Article
Mathematics, Applied
Xiaowei Liu, Min Yang, Jin Zhang
Summary: In this paper, the weak Galerkin method on a Shishkin rectangular mesh is analyzed for a singularly perturbed convection-diffusion problem in two dimensions. The method achieves supercloseness through a specially constructed interpolant, which consists of vertices-edges-element interpolant inside the layers and modified Gauss-Radau interpolant outside the layers in the interior of each element, and vertices-edges-element interpolant inside the layers and weighted L2 projection outside the layers on the boundary of each element. Additionally, over-penalization technique is used inside the layers, and supercloseness of order k + 1/2 is proved, even up to almost k + 1 under appropriate assumptions. Numerical experiments validate the theoretical result.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Jin Zhang, Xiaowei Liu
Summary: This study provides a convergence analysis for finite element methods of any order applied to singularly perturbed reaction-diffusion problems using Shishkin and Bakhvalov-Shishkin meshes. A new interpolant is introduced for analysis in the balanced norm, which along with superconvergence estimations, proves supercloseness results. Numerical experiments confirm these theoretical findings.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics, Applied
Xiaoqi Ma, Jin Zhang
Summary: In this study, we investigate the supercloseness property of the nonsymmetric interior penalty Galerkin (NIPG) method on a Bakhvalov-type mesh for a singularly perturbed convection diffusion problem. We propose a new composite interpolation method that combines Gaul3 Radau projection outside the layer and Gaul3 Lobatto projection inside the layer. By selecting appropriate penalty parameters at different mesh points, we obtain the supercloseness of k + 21th order (k >= 1) in an energy norm.
APPLIED MATHEMATICS LETTERS
(2023)
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
Xiaowei Liu, Jin Zhang
Summary: This paper investigates singularly perturbed convection-diffusion equations and proposes a streamline diffusion finite element method for the last element mesh analysis, achieving an almost second-order supercloseness that is consistent with numerical experiments.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics, Applied
Jin Zhang, Xiaowei Liu
Summary: In this article, a singularly perturbed convection-diffusion equation is solved using a linear finite element method on Bakhvalov-type meshes. A supercloseness result is obtained for the first time on Bakhvalov-type meshes through a novel interpolation of the solution.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
Yue Wang, Yonghai Li, Xiangyun Meng
Summary: This paper presents the construction and analysis of an upwind finite volume element method on a Shishkin mesh for singularly perturbed convection-diffusion problems. The stability of the method is proven under the assumption of the convection and reaction term coefficients. The error estimate in the energy norm is provided on the Shishkin mesh, and an optimal error bound of O(N-1(ln N)3/2) is obtained. Numerical examples are used to illustrate the theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Yanhui Lv, Jin Zhang
Summary: In this paper, a singularly perturbed elliptic problem with two parameters in two dimensions is considered. The uniform convergence and supercloseness in an energy norm are proved using the linear finite element method on a Shishkin triangular mesh. Integral inequalities play a crucial role in the analysis, and the theoretical results are verified through numerical tests.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics, Applied
A. F. Hegarty, E. O'Riordan
Summary: A finite difference method is used to solve singularly perturbed convection-diffusion problems on smooth domains, with constraints imposed on the data to ensure regular exponential boundary layers in the solution. A domain decomposition method is employed, with a rectangular grid outside the boundary layer and a Shishkin mesh near the boundary layer aligned to the curvature of the outflow boundary. Numerical results are presented to demonstrate the effectiveness of the proposed algorithm.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Xiaoqi Ma, Jin Zhang
Summary: In this paper, a singularly perturbed convection-diffusion problem with a discontinuous convection is discussed. The interior layer appearing in the solution due to this discontinuity is solved using a streamline diffusion finite element method on Shishkin mesh, and the optimal order of convergence in a modified streamline diffusion norm is derived. Numerical results are presented to validate the theoretical conclusion.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Abhay Kumar Chaturvedi, S. Chandra Sekhara Rao
Summary: This article investigates a two-dimensional singularly perturbed convection-reaction-diffusion interface problem with discontinuities in the coefficients and source term. A Local Discontinuous Galerkin method is constructed on a Shishkin mesh, and the error in the computed solution converges at a rate of O((N-1ln N)r+12). Numerical results are provided to validate the theoretical findings.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Jin Zhang, Xiaowei Liu
