Article
Mathematics, Applied
L. S. Senthilkumar, Subburayan Veerasamy, Ravi P. Agarwal
Summary: This article presents an asymptotic streamline diffusion finite element method (SDFEM) for singularly perturbed convection-diffusion-type differential difference equations. The method achieves almost second-order or first-order convergence in different norms. The theoretical results are validated through numerical experiments.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
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
D. Avijit, S. Natesan
Summary: This article discusses the residual-based a posteriori error estimation in the standard energy norm for the streamline-diffusion finite element method (SDFEM) for singularly perturbed convection-diffusion equations. The dual-weighted residual (DWR) technique is used to enhance the accuracy of the error estimator. The main contribution of the study is the development of an efficient computable DWR-type robust residual-based a posteriori error bound for the SDFEM.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
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
J. L. Gracia, E. O'Riordan
Summary: In this study, a singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. An analytic function is identified to match the discontinuity in the initial condition and satisfy the homogenous parabolic differential equation associated with the problem. By using an upwind finite difference operator with a layer-adapted mesh, the numerical approximation of the difference between the analytical function and the solution of the parabolic problem is shown to be parameter-uniform, with numerical results illustrating the theoretical error bounds established in the paper.
NUMERICAL ALGORITHMS
(2021)
Article
Mathematics, Applied
J. L. Gracia, E. O'Riordan
Summary: This article investigates a singular perturbed parabolic problem of convection-diffusion type with incompatible inflow boundary and initial conditions. When the coefficients are constant, a set of singular functions is identified to match the incompatibilities in the data and satisfy the associated homogeneous differential equation. In the case of variable coefficients and continuous boundary/initial data, a numerical method is developed with its convergence rate depending on the level of compatibility satisfied by the data. Numerical results are provided to validate the theoretical error bounds for both approaches.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Relja Vulanovic, Thai Anh Nhan
Summary: This study considers the Kellogg-Tsan decomposition of the solution to the linear one-dimensional singularly perturbed convection-diffusion problem and improves it by including the solution of the corresponding reduced problem. The upwind scheme on a modified Shishkin-type mesh is used to approximate the unknown component of the decomposition. It is proved that the error is O(epsilon(ln epsilon)N-2(-1)), where epsilon is the perturbation parameter and N is the number of mesh steps, demonstrating high accuracy of the method through numerical examples.
APPLIED NUMERICAL MATHEMATICS
(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
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
Avijit Das, Srinivasan Natesan
Summary: This article discusses a fully-discrete numerical method for the system of 2D singularly perturbed parabolic convection-diffusion problems. The method combines the implicit Backward-Euler scheme for the temporal derivative and the streamline-diffusion finite element method (SDFEM) for the spatial derivatives. The stability of the method is discussed based on a stabilization parameter, and the theoretical results are validated through numerical experiments.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
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
Computer Science, Interdisciplinary Applications
S. Chandra Sekhara Rao, Abhay Kumar Chaturvedi
Summary: This article investigates a two-dimensional singularly perturbed convection-reaction-diffusion problem with discontinuities, proposing a decomposition of the solution and constructing a finite difference scheme on an appropriate Shishkin mesh. The numerical results support the theoretical conclusions.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2021)
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
D. Avijit, S. Natesan
Summary: This article examines a fully discrete numerical method for solving the system of singularly perturbed parabolic convection-diffusion problem, incorporating layer-adapted mesh and stabilization parameter to ensure stability and accuracy. The numerical experiments validate the theoretical estimates, demonstrating the effectiveness of the method.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2022)
Article
Mathematics, Applied
Ram Shiromani, Vembu Shanthi, J. Vigo-Aguiar
Summary: This paper investigates a class of singularly perturbed 2-D elliptic convection-diffusion partial differential equations with non-smooth convection and source terms. An efficient numerical method is developed to approximate the linear problem, with spatial discretization based on a finite difference scheme. The theoretical outcomes are supported by extensive numerical experiments, including a comparison of accuracy and computational cost of the proposed numerical method.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Xiaowei Liu, Martin Stynes, Jin Zhang
IMA JOURNAL OF NUMERICAL ANALYSIS
(2018)
Article
Mathematics, Applied
Jin Zhang, Xiaowei Liu
APPLIED NUMERICAL MATHEMATICS
(2018)
Article
Mathematics, Applied
Xiaowei Liu, Jin Zhang
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2018)
Article
Mathematics, Applied
Jin Zhang, Xiaowei Liu
JOURNAL OF SCIENTIFIC COMPUTING
(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)