Article
Mathematics, Applied
M. Brdar, G. Radojev, H-G Roos, Lj Teofanov
Summary: This paper investigates a singularly perturbed convection-diffusion boundary value problem with exponential and characteristic boundary layers. The problem is numerically solved using the FEM and SDFEM method on a graded mesh, showing almost uniform convergence and superconvergence. The use of a graded mesh allows for almost uniform estimates in the SD norm for the SDFEM method, supported by numerical results to validate theoretical bounds.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Multidisciplinary Sciences
Jan Jaskowiec, Jerzy Pamin
Summary: The paper introduces a posteriori error approximation concept based on residuals in the two-dimensional discontinuous Galerkin (DG) method. The approach takes advantage of the hierarchical nature of the basis functions, constructing the error function in an enriched approximation space. A DG method with finite difference (DGFD) is utilized instead of the popular interior penalty approach, enforcing continuity of the approximate solution through finite difference conditions on the mesh skeleton. Various benchmark examples are presented, demonstrating good correlation between error estimation maps and exact errors. The concept is also applied for adaptive hp mesh refinement.
SCIENTIFIC REPORTS
(2023)
Article
Mathematics, Applied
Lixiu Wang, Qian Zhang, Zhimin Zhang
Summary: In this paper, we provide a theoretical justification for the previously observed superconvergence phenomena of the curlcurl-conforming finite elements on rectangular domains. We establish a superconvergence theory for these elements on rectangular meshes and show that the convergence rates are one-order higher than the optimal rates. Numerical experiments are conducted to confirm our theoretical results.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Maria Gabriela Armentano, Ariel L. Lombardi, Cecilia Penessi
Summary: The aim of this paper is to provide robust approximations of singularly perturbed reaction-diffusion equations in two dimensions using finite elements on graded meshes. By appropriately choosing the mesh grading parameter, quasioptimal error estimations for piecewise bilinear elements are obtained using a weighted variational formulation introduced by N. Madden and M. Stynes in Calcolo 58(2) 2021. A supercloseness result is also proven, indicating that the difference between the finite element solution and the Lagrange interpolation of the exact solution, in the weighted balanced norm, is of higher order than the error itself. Numerical examples are presented to demonstrate the good performance of the approach.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Yingying Xie, Daopeng Yin, Liquan Mei
Summary: In this paper, an efficient numerical scheme based on graded meshes is proposed for solving the fractional neutron diffusion equation with delayed neutrons and non-smooth solutions. The method is shown to be effective and convergent through theoretical analysis and numerical experiments.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics, Applied
Xiaobin Liu, Dazhi Zhang, Xiong Meng, Boying Wu
Summary: This paper investigates the superconvergence properties of the local discontinuous Galerkin methods for solving nonlinear convection-diffusion equations in one-dimensional space. The study introduces a new projection and proves superconvergence of order (2k + 1) for cell averages and numerical flux based on elaborate error estimates. Additionally, improvements in convergence orders for errors at Radau points are discussed, and the theoretical findings are confirmed through numerical experiments.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Interdisciplinary Applications
Yabing Wei, Yanmin Zhao, Shujuan Lu, Fenling Wang, Yayun Fu
Summary: In this paper, a numerical approximation is proposed based on the L2-1(sigma) scheme and nonconforming EQ(1)(rot) finite element method (FEM) for solving two-dimensional time-fractional diffusion equations with variable coefficients. A novel and detailed analysis of the equations with an initial singularity is conducted on anisotropic meshes. The fully discrete scheme is proven to be unconditionally stable and achieves optimal second-order accuracy for convergence and superconvergence in both time and space directions. The accuracy of the proposed method is verified through comparison with theoretical analysis.
FRACTAL AND FRACTIONAL
(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
Bin Lan, Haiyan Li, Jianqiang Dong
Summary: We propose a finite volume scheme for 2D steady convection-diffusion equations on arbitrary convex polygonal meshes. The scheme utilizes auxiliary unknowns on the cell-edge and cell-node of the mesh, and computes the continuous flux using a two-point nonlinear flux. The positivity of the scheme is guaranteed by the M-matrix property of the transpose of the coefficient matrix. We also introduce a new strategy to compute the value on the cell-edge without previous reconstruction. Numerical results demonstrate the positivity of our scheme on polygonal meshes for diffusion-dominated and convection-dominated problems, with second-order accuracy.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Yabing Wei, Shujuan Lu, Hu Chen, Yanmin Zhao, Fenling Wang
Summary: In this paper, an unconditionally stable fully discrete numerical scheme for 2D time fractional variable coefficient diffusion equations with non-smooth solutions is constructed and analyzed. The scheme utilizes the L2-1(sigma) scheme for time fractional derivative discretization and anisotropic FEM for spatial discretization, with rigorous proof of unconditional stability and convergence. The error analysis is validated through a numerical example presented in the paper.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
M. Arrutselvi, E. Natarajan, S. Natarajan
Summary: This paper analyzes the virtual element method for the quasilinear convection-diffusion-reaction equation, proving the existence and uniqueness of solution branches of the discrete problem, and verifying theoretical estimates with numerical experiments.
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Petr Knobloch
Summary: This paper investigates algebraic stabilization of finite element discretizations of convection-diffusion-reaction equations and designs a new method called Symmetrized Monotone Upwind-type Algebraically Stabilized (SMUAS) method. It is proved that the SMUAS method preserves linearity and satisfies the discrete maximum principle on arbitrary simplicial meshes. Numerical results indicate that the SMUAS method has optimal convergence rates on general simplicial meshes.
NUMERICAL ALGORITHMS
(2023)
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
Davide Cortellessa, Nicola Ferro, Simona Perotto, Stefano Micheletti
Summary: We propose a new algorithm for designing topologically optimized lightweight structures that meet the minimum compliance requirement. The algorithm improves the computational efficiency of the standard level set formulation by using a customized computational mesh. The algorithm aims to deliver a final layout with a smooth contour and reliable mechanical properties. The effectiveness of the algorithm is confirmed through numerical investigations in both two-and three-dimensional contexts.
APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Mathematics, Applied
Gang Peng
Summary: A new positivity-preserving finite volume scheme is proposed for solving the convection-diffusion equation on distorted meshes in 2D or 3D. The scheme utilizes a nonlinear two-point flux approximation for diffusion flux and a second-order upwind method with a slope limiter for convection flux. The scheme exhibits second-order convergence rate and is effective in solving the convection-diffusion problem based on numerical results.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2022)
