Article
Engineering, Multidisciplinary
Troy Shilt, Patrick J. O'Hara, Jack J. McNamara
Summary: The article explores the alleviation of spurious oscillations introduced by traditional finite element methods in advection dominated problems through a generalized finite element formulation. This method is demonstrated to effectively capture boundary layer development and provide smooth numerical solutions with improved error levels compared to traditional formulations. Insights into the improvements offered by the generalized finite element method are further illuminated through a consistent decomposition of the variational multiscale method for comparison with classical stabilized methods.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2021)
Article
Mathematics, Applied
Vivek Kumar, Guenter Leugering
Summary: This article focuses on the study of singular perturbed static convection-diffusion equations with varying coefficients on a metric graph G = (V, E). The emphasis is on the convection dominated situation where a small parameter epsilon > 0 appears in front of the diffusion term. The reduced problem in the limit epsilon -> 0 may exhibit boundary layers at multiple vertices and simple nodes. Various scenarios are analyzed and validated in several test cases, using exemplary graphs and an upwind finite difference method on a piece-wise Shishkin mesh. Error estimates are discussed to demonstrate epsilon-uniform convergence. (c) 2023 Elsevier B.V. All rights reserved.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(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
R. Soundararajan, V. Subburayan, Patricia J. Y. Wong
Summary: This article presents a study on 1D parabolic Dirichlet's type differential equations with discontinuous source terms. The time derivative is discretized using the Euler backward method and the streamline-diffusion finite element method is used to solve locally one-dimensional stationary problems on a Shishkin mesh. The proposed method achieves first-order convergence in time and second-order convergence in space. It offers advantages such as more accurate approximations of the solution on the boundary layer region, better efficiency, and robustness in dealing with discontinuous line source terms. Numerical examples demonstrate the effectiveness and efficiency of the method, making it a valuable tool in various fields.
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
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
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
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
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
Engineering, Multidisciplinary
Ram Prasad Yadav, Pratima Rai, Kapil K. Sharma
Summary: This paper presents the nonsymmetric interior penalty Galerkin (NIPG) finite element method for a class of one-dimensional convection dominated diffusion problems with discontinuous coefficients. The solution of the considered class of problem exhibits boundary and interior layers. Piecewise uniform Shishkin-type meshes are used for the spatial discretization. The error estimates in the energy norm have been derived for the proposed schemes. Theoretical results are supported by conducting numerical experiments. It is established that the errors are uniform with respect to the perturbation parameter epsilon. The uniformness of the error estimates with the perturbation parameter epsilon has also been established numerically for L-infinity- norm.
INTERNATIONAL JOURNAL OF COMPUTATIONAL METHODS
(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
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
Computer Science, Interdisciplinary Applications
Pengcong Mu, Xinming Wu, Weiying Zheng
Summary: In this study, a finite element method is proposed to solve the coupled model of classical drift-diffusion equations and Schrodinger-Poisson equations in simulating a resonant tunneling diode. An energy-adaptive algorithm is introduced to accurately compute the coupling coefficient and quantum electron density while reducing the number of nodes. Numerical experiments show that the proposed method based on energy-adaptive grids is robust and converges quickly.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
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
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, Applied
Pengtao Sun, Cheng Wang
APPLIED NUMERICAL MATHEMATICS
(2020)
Article
Physics, Mathematical
Hua Wang, Jinru Chen, Pengtao Sun, Rihui Lan
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2020)
Article
Mathematics, Applied
Rihui Lan, Pengtao Sun
JOURNAL OF SCIENTIFIC COMPUTING
(2020)
Article
Mathematics, Applied
Rihui Lan, Pengtao Sun
Summary: This paper develops a monolithic arbitrary Lagrangian-Eulerian (ALE)-finite element method for a type of moving interface problem with jump coefficients, based on a novel ALE mapping. The stability and error estimate analyses are conducted in the ALE frame, and numerical experiments are carried out to validate theoretical results in various cases. The developed novel ALE-FEM can potentially be extended to solve moving interface problems involving the pore fluid equation or Biot's model in the future.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Pengtao Sun, Chen-Song Zhang, Rihui Lan, Lin Li
Summary: An advanced mixed finite element method was developed for solving a cardiovascular fluid-structure interaction problem with multiple moving interfaces. Numerical experiments were conducted to demonstrate the effectiveness and strength of this method in the cardiovascular environment with multi-interface problems.
