxwell eigenproblem by discretizing and constructing a penalty term to deal with its huge kernel. The algorithm is efficient and capable as demonstrated by numerical examples.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Qi Tao, Liangyue Ji, Jennifer K. Ryan, Yan Xu
Summary: In this paper, the authors discuss the enhancement of accuracy in discontinuous Galerkin methods for solving PDEs with high order spatial derivatives. They introduce the use of a Smoothness-Increasing Accuracy-Conserving (SIAC) filter to create a superconvergence filtered solution. The authors provide theoretical proofs for the accuracy improvement and demonstrate the applicability of the SIAC filter for both linear and nonlinear higher order PDEs.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Computer Science, Interdisciplinary Applications
Jiahui Zhang, Yinhua Xia, Yan Xu
Summary: This paper develops ALE-WENO schemes for the shallow water equations under the ALE framework, adopting the WENO hybrid reconstruction on moving meshes. The schemes demonstrate high order accuracy and positivity-preserving property, as well as the preservation of equilibrium and capturing of small perturbations without numerical oscillations. The ALE-WENO hybrid schemes have an advantage over static mesh simulations due to higher resolution interface tracking.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Jinyang Lu, Yan Xu, Chao Zhang
Summary: This paper presents a uniform framework of local discontinuous Galerkin (LDG) methods for two-dimensional Camassa-Holm equations and two-dimensional mu-Camassa-Holm equations. The energy stability and semi-discrete error estimates are derived based on this framework. Numerical experiments demonstrate the accuracy and stability of these schemes.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Jiahui Zhang, Yinhua Xia, Yan Xu
Summary: In this paper, a new well-balanced discontinuous Galerkin (DG) method is proposed to preserve moving-water equilibria in the shallow water equations. Instead of approximating the conservative variables, the scheme approximates the equilibrium variables in the DG piecewise polynomial space. The numerical fluxes are modified based on the generalized hydrostatic reconstruction to achieve moving water equilibrium preservation.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Qian Zhang, Yan Xu, Yue Liu
Summary: This paper develops a high-order discontinuous Galerkin (DG) method for the Camassa-Holm-Kadomtsev-Petviashvili (CH-KP) type equations on Cartesian meshes. The proposed method efficiently deals with the integration operator partial differential delta(-1) element by element and is applicable to most solutions. The DG scheme is proven to be an energy stable numerical scheme using the instinctive energy of the original PDE as a guiding principle, and the semi-discrete error estimates are derived for the nonlinear case without any priori assumption. Several numerical experiments demonstrate the capability of the proposed schemes for various types of solutions.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Wei Zheng, Yan Xu
Summary: In this paper, bound preserving and high-order accurate LDG schemes are used to solve chemotaxis models. First-order temporal accurate schemes are constructed using the gradient flow structure of the models, and can be extended to high accuracy schemes. Bound preserving is achieved by using Lagrange multipliers and an efficient active set semi-smooth Newton method.
COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Mathematics, Applied
Weijie Zhang, Yulong Xing, Yinhua Xia, Yan Xu
Summary: In this study, high-order arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) methods for the Euler equations with gravitational fields on moving meshes are proposed. The goal is to show that ALE-DG methods can achieve structure-preserving properties of DG methods for the Euler equations on arbitrary moving meshes. Two well-balanced and positivity-preserving ALE-DG schemes are presented, which can preserve the explicitly given equilibrium state on any moving grids. The schemes are established in both one and two dimensions on unstructured triangular meshes.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Ruihan Guo, Yan Xu
Summary: In this paper, we propose a class of semi-implicit spectral deferred correction (SDC) methods based on second-order time integration methods, where the order of accuracy is increased by two for each additional iteration. The spatial discretization is done using the local discontinuous Galerkin (LDG) method to obtain fully-discrete schemes that are high-order accurate in both space and time. Numerical experiments are conducted to demonstrate the accuracy, efficiency, and robustness of the proposed semi-implicit SDC methods in solving complex nonlinear PDEs.
JOURNAL OF COMPUTATIONAL MATHEMATICS
(2023)
Article
Engineering, Mechanical
Xiangyi Meng, Yan Xu
Summary: In this paper, a mesh adaptation algorithm for the unsteady compressible Navier-Stokes equations is presented. The algorithm combines local discontinuous Galerkin methods with implicit-explicit Runge-Kutta or spectral deferred correction time discretization methods. The algorithm shows high accuracy, efficiency, and capability in capturing flow structures through numerical experiments.
ADVANCES IN AERODYNAMICS
(2022)
Article
Mathematics, Applied
Qi Tao, Yan Xu, Xiaozhou Li
Summary: In this paper, the important of negative norm estimates for the arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method solving nonlinear hyperbolic equations with smooth solutions is presented. The smoothness-increasing accuracy-conserving (SIAC) filter is used as a post-processing technique to enhance the accuracy of the discontinuous Galerkin (DG) solutions. This work is a crucial step in extending the SIAC filter to moving mesh for nonlinear problems.
COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION
(2022)
