Article
Mathematics, Applied
Kaifang Liu, Lunji Song
Summary: Interior-penalized weak Galerkin (IPWG) finite element methods are proposed and analyzed for solving second order elliptic equations. Numerical experiments demonstrate the effectiveness of the method, showing optimal convergence rates in L-2 norm.
Article
Mathematics, Applied
Dan Li, Chunmei Wang, Junping Wang
Summary: The generalized weak Galerkin (gWG) finite element method is proposed and analyzed for the biharmonic equation. A new generalized discrete weak second order partial derivative is introduced in the gWG scheme to allow arbitrary combinations of piecewise polynomial functions defined in the interior and on the boundary of general polygonal or polyhedral elements. The error estimates are established for the numerical approximation in a discrete H2 norm and a L2 norm. The numerical results demonstrate the accuracy and flexibility of the proposed gWG method for the biharmonic equation.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Dan Li, Chunmei Wang, Junping Wang
Summary: This article presents a new primal-dual weak Galerkin (PDWG) finite element method for solving transport equations in non-divergence form. The method employs locally reconstructed differential operators and stabilizers in the weak Galerkin framework, and results in a symmetric discrete linear system involving the primal variable and the dual variable (Lagrangian multiplier) for the adjoint equation. The article establishes optimal order error estimates in various discrete Sobolev norms for the corresponding numerical solutions, and provides numerical results to demonstrate the accuracy and efficiency of the new PDWG method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Emmanuil H. Georgoulis
Summary: This work focuses on developing a family of Galerkin finite element methods for the classical Kolmogorov equation, with the key attribute of admitting decay properties at the (semi)discrete level for general families of triangulations. The method construction combines ideas from the general theory of hypocoercivity developed by Villani and a judicious choice of numerical flux functions, allowing for robust error analysis for final times tending to infinity. The extension to three spatial dimensions is also briefly discussed.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2021)
Article
Mathematics, Applied
J. Zhang, X. Liu
Summary: This paper presents a weak Galerkin finite element method for solving the singularly perturbed convection-diffusion equation in 2D. The method utilizes polynomial approximations of different degrees on each mesh element, ensuring uniform convergence. Numerical experiments confirm the method's uniform convergence and optimal order.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics, Applied
Chun-Mei Xie, Min-Fu Feng, Hua-Yi Wei
Summary: In this paper, a new H1 weak Galerkin mixed finite element method is proposed for the Sobolev equation, which includes the exact solution u and the intermediate solution p. The method adopts discontinuous finite elements Pk/Pk and arbitrary shape of polygons for the approximation solution pair (uh, ph) on finite element partitions. The semi-discrete and full-discrete formulations are proven to be stable and parameter-free with optimal error estimates. Numerical experiments demonstrate the efficiency of the proposed methods.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Chunmei Wang, Junping Wang, Shangyou Zhang
Summary: This article presents a weak Galerkin finite element method for quad-curl problems in three dimensions. It is demonstrated that this method achieves stability and optimal error estimates in discrete norms for the exact solution. Furthermore, an optimal order L2 error estimate, excluding the lowest order k = 2, is derived for the WG solution. Numerical experiments confirm the efficiency, accuracy, and superconvergence of the WG method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Christoph Erath, Lorenzo Mascotto, Jens M. Melenk, Ilaria Perugia, Alexander Rieder
Summary: In this paper, we present a coupling method that combines the discontinuous Galerkin finite element method with the boundary element method to solve the three-dimensional Helmholtz equation with variable coefficients. The coupling is achieved through a mortar variable related to an impedance trace on a smooth interface. The method has a block structure with nonsingular subblocks, and we prove the quasi-optimality of both the h and p versions of the scheme under certain conditions.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Yingying Xie, Liuqiang Zhong
Summary: We investigated the AWG finite element method for second order elliptic problems and showed that the error between two consecutive adaptive loops is a contraction. Numerical experiments were conducted to support the theoretical findings.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Lin Mu
Summary: This study focuses on the development of a high-order weak Galerkin finite element method (WGFEM) on a curved mesh for curved two-dimensional domains. Numerical experiments demonstrate that the proposed curved WGFEM achieves an optimal convergence rate for errors measured in L-2-norm and energy-norm.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Xiaoshen Wang, Xiu Ye, Shangyou Zhang
Summary: The paper investigates the connections between weak Galerkin methods with and without stabilizers, revealing that the choice of stabilizers does not affect the convergence rates for WG elements with optional stabilizers. This phenomenon is verified both theoretically and numerically in the study.