JOURNAL OF COMPUTATIONAL SCIENCE
(2021)
Article
Computer Science, Interdisciplinary Applications
Wenrui Hao, Pengtao Sun, Jinchao Xu, Lian Zhang
Summary: In this paper, a multiscale and monolithic arbitrary Lagrangian-Eulerian finite element method (ALE-FEM) is developed for a multiscale hemodynamic fluid-structure interaction (FSI) problem involving an aortic aneurysm growth to predict the long-term aneurysm risk in the cardiovascular environment. Different time scales and methods are employed to handle the complex process, and simulations are conducted to validate the accuracy and efficiency of the developed model.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Mingyan He, Pengtao Sun
Summary: This paper develops a complete mixed finite element method for a modified Poisson-Nernst-Planck/Navier-Stokes (PNP/NS) coupling system, presenting stabilized mixed weak forms and semi- and fully discrete schemes to achieve optimal convergence rates. Numerical experiments validate the theoretical results obtained for the entire modified PNP/NS model.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Physics, Mathematical
Jianhong Chen, Wenrui Hao, Pengtao Sun, Lian Zhang
Summary: The study introduces an efficient blood pressure measurement model that integrates a fluid-structure interaction model with the photoplethysmogram (PPG) signal and develops a data-driven computational approach to fit two optimization parameters in the proposed model for each individual.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2022)
Article
Physics, Mathematical
Cheng Wang, Pengtao Sun
Summary: An augmented Lagrangian Uzawa iterative method is proposed for solving double saddle-point systems with semi definite (2,2) block, with convergence proven under the assumption of a unique solution. The method is also applied to elliptic interface problems, with numerical experiments validating theoretical results and studying the method's performance.
COMMUNICATIONS IN COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Mingyan He, Pengtao Sun, Hui Zhao
Summary: This paper develops both the standard finite element method (FEM) and the mixed FEM to solve the fourth-order modified Poisson-Fermi equation resulting from the BSK theory, for accounting for electrostatic correlations in concentrated electrolytes. Theoretical results are validated by numerical experiments and a practical example, which demonstrate the necessity and accuracy of the modified model over the classical model in describing electrostatic potential fields due to correlation effects.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Mingyan He, Jia Tian, Pengtao Sun, Zhengfang Zhang
Summary: In this paper, an energy-conserving finite element method is developed for a class of nonlinear fourth-order wave equations. The method achieves accurate discretization in both time and space, and maintains the discrete energy constant at each time step. Numerical experiments are conducted to validate the effectiveness of the method, and the spatial and temporal convergence properties of the finite element solution are obtained.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Xingwen Zhu, Mingyan He, Pengtao Sun
Summary: This paper compares the finite element method (FEM) and the mesh-free deep neural network (DNN) approach for solving coupled nonlinear hyperbolic/wave partial differential equations (PDEs). The mesh-free DNN approach is found to be easily applicable to different problems, but it lacks a definite convergence pattern and energy dissipation property compared to the problem-dependent FEM. Numerical experiments validate these findings by comparing the convergent accuracies and energy performances of both approaches.
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING
(2022)
Article
Mathematics, Applied
Ian Kesler, Rihui Lan, Pengtao Sun
Summary: In this paper, a nonconservative arbitrary Lagrangian-Eulerian (ALE) finite element method is developed for a transient Stokes/parabolic moving interface problem with jump coefficients. The stability and optimal convergence properties of both semi- and full discretizations are analyzed in terms of the energy norm, demonstrating its effectiveness for fluid-structure interaction (FSI) problems.
INTERNATIONAL JOURNAL OF NUMERICAL ANALYSIS AND MODELING
(2021)
Article
Mathematics, Applied
Rihui Lan, Michael J. Ramirez, Pengtao Sun
RESULTS IN APPLIED MATHEMATICS
(2020)
Article
Mathematics, Applied
Mingyan He, Pengtao Sun
APPLIED NUMERICAL MATHEMATICS
(2020)
