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multigrid method for noncoercive parabolic variational inequality

JOURNAL OF INEQUALITIES AND APPLICATIONS 2025年第1期2025卷 1-14页

作者： Bahi, Mostafa Beggas, Mohammed Haiour, Mohamed Boulaaras, Salah Uni EL OUED Fac Exact Sci Dept Math El Oued 39000 Algeria Univ Annaba Fac Sci Dept Math Annaba 23000 Algeria Qassim Univ Coll Sci Dept Math Buraydah 51452 Saudi Arabia

In this article, our work is focused on the proof of the uniform convergence of the multigrid method for parabolic variational inequality with a noncoercive operator and its numerical solution. To discretize the problem, we utilize a finite element scheme for the noncoercive operator and Euler scheme for the time. To obtain the system discretization of our problem, we reformulate the parabolic variational inequality as a Hamilton-Jacobi-Bellman equation. On the smooth grid, we apply the multigrid method as an interior iteration on the linear system. Finally, we provide a proof of the uniform convergence of the multigrid method for parabolic variational inequality with a noncoercive operator, providing a numerical example of this problem.

关键词： multigrid method Parabolic variational inequality Noncoersive operator Hamilton-Jacobi-Bellman equation Numerical methods

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Three-dimensional simulations of double-diffusive convection of nanofluids and conjugate heat transfer in an n-shaped cavity with non-uniform boundary conditions using the multigrid method

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INTERNATIONAL COMMUNICATIONS IN HEAT AND MASS TRANSFER 2025年 162卷

作者： Chou, Yen-De Hwang, Wei-Shien Solovchuk, Maxim Natl Taiwan Univ Dept Engn Sci & Ocean Engn 1Sec 4Roosevelt Rd Taipei 10617 Taiwan Natl Hlth Res Inst Inst Biomed Engn & Nanomed 35 Keyan Rd Miaoli Cty 35053 Taiwan

Nanofluids are a new type of fluid designed to enhance heat transfer. Brownian motion is one of the key mechanisms by which nanofluids enhance heat transfer. In engineering applications involving double-diffusive convection, the temperature and concentration distributions on the surfaces of objects are often non-uniform. The aim of this study is to develop a fast solver to investigate: (1) the effects of non-uniform heating, non-uniform concentration, and Brownian motion on the heat and mass transfer in nanofluids within a threedimensional n-shaped cavity, and (2) the effects of the composition and arrangement of multi-layer solids on the conjugate heat transfer. The results show that the multigrid method can accelerate the computations by a factor of 1000. Compared to uniform heating and uniform concentration, non-uniform heating and non-uniform concentration can enhance the heat transfer rate by 23.73% and the mass transfer rate by 28.04%. The heat transfer rate of the 5-layer solid is 6.91% higher than that of the 3-layer solid. This study provides important guidance for improving heat and mass transfer efficiency, with potential applications in cooling of electronic devices, solar collectors, and chemical reactors.

关键词： Double-diffusive convection Conjugate heat transfer Nanofluid Brownian motion Non-uniform heating multigrid method

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multigrid method for solving convection-diffusion problems with dominant convection

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2009年第1期226卷 77-83页

作者： Muratova, Galina V. Andreeva, Evgeniya M. S Fed Univ Rostov Na Donu Russia

A modification of the multigrid method for the solution of linear algebraic equation systems with a strongly nonsymmetric matrix obtained after difference approximation of the convection-diffusion equation with dominant convection is proposed. Specially created triangular iterative methods have been used as the smoothers of the multigrid method. Some theoretical and numerical results are presented. (c) 2008 Elsevier B.V. All rights reserved.