NUMERICAL ALGORITHMS
(2021)
Article
Mathematics, Applied
Naresh Kumar, Jogen Dutta, Bhupen Deka
Summary: In this paper, we present weak Galerkin finite element methods for solving hyperbolic problems on polygonal meshes. We propose both semidiscrete and fully discrete schemes to numerically solve the second-order linear wave equation. Optimal order error estimate in the L-2 norm is shown to hold as O(h(k+1) + t(2)) for sufficiently smooth solutions, where h is the mesh size and t is the time step. Extensive numerical experiments are conducted to demonstrate the robustness, reliability, flexibility, and accuracy of the proposed method.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Shuhao Cao, Chunmei Wang, Junping Wang
Summary: This paper presents a new numerical method for div-curl systems with the normal boundary condition. The method provides accurate and reliable numerical solutions under the assumption of low H-alpha-regularity for the true solution, and effectively approximates normal harmonic vector fields on domains with complex topology.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Peng Zhu, Shenglan Xie
Summary: This paper presents a fully discrete stabilizer-free weak Galerkin finite element scheme for approximating parabolic equations. The temporal variable is discretized using the second order Crank-Nicolson scheme, while the spatial variables are discretized using a stabilizer-free weak Galerkin finite element method. The stability and supercloseness convergence of both the semi-discrete and fully discrete methods are established. Furthermore, a postprocessing technique is proposed to obtain a global superconvergence finite element approximation with higher accuracy. Numerical experiments are conducted to validate the theoretical findings.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Waixiang Cao, Chunmei Wang
Summary: This article introduces a new primal-dual weak Galerkin finite element method for the convection-diffusion equation, with optimal error estimates established in various discrete norms and standard L-2 norms. A series of numerical experiments were conducted to validate the theoretical findings.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Lin Mu, Xu Zhang
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2019)
Article
Mathematics, Applied
Lin Mu
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2019)
Article
Mathematics, Applied
James H. Adler, Xiaozhe Hu, Lin Mu, Xiu Ye
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2019)
Article
Mathematics, Applied
Feng Bao, Lin Mu, Jin Wang
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2019)
Article
Mathematics, Applied
Lin Mu
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2019)
Article
Engineering, Multidisciplinary
Gang Wang, Lin Mu, Ying Wang, Yinnian He
Summary: This paper introduces a pressure-robust virtual element method for solving the Stokes problem on convex polygonal meshes. By enhancing the approximation methods for velocity and pressure, pressure-independent velocity approximation is achieved, with numerical experiments validating the theoretical conclusions.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Mathematics, Applied
Lin Mu, Xiu Ye, Shangyou Zhang
Summary: Pressure-robustness is crucial for incompressible fluid simulations. Enhancements to the discontinuous Galerkin finite element methods in the primary velocity-pressure formulation for solving Stokes equations have been developed to achieve pressure-robustness. The new schemes show improvements in source term modifications and have been validated through numerical experiments. Optimal-order error estimates have been established for the numerical approximations in various norms.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Computer Science, Interdisciplinary Applications
Bin Wang, Ingo Wald, Nate Morrical, Will Usher, Lin Mu, Karsten Thompson, Richard Hughes
Summary: A novel efficient and robust particle tracking method (RT method) is presented to accelerate Eulerian-Lagrangian simulations using hardware ray tracing cores and GPU parallel computing technology. The method includes a hardware-accelerated hosting cell locator and a robust treatment of particle-wall interaction, demonstrated through numerical simulations and experimental observations. Benchmark results show a significant performance improvement compared to the reference method for large-scale simulations.
COMPUTER PHYSICS COMMUNICATIONS
(2022)
Article
Mathematics, Applied
Guannan Zhang, Lin Mu
Summary: A non-intrusive domain-decomposition model reduction method has been developed for linear steady-state PDEs with random-field coefficients, enabling the construction of a reduced model without intrusive implementation from scratch by accessing the final linear system of a deterministic PDE solver.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2022)
Article
Computer Science, Interdisciplinary Applications
David Green, Xiaozhe Hu, Jeremy Lore, Lin Mu, Mark L. Stowell
Summary: In this paper, an interior penalty discontinuous Galerkin finite element scheme is presented for solving diffusion problems with strong anisotropy arising in magnetized plasmas for fusion applications. The authors demonstrate the accuracy of the high-order scheme and develop an efficient preconditioning technique that is robust to the mesh size and anisotropy of the problem. Several numerical tests are provided to validate the accuracy and efficiency of the proposed algorithm.
COMPUTER PHYSICS COMMUNICATIONS
(2022)
Article
Mathematics, Applied
Lin Mu
Summary: In this article, a novel numerical scheme for solving the steady incompressible Navier-Stokes equations is developed and analyzed using the weak Galerkin methods. The algorithm achieves pressure-robustness by employing a divergence-preserving velocity reconstruction operator, which ensures that the velocity error is independent of the pressure and irrotational body force. Error analysis is conducted to establish the convergence rate, and numerical experiments are presented to validate the theoretical conclusions.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Lin Mu, Xiu Ye, Shangyou Zhang
Summary: This paper proposes a new stabilizer-free and pressure-robust weak Galerkin finite element method for the Stokes equations with superconvergence, which is validated through numerical experiments for its effectiveness and robustness.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Article
Computer Science, Interdisciplinary Applications
Eduardo D'Azevedo, David L. Green, Lin Mu
COMPUTER PHYSICS COMMUNICATIONS
(2020)
Article
Mathematics, Applied
Lin Mu
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2020)
Article
Mathematics, Applied
Lin Mu
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2020)
