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A higher-order uniformly convergent defect correction method for singularly perturbed convection-diffusion problems on an adaptive mesh

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ALEXANDRIA ENGINEERING JOURNAL 2022年第12期61卷 9911-9920页

作者： Kaushik, Aditya Choudhary, Monika Delhi Technol Univ Dept Appl Math Delhi 110042 India Panjab Univ Univ Inst Engn & Technol Chandigarh 160014 India

This paper presents a defect correction scheme based on finite-difference discretization for a singularly perturbed convection-dominated diffusion problem. The solution of this class of problems exhibits a multiscale character. There are narrow regions in which the solution grows exponentially and reveals layer behaviour. The defect correction method that we propose improves the efficiency of a numerical solution through iterative improvement and generates a stable higher-order method over an adaptive non-uniform polynomial-Shishkin mesh. An extensive theoretical analysis is presented, which establishes that the method is second-order uniformly convergent and highly stable. The convergence obtained is optimal because it is free from any logarithmic term. The numerical result for two model problems is presented, agreeing with the theoretical estimates. Furthermore, we compare the results with those of other non-uniform mesh found in the literature. (C) 2022 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University

关键词： Singular perturbation Upwind method Finite difference Non-uniform mesh defect-correction method

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A new two-level defect-correction method for the steady Navier-Stokes equations

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2021年 381卷 113009-113009页

作者： Shang, Yueqiang Southwest Univ Sch Math & Stat Chongqing 400715 Peoples R China

A new defect-correction method based on subgrid stabilization for the simulation of steady incompressible Navier-Stokes equations with high Reynolds numbers is proposed and studied. This method uses a two-grid finite element discretization strategy and consists of three steps: in the first step, a small nonlinear coarse mesh system is solved, and then, in the following two steps, two Newton-linearized fine mesh problems which have the same stiffness matrices with only different right-hand sides are solved. The nonlinear coarse mesh system incorporates an artificial viscosity term into the Navier-Stokes system as a stabilizing factor, making the nonlinear system easier to resolve. While the linear fine mesh problems are stabilized by a subgrid model defined by an elliptic projection into lower-order finite element spaces for the velocity. Error bounds of the approximate solutions are estimated. Algorithmic parameter scalings are derived from the analysis. Effectiveness of the proposed method is also illustrated by some numerical results. (C) 2020 Elsevier B.V. All rights reserved.

关键词： Navier-Stokes equations Finite element method Subgrid stabilization method defect-correction method Two-grid method

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A new defect-correction method for the stationary Navier-Stokes equations based on pressure projection

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MATHEMATICAL methodS IN THE APPLIED SCIENCES 2018年第1期41卷 250-260页

作者： Wen, Juan He, Yinnian Xian Univ Technol Sch Sci Xian 710048 Shaanxi Peoples R China Xi An Jiao Tong Univ Ctr Computat Geosci Sch Math & Stat Xian 710049 Peoples R China

A new defect-correction method based on the pressure projection for the stationary Navier-Stokes equations is proposed in this paper. A local stabilized technique based on the pressure projection is used in both defect step and correction step. The stability and convergence of this new method is analyzed detailedly. Finally, numerical examples confirm our theory analysis and validate high efficiency and good stability of this new method.

关键词： defect-correction method error analysis pressure projection the Navier-Stokes equations

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Parallel defect-correction methods for incompressible flows with friction boundary conditions

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COMPUTERS & FLUIDS 2023年 251卷

作者： Zheng, Bo Shang, Yueqiang Southwest Univ Sch Math & Stat Chongqing 400715 Peoples R China