Summary: Supercloseness and postprocessing of the linear finite element method on Bakhvalov-type mesh for a singularly perturbed convection diffusion problem are studied. By introducing a novel interpolation and a new postprocessing operator, second order supercloseness and convergence are obtained.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics, Applied
Jin Zhang, Xiaoqi Ma, Yanhui Lv
Summary: In this paper, a singularly perturbed convection diffusion problem is discussed, and a finite element method on Shishkin mesh is constructed to address the issue of uniform convergence. The paper proves the minimum principle and stability result, and derives asymptotic expansion of the solution to establish a priori estimates. Uniform convergence of almost order k in the energy norm is proven, with k being the order of piecewise polynomials.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
Jin Zhang, Yanhui Lv
Summary: This paper applies the linear finite element method to a singularly perturbed problem with two parameters on a Bakhvalov-type mesh. The solution of this problem exhibits both a strong exponential boundary layer and a weak exponential boundary layer. Through analysis of three cases using different techniques, the study obtains uniform convergence and supercloseness results, which are confirmed by numerical tests.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Jin Zhang, Xiaowei Liu
Summary: The study reported convergence stalls on Bakhvalov-Shishkin mesh when N-1 <= epsilon, presented uniform convergence analysis of finite element methods on Bakhvalov-type meshes related to Bakhvalov-Shishkin mesh, proved an optimal order of convergence, and used the results for mesh improvement. These theoretical results were verified by numerical experiments.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Xiaowei Liu, Martin Stynes, Jin Zhang
IMA JOURNAL OF NUMERICAL ANALYSIS
(2018)
Article
Mathematics, Applied
Xiaowei Liu, Jin Zhang
NUMERICAL ALGORITHMS
(2018)
Article
Mathematics, Applied
Xiaowei Liu, Jin Zhang
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2018)
Article
Mathematics, Applied
Xiaowei Liu, Jin Zhang
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2018)
Article
Mathematics, Applied
Jin Zhang, Xiaowei Liu
APPLIED MATHEMATICS LETTERS
(2019)
Article
Computer Science, Interdisciplinary Applications
Cong Xu, Min Yang, Jin Zhang
JOURNAL OF STATISTICAL COMPUTATION AND SIMULATION
(2020)
Article
Mathematics, Applied
Jin Zhang, Xiaowei Liu
JOURNAL OF SCIENTIFIC COMPUTING
(2020)
Article
Mathematics, Applied
Jin Zhang, Xiaowei Liu
Summary: This article focuses on the convergence analysis of finite element methods for singularly perturbed reaction-diffusion problems using balanced norms on Bakhvalov-type rectangular meshes. A novel interpolation operator is introduced for optimal order convergence in the balanced norm. The stability of the L-2 projection and characteristics of Bakhvalov-type meshes play a crucial role in the analysis. Additionally, a supercloseness result in the balanced norm is obtained using a novel interpolant with corrections on the boundary.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Jin Zhang, Xiaowei Liu
Summary: This study provides a convergence analysis for finite element methods of any order applied to singularly perturbed reaction-diffusion problems using Shishkin and Bakhvalov-Shishkin meshes. A new interpolant is introduced for analysis in the balanced norm, which along with superconvergence estimations, proves supercloseness results. Numerical experiments confirm these theoretical findings.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics, Applied
Jin Zhang, Xiaowei Liu
Summary: Supercloseness and postprocessing of the linear finite element method on Bakhvalov-type mesh for a singularly perturbed convection diffusion problem are studied. By introducing a novel interpolation and a new postprocessing operator, second order supercloseness and convergence are obtained.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics, Applied
Jin Zhang, Xiaowei Liu
Summary: A finite element method of any order is used to solve a singularly perturbed convection-diffusion equation in 2D, with exponential boundary layers. The method is applied on a Bakhvalov-type mesh and a uniform convergence of (almost) optimal order is proved using a carefully defined interpolant.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Xiaowei Liu, Jin Zhang
Summary: Recently, Zhang and Liu (2021) [26] proved a supercloseness result in a balanced norm of a finite element method on a Bakhvalov-type mesh for singularly perturbed reaction-diffusion problem. However, a drawback is that a post-processing operator cannot be proposed due to the complexity of the composite interpolation. In this manuscript, two simple interpolations, i.e., the vertices-edges-element interpolation and the Lagrange interpolation based on Gauss-Lobatto points, are provided, which are both superclose to the numerical solution. Moreover, a post-processing operator is proposed based on the supercloseness result of the Lagrange interpolation, taking into account the structure of the Bakhvalov-type mesh. For this operator, a new analysis and a special anisotropic interpolation bound are presented to ensure its stability and approximation properties. It is also proven that the post-processing operator can be utilized to improve the accuracy of the numerical solution.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Jin Zhang, Xiaowei Liu
APPLIED MATHEMATICS AND COMPUTATION
(2020)
Article
Computer Science, Software Engineering
Jin Zhang, Xiaowei Liu
BIT NUMERICAL MATHEMATICS
(2018)