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, 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
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
Ram Prasad Yadav, Pratima Rai, Kapil K. Sharma
Summary: This paper presents a non-symmetric interior penalty Galerkin (NIPG) finite element scheme for one-dimensional singularly perturbed reaction-diffusion equations with discontinuous diffusion. The solution of the considered class of problem is known to exhibit boundary and interior layers. Error estimates for the proposed scheme are derived in the energy norm as well as the balanced norm.
COMPUTATIONAL & APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Xiaowei Liu, Min Yang
Summary: This study introduces a balanced norm to accurately reflect the behavior of layers in the finite element method for singularly perturbed reaction-diffusion problems. It proves the convergence of optimal order in the balanced norm for rectangular finite elements, and also in the case of Bakhvalov-Shishkin triangular meshes. Numerical experiments support the theoretical results.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Computer Science, Interdisciplinary Applications
Aditya Kaushik, Vijayant Kumar, Manju Sharma, Nitika Sharma
Summary: This paper introduces a modified graded mesh for solving singularly perturbed reaction-diffusion problems using a recursive generation method. The numerical solution is based on finite element method with polynomials of degree at least p. The parameter uniform convergence of optimal order in epsilon-weighted energy norm is proven. Test examples and comparative analysis with other adaptive meshes validate the effectiveness of the proposed method.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2021)
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
Yanhui Lv, Jin Zhang
Summary: This paper considers the two-parameter singularly perturbed problems and proposes a balanced norm to capture each exponential layer. The uniform convergence and optimal order of the finite element method on a Shishkin mesh are proved using this norm.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
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
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
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
Yue Wang, Xiangyun Meng, Yonghai Li
Summary: This paper presents a finite volume element method (FVEM) on the Shishkin mesh for solving a singularly perturbed reaction-diffusion problem, and establishes the stability of the method in energy norm. Furthermore, optimal error estimate in energy norm is derived under the decomposition of solution. Numerical experiments are provided to confirm the theoretical results.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics
Manikandan Mariappan, Chandru Muthusamy, Higinio Ramos
Summary: This article develops and analyzes a numerical scheme to solve a singularly perturbed parabolic system of n reaction-diffusion equations. The scheme considers m equations with a perturbation parameter and the rest without it. It uses finite difference approximations on a uniform mesh in the temporal variable and a piecewise uniform Shishkin mesh in the spatial variable. Convergence properties and error analyses are derived, and numerical experiments are presented to support the theoretical results.
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, Interdisciplinary Applications
Sekar Elango, Bundit Unyong
Summary: This article investigates the use of two non-uniform meshes in the finite difference method for solving singularly perturbed mixed-delay differential equations. The multiplication of the second-order derivative by a small parameter creates boundary layers at x=0 and x=3, as well as strong interior layers at x=1 and x=2 due to the delay terms. It is proven that the method has almost first-order convergence on the Shishkin mesh and first-order convergence on the Bakhvalov-Shishkin mesh. Error estimates are derived in the discrete maximum norm, and practical examples are provided for validation.
FRACTAL AND FRACTIONAL
(2023)