We study three parallel defect-correction methods based on finite element approximations for the incompress-ible Navier-Stokes problem with friction boundary conditions and high Reynolds numbers in this work, where a fully overlapping domain decomposition is considered for parallelization. In the proposed methods, with a global multiscale grid that builds a fine grid around its own subdomain and coarse elsewhere, we iteratively solve an artificial viscosity nonlinear variational inequality problem in a defect step, and then compute the residual by the linearized variational inequality problems in the r-step corrections. The studied methods are easy to implement on the basis of the existing Navier-Stokes solver and possess less communication complexity. We provide a rigorously theoretical derivation for the error estimates of the one-step correction solutions from the proposed methods under some stable conditions, and derive scalings of the algorithmic parameters. We demonstrate by a series of numerical experiments that the velocity and pressure errors computed by our parallel defect-correction methods are comparable to those of the standard defect-correction method, while our present methods reduce the computational cost.

关键词： Incompressible Navier-Stokes problem Friction boundary conditions defect-correction method Finite element Parallelization High Reynolds numbers

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Two-Level defect-correction Stabilized Finite Element method for the Incompressible Navier-Stokes Equations Based on Pressure Projection

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INTERNATIONAL JOURNAL OF COMPUTATIONAL methodS 2021年第8期18卷

作者： Wen, Juan Wang, Yu Xi An Jiao Tong Univ Sch Energy & Power Key Lab Thermofluid Sci & Engn Minist Educ Xian 710049 Shaanxi Peoples R China Xian Univ Technol Sch Sci Xian 710048 Shaanxi Peoples R China

In this paper, we propose the two-level defect-correction stabilized finite element method based on pressure projection for solving the incompressible Navier-Stokes equations. The new method combines the two-level method and the defect-correction strategy with the stabilized method based on pressure projection. It has the good properties of these three methods, such as high efficiency, good stability and saving the computing time. The stability and convergence of this new method are analyzed. Finally, numerical example confirms our theory analysis.

关键词： Two-level defect-correction method pressure projection error analysis the Navier-Stokes equations

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Deferred defect-correction finite element method for the Darcy-Brinkman model

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ZAMM-ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND MECHANIK 2021年第11期101卷

作者： Zeng, Yunhua Huang, Pengzhan He, Yinnian Xinjiang Univ Coll Math & Syst Sci Urumqi 830046 Peoples R China Xi An Jiao Tong Univ Sch Math & Stat Xian 710049 Peoples R China

This paper proposes a deferred defect-correction method for the time-dependent nonlinear Darcy-Brinkman model based on the mixed finite element method. The presented method combines the deferred correction strategy with the defect-correction method and involves two steps. In the first step, a defect Darcy-Brinkman model is solved. Then in the second step, a correction Darcy-Brinkman model based on the deferred correction approach is solved. Moreover, unconditional stability and convergence of the presented method are deduced. Finally, several numerical examples are given to support the theoretical analysis.

关键词： Darcy-Brinkman model defect-correction method deferred correction method finite element method

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A three-step defect-correction algorithm for incompressible flows with friction boundary conditions

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NUMERICAL ALGORITHMS 2022年第4期91卷 1483-1510页

作者： Zheng, Bo Shang, Yueqiang Southwest Univ Sch Math & Stat Chongqing 400715 Peoples R China

Based on finite element discretization and a recent variational multiscale-stabilized method, we propose a three-step defect-correction algorithm for solving the stationary incompressible Navier-Stokes equations with large Reynolds numbers, where nonlinear slip boundary conditions of friction type are considered. This proposed algorithm consists of solving one nonlinear Navier-Stokes type variational inequality problem on a coarse grid in a defect step, and solving two stabilized and linearized Navier-Stokes type variational inequality problems which have the same stiffness matrices with only different right-hand sides on a fine grid in correction steps. In the defect step, an artificial viscosity term is used as a stabilizing factor, making the nonlinear system easier to solve. Error bounds of the approximate solutions in L-2 norms for the velocity gradient and pressure are estimated. Scalings of the algorithmic parameters are derived. Some numerical results are given to support the theoretical predictions and test the validity of the present algorithm.