关键词： Convection-diffusion equation multigrid method Smoothing procedure Triangular iterative method

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multigrid method for the solution of EHL point contact with bio-based oil as lubricants for smooth and rough asperity

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INDUSTRIAL LUBRICATION AND TRIBOLOGY 2018年第4期70卷 599-611页

作者： Awati, Vishwanath B. Naik, Shankar Kumar, Mahesh N. Rani Channamma Univ Dept Math Belagavi India

Purpose The purpose of this paper is to study the elastohydrodynamic lubrication point contact problem with bio-based oil as lubricants for an isothermal case. The simulation of the problem is analyzed on smooth and rough asperity. Design/methodology/approach The modified Reynolds equation is discretized using finite difference and multigrid method with full approximation scheme (FAS), applied for its solution with varying load and speed. Findings This paper traces out the comparison of minimum and central film thickness with the standard formulation of Hamrock and Dowson. The effect of longitudinal roughness on surfaces is investigated by means of numerical simulations. Originality/value The results obtained are comparable with the standard results, and are shown by graphs and tables. Bio-based products bring out an alternative source of lubricant to reduce energy crises.

关键词： Elastohydrodynamic lubrication (EHL) Point contact Bio-based oil Longitudinal roughness multigrid method FAS

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multigrid method for fractional diffusion equations

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JOURNAL OF COMPUTATIONAL PHYSICS 2012年第2期231卷 693-703页

作者： Pang, Hong-Kui Sun, Hai-Wei Univ Macau Dept Math Macau Peoples R China Xuzhou Normal Univ Sch Math Sci Xuzhou 221116 Jiangsu Peoples R China

The fractional diffusion equation is discretized by the implicit finite difference scheme with the shifted Grunwald formula. The scheme is unconditionally stable and the coefficient matrix possesses the Toeplitz-like structure. A multigrid method is proposed to solve the resulting system. Meanwhile, the fast Toeplitz matrix-vector multiplication is utilized to lower the computational cost with only O(N log N) complexity, where N is the number of the grid points. Numerical experiments are given to demonstrate the efficiency of the method. (C) 2011 Elsevier Inc. All rights reserved.

关键词： Fractional diffusion equation Toeplitz matrices Fast Fourier transform multigrid method Damped-Jacobi method

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multigrid method based on the transformation-free HOC scheme on nonuniform grids for 2D convection diffusion problems

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JOURNAL OF COMPUTATIONAL PHYSICS 2011年第10期230卷 4051-4070页

作者： Ge, Yongbin Cao, Fujun Ningxia Univ Inst Appl Math & Mech Ningxia 750021 Peoples R China Inner Mongolia Univ Sci & Technol Sch Math Phys & Biol Engn Baotou Inner Mongolia Peoples R China

In this paper, a multigrid method based on the high order compact (HOC) difference scheme on nonuniform grids, which has been proposed by Kalita et al. [J.C. Kalita, A.K. Dass, D.C. Dalal, A transformation-free HOC scheme for steady convection-diffusion on nonuniform grids, Int. J. Numer. methods Fluids 44 (2004) 33-53], is proposed to solve the two-dimensional (2D) convection diffusion equation. The HOC scheme is not involved in any grid transformation to map the nonuniform grids to uniform grids, consequently, the multigrid method is brand-new for solving the discrete system arising from the difference equation on nonuniform grids. The corresponding multigrid projection and interpolation operators are constructed by the area ratio. Some boundary layer and local singularity problems are used to demonstrate the superiority of the present method. Numerical results show that the multigrid method with the HOC scheme on nonuniform grids almost gets as equally efficient convergence rate as on uniform grids and the computed solution on nonuniform grids retains fourth order accuracy while on uniform grids just gets very poor solution for very steep boundary layer or high local singularity problems. The present method is also applied to solve the 2D incompressible Navier-Stokes equations using the stream function-vorticity formulation and the numerical solutions of the lid-driven cavity flow problem are obtained and compared with solutions available in the literature. (C) 2011 Elsevier Inc. All rights reserved.

关键词： Convection diffusion equation Nonuniform grids multigrid method Boundary layer Local singularity Navier-Stokes equations Stream function-vorticity formulation

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A multigrid method FOR RECIRCULATING-FLOWS

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INTERNATIONAL JOURNAL FOR NUMERICAL methodS IN FLUIDS 1988年第4期8卷 417-440页

作者： SIVALOGANATHAN, S SHAW, GJ Oxford University Computing Laboratory 8–11 Keble Rd Oxford U.K.

