• Title, Summary, Keyword: multigrid methods

### MULTIGRID METHOD FOR AN ACCURATE SEMI-ANALYTIC FINITE DIFFERENCE SCHEME

• Lee, Jun-S.
• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.7 no.2
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• pp.75-81
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• 2003
• Compact schemes are shown to be effective for a class of problems including convection-diffusion equations when combined with multigrid algorithms [7, 8] and V-cycle convergence is proved[5]. We apply the multigrid algorithm for an semianalytic finite difference scheme, which is desinged to preserve high order accuracy despite of singularities.

### MULTIGRID CONVERGENCE THEORY FOR FINITE ELEMENT/FINITE VOLUME METHOD FOR ELLIPTIC PROBLEMS:A SURVEY

• Kwak, Do-Y.
• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.12 no.2
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• pp.69-79
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• 2008
• Multigrid methods finite element/finite volume methods and their convergence properties are reviewed in a general setting. Some early theoretical results in simple finite element methods in variational setting method are given and extension to nonnested-noninherited forms are presented. Finally, the parallel theory for nonconforming element[13] and for cell centered finite difference methods [15, 23] are discussed.

### A MULTIGRID METHOD FOR AN OPTIMAL CONTROL PROBLEM OF A DIFFUSION-CONVECTION EQUATION

• Baek, Hun-Ki;Kim, Sang-Dong;Lee, Hyung-Chun
• Journal of the Korean Mathematical Society
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• v.47 no.1
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• pp.83-100
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• 2010
• In this article, an optimal control problem associated with convection-diffusion equation is considered. Using Lagrange multiplier, the optimality system is obtained. The derived optimal system becomes coupled, non-symmetric partial differential equations. For discretizations and implementations, the finite element multigrid V-cycle is employed. The convergence analysis of finite element multigrid methods for the derived optimal system is shown. Some numerical simulations are performed.

### MULTIGRID METHODS FOR THE PURE TRACTION PROBLEM OF LINEAR ELASTICITY: FOSLS FORMULATION

• Lee, Chang-Ock
• Communications of the Korean Mathematical Society
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• v.12 no.3
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• pp.813-827
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• 1997
• Multigrid methods for two first-order system least squares (FOSLS) using bilinear finite elements are developed for the pure traction problem of planar linear elasticity. They are two-stage algorithms that first solve for the gradients of displacement, then for the displacement itself. In this paper, concentration is given on solving for the gradients of displacement only. Numerical results show that the convergences are uniform even as the material becomes nearly incompressible. Computations for convergence rates are included.

### EXPERIMENTAL RESULTS OF W-CYCLE MULTIGRID FOR PLANAR LINEAR ELASTICITY

• Yoo, Jae-Chil
• East Asian mathematical journal
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• v.14 no.2
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• pp.399-410
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• 1998
• In [3], Franca and Stenberg developed several Galerkin least squares methods for the solution of the problem of linear elasticity. That work concerned itself only with the error estimates of the method. It did not address the related problem of finding effective methods for the solution of the associated-linear systems. In this work, we present computational experiments of W-cycle multigrid method. Computational experiments show that the convergence is uniform as the parameter, $\nu$, goes to 1/2.

### Preconditioned Multistage Time Stepping for the Multigrid Method (다중 격자 기법을 위한 예조건화된 다단계 시간 전진 기법)

• Kim Yoonsik;Kwon Jang Hyuk
• 한국전산유체공학회:학술대회논문집
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• pp.127-133
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• 2001
• In this paper, the preconditioned multistage time stepping methods which are popular multigrid smoothers is studied for the compressible flow calculations. Fourier analysis on the local time stepping and block-Jacobi preconditioned residual operators is performed using the linearized 2-D Navier-Stokes equations. It fumed out that block-Jacobi preconditioner has better performance in eigenvalue clustering. They are implemented in the 2-D compressible Euler and Wavier-Stokes calculations with multigrid methods to verify that the block-Jacobi preconditioned multistage time stepping shows better performance in convergence acceleration.

### Aggregation multigrid method for schur complement system in FE analysis of continuum elements

• Ko, Jin-Hwan;Lee, Byung Chai
• Structural Engineering and Mechanics
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• v.30 no.4
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• pp.467-480
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• 2008
• An aggregation multigrid method (AMM) is a leading iterative solver in solid mechanics. Recently, AMM is applied for solving Schur Complement system in the FE analysis of shell structures. In this work, an extended application of AMM for solving Schur Complement system in the FE analysis of continuum elements is presented. Further, the performance of the proposed AMM in multiple load cases, which is a challenging problem for an iterative solver, is studied. The proposed method is developed by combining the substructuring and the multigrid methods. The substructuring method avoids factorizing the full-size matrix of an original system and the multigrid method gives near-optimal convergence. This method is demonstrated for the FE analysis of several elastostatic problems. The numerical results show better performance by the proposed method as compared to the preconditioned conjugate gradient method. The smaller computational cost for the iterative procedure of the proposed method gives a good alternative to a direct solver in large systems with multiple load cases.

### Convergence Acceleration Methods for the Multigrid Navier-Stokes Computation (다중 격자 Wavier-Stokes 해석의 수렴성 증진 기법)

• Kim Yoonsik;Kwon Jang Hyuk;Choi Yun Ho;Lee Seungsoo
• Proceedings of the KSME Conference
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• pp.35-38
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• 2002
• The convergence acceleration methods for the compressible Wavier-Stokes equations are studied ,which are multigrid method and implicit preconditioned multistage time stepping method. In this paper, the performance of implicit preconditioning methods are studied for the full-coarsening multigrid methods on the high Reynolds number compressible flow computations. The effect of numerical flux on the convergence are investigated for the inviscid and viscous calculations.

### Comparison of multigrid performance for higher order scheme with 5-point scheme

• Han, Mun. S.;Kwak, Do Y.;Lee, Jun S.
• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.4 no.2
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• pp.135-142
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• 2000
• We consider a multigrid algorithm for higher order finite difference scheme for the Poisson problem on rectangular domain. Several smoothers including Jacobi, Red-black Gauss-Seidel are tested and compared. Since higher order scheme gives much more accurate result then 5-point scheme, one may use small number of levels with higher order scheme and thus the overall cost is reduced quite a lot. The numerical experiment compares the two cases.

### MULTIGRID SOLUTION OF THREE DIMENSIONAL BIHARMONIC EQUATIONS WITH DIRICHLET BOUNDARY CONDITIONS OF SECOND KIND

• Ibrahim, S.A. Hoda;Hassan, Naglaa Ameen
• Journal of applied mathematics & informatics
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• v.30 no.1_2
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• pp.235-244
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• 2012
• In this paper, we solve the three-dimensional biharmonic equation with Dirichlet boundary conditions of second kind using the full multigrid (FMG) algorithm. We derive a finite difference approximations for the biharmonic equation on a 18 point compact stencil. The unknown solution and its second derivatives are carried as unknowns at grid points. In the multigrid methods, we use a fourth order interpolation to producing a new intermediate unknown functions values on a finer grid, and the full weighting restriction operators to calculating the residuals at coarse grid points. A set of test problems gives excellent results.