关键词： Incompressible Navier-Stokes equations Nonlinear slip boundary conditions Finite element Three-step method defect-correction method Variational multiscale method

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Two-level defect-correction stabilized algorithms for the simulation of 2D/3D steady Navier-Stokes equations with damping

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APPLIED NUMERICAL MATHEMATICS 2021年 163卷 182-203页

作者： Zheng, Bo Shang, Yueqiang Southwest Univ Sch Math & Stat Chongqing 400715 Peoples R China

By combining the defect-correction method with the two-level discretization strategy and the local pressure projection stabilized method, this paper presents and studies two kinds of two-level defect-correction stabilized algorithms for the simulation of 2D/3D steady Navier-Stokes equations with damping, where the lowest equal-order P-1 - P-1 finite elements are used for the velocity and pressure approximations. In the proposed algorithms, an artificial viscosity stabilized nonlinear Navier-Stokes problem with damping is first solved in the coarse grid defect step, and then corrections are computed in the fine grid correction step by solving a linear problem based on Oseen-type and Newton-type iterations, respectively. Under the uniqueness condition, stability of the proposed algorithms is analyzed, and optimal error estimates of the approximate solutions are deduced. The correctness of the theoretical predictions and the effectiveness of the proposed algorithms are illustrated by some 2D and 3D numerical results. (C) 2021 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Navier-Stokes equations with damping Two-level method defect-correction method Stabilized finite element method Stability and error estimates

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A three-step defect-correction stabilized algorithm for incompressible flows with non-homogeneous Dirichlet boundary conditions

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2024年第1期50卷 3-3页

作者： Zheng, Bo Shang, Yueqiang Southwest Univ Sch Math & Stat Chongqing 400715 Peoples R China

Based on two-grid discretizations and quadratic equal-order finite elements for the velocity and pressure approximations, we develop a three-step defect-correction stabilized algorithm for the incompressible Navier-Stokes equations, where non-homogeneous Dirichlet boundary conditions are considered and high Reynolds numbers are allowed. In this developed algorithm, we first solve an artificial viscosity stabilized nonlinear problem on a coarse grid in a defect step and then correct the resulting residual by solving two stabilized and linearized problems on a fine grid in correction steps. While the fine grid correction problems have the same stiffness matrices with only different right-hand sides. We use a variational multiscale method to stabilize the system, making the algorithm has a broad range of potential applications in the simulation of high Reynolds number flows. Under the weak uniqueness condition, we give a stability analysis of the present algorithm, analyze the error bounds of the approximate solutions, and derive the algorithmic parameter scalings. Finally, we perform a series of numerical examples to demonstrate the promise of the proposed algorithm.

关键词： Navier-Stokes equations Stabilized finite element method Two-grid method defect-correction method Variational multiscale method

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defect-correction finite element method based on Crank-Nicolson extrapolation scheme for the transient conduction-convection problem with high Reynolds number

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INTERNATIONAL COMMUNICATIONS IN HEAT AND MASS TRANSFER 2017年 81卷 229-249页

作者： Su, Haiyan Feng, Xinlong He, Yinnian Xinjiang Univ Coll Math & Syst Sci Urumqi 830046 Peoples R China Xi An Jiao Tong Univ Ctr Computat Geosci Sch Math & Stat Xian 710049 Peoples R China

A defect-correction finite element (FE) method is designed and analyzed for solving the two-dimensional (2D) transient conduction-convection problem at high Reynolds number. The method combines the merits of Crank-Nicolson (CN) extrapolation discretization and defect-correction scheme, which consists of solving a linearized problem with an added artificial viscosity term and then correcting the previous numerical solutions by a linearized defect-correction technique. The stability and optimal error estimate of the fully discrete scheme are derived. Finally, performance of the proposed method is investigated by numerical experiments. (C) 2016 Elsevier Ltd. All rights reserved.

关键词： Transient conduction-convection problem defect-correction method Crank-Nicolson extrapolation Finite element method High Reynolds number Error estimate

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