AbstractThe use of multigrid methods in complex fluid flow problems is recent and still under development. In this paper we present a multigrid method for the incompressible Navier‐Stokes equations. The distinctive features of the method are the use of a pressure‐correction method as a smoother and a novel continuity‐preserving manner of grid coarsening. The shear‐driven cavity problem is used as a test case to demonstrate the efficiency of the

关键词： multigrid method Pressure correction

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multigrid method and fourth-order compact scheme for 2D Poisson equation with unequal mesh-size discretization

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JOURNAL OF COMPUTATIONAL PHYSICS 2002年第1期179卷 170-179页

作者： Zhang, J Univ Kentucky Dept Comp Sci Lab High Performance Sci Comp & Comp Simulat Lexington KY 40506 USA

A fourth-order compact difference scheme with unequal mesh sizes indifferent coordinate directions is employed to discretize a two-dimensional Poisson equation in a rectangular domain. multigrid methods using a partial semicoarsening strategy and line Gauss-Seidel relaxation are designed to solve the resulting sparse linear systems. Numerical experiments are conducted to test the accuracy of the fourth-order compact difference scheme and to compare it with the standard second-order difference scheme, Convergence behavior of the partial semicoarsening and line Gauss-Seidel relaxation multigrid methods is examined experimentally. (C) 2002 Elsevier Science (USA).

关键词： Poisson equation fourth-order compact scheme unequal mesh size multigrid method

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multigrid method for coupled semilinear elliptic equation

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MATHEMATICAL methodS IN THE APPLIED SCIENCES 2019年第8期42卷 2707-2720页

作者： Xu, Fei Ma, Hongkun Zhai, Jian Beijing Univ Technol Coll Appl Sci Beijing Inst Sci & Engn Comp Beijing Peoples R China Sun Yat Sen Univ Sun Yat Sen Business Sch Guangzhou 510275 Guangdong Peoples R China Natl Res Ctr High Performance Comp Engn Technol Beijing Peoples R China

This paper introduces a kind of multigrid finite element method for the coupled semilinear elliptic equations. Instead of the common way of directly solving the coupled semilinear elliptic problems on some fine spaces, the presented method transforms the solution of the coupled semilinear elliptic problem into a series of solutions of the corresponding decoupled linear boundary value problems on the sequence of multilevel finite element spaces and some coupled semilinear elliptic problems on a very low dimensional space. The decoupled linearized boundary value problems can be solved by some multigrid iterations efficiently. The optimal error estimate and optimal computational work are proved theoretically and demonstrated numerically. Moreover, the requirement of bounded second-order derivatives of the nonlinear term in the existing multigrid method is reduced to a Lipschitz continuous condition in the proposed method.

关键词： adaptive finite element method coupled semilinear elliptic equation multigrid method parallel computing

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multigrid method for ill-conditioned symmetric Toeplitz systems

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 1998年第2期19卷 516-529页

作者： Chan, RH Chang, QS Sun, HW Chinese Univ Hong Kong Dept Math Shatin Hong Kong Acad Sinica Inst Appl Math Beijing 100080 Peoples R China

In this paper, we consider solutions of Toeplitz systems A(n)u = b where the Toeplitz matrices A(n) are generated by nonnegative functions with zeros. Since the matrices A(n) are ill conditioned, the convergence factor of classical iterative methods, such as the damped Jacobi method, will approach one as the size n of the matrices becomes large. Here we propose to solve the systems by the multigrid method. The cost per iteration for the method is of O(n log n) operations. For a class of Toeplitz matrices which includes weakly diagonally dominant Toeplitz matrices, we show that the convergence factor of the two-grid method is uniformly bounded below one independent of n, and the full multigrid method has convergence factor depending only on the number of levels. Numerical results are given to illustrate the rate of convergence.

关键词： multigrid method Toeplitz matrices

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