• Title/Summary/Keyword: Multigrid

### Reduction factor of multigrid iterations for elliptic problems

• Kwak, Do-Y.
• Journal of the Korean Mathematical Society
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• v.32 no.1
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• pp.7-15
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• 1995
• Multigrid method has been used widely to solve elliptic problems because of its applicability to many class of problems and fast convergence ([1], [3], [9], [10], [11], [12]). The estimate of convergence rate of multigrid is one of the main objectives of the multigrid analysis ([1], [2], [5], [6], [7], [8]). In many problems, the convergence rate depends on the regularity of the solutions([5], [6], [8]). In this paper, we present an improved estimate of reduction factor of multigrid iteration based on the proof in [6].

### A Study of n Multigrid Finite-Volume Method for Radiation (다중격자 유한체적법에 의한 복사열전달 해석)

• Kim, Man-Young;Do, Young-Byun;Baek, Seung-Wook
• Transactions of the Korean Society of Mechanical Engineers B
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• v.27 no.1
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• pp.135-140
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• 2003
• The convergence of finite volume method (FVM) or discrete ordinate method (DOM) is known to degrade for optical thickness greater than unity and large scattering albedo. The present article presents a convergence acceleration procedure for the FVM based on a full approximation storage (FAS) multigrid method. Among a variety of multigrid cycles, the V-cycle is used and the full multigrid algorithm (FMG) is applied to an analysis of radiation in irregular two-dimensional geometry. Solution convergence is discussed for the several cases of various optical thickness and scattering albedo. At small scattering albedo and optical thickness, there is no advantage to using the multigrid method for calculation CPU time. For large scattering albedo greater than 0.5 and optical thickness greater than unity, however, the multigrid method improves the convergence and the solution is rapidly obtained.

### THE ANANLYSIS OF WILSON'S NONCONFORMING MULTIGRID ALGORITHM FOR SOLVING THE ELASTICITY PROBLEMS

• KANG, KAB SEOK;KWAK, DO YOUNG
• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.1 no.1
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• pp.105-125
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• 1997
• In this paper we consider multigrid algorithms for solving elasticity problems by using Wilson's nonconforming finite element method. We consider two types of intergrid transfer operators which is needed to define the multigrid algorithm and prove convergence of $\mathcal{W}$-cycle mutigrid algorithm and uniform condition number estimates for the variable $\mathcal{V}$-cycle multigrid preconditioner.

### 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.

### 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.

### An Efficient Multigrid Algorithm for the Reactor Eigenvalue Problems

• Cho, Nam-Zin;Lee, Kang-Hyun;Kim, Yong-Hee
• Proceedings of the Korean Nuclear Society Conference
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• pp.27-32
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• 1997
• In this paper, a new multigrid method is developed to solve the reactor eigenvalue problems. The new algorithm can be used in any matrix equation concerned with the eigenvalue problem. The finite difference neutron diffusion problem is considered demonstration of the performance of the new multigrid algorithm. The numerical results show that the new multigrid algorithm works well and requires much shorter (7~10 times) computing time compaired to the production code VENTURE.

### SMOOTHING ANALYSIS IN MULTIGRID METHOD FOR THE LINEAR ELASTICITY FOR MIXED FORMULATION

• KANG, KAB SEOK
• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.5 no.1
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• pp.11-24
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• 2001
• We introduce an assumption about smoothing operator for mixed formulations and show that convergence of Multigrid method for the mixed finite element formulation for the Linear Elasticity. And we show that Richardson and Kaczmarz smoothing satisfy this assumption.

### AN ADAPTIVE MULTIGRID TECHNIQUE FOR OPTION PRICING UNDER THE BLACK-SCHOLES MODEL

• Jeong, Darae;Li, Yibao;Choi, Yongho;Moon, Kyoung-Sook;Kim, Junseok
• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.17 no.4
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• pp.295-306
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• 2013
• In this paper, we consider the adaptive multigrid method for solving the Black-Scholes equation to improve the efficiency of the option pricing. Adaptive meshing is generally regarded as an indispensable tool because of reduction of the computational costs. The Black-Scholes equation is discretized using a Crank-Nicolson scheme on block-structured adaptively refined rectangular meshes. And the resulting discrete equations are solved by a fast solver such as a multigrid method. Numerical simulations are performed to confirm the efficiency of the adaptive multigrid technique. In particular, through the comparison of computational results on adaptively refined mesh and uniform mesh, we show that adaptively refined mesh solver is superior to a standard method.

### Convergence Study of Multigrid Method for K-$\omega$ Turbulence Equations (K-$\omega$ 난류방정식을 위한 다중격자기법의 수렴성 연구)

• Park Soo Hvung;Sung Chun-ho;Kwon Jang Hyuk;Lee Seungsoo
• Journal of computational fluids engineering
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• v.7 no.4
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• pp.19-27
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• 2002
• An efficient implicit multigrid method is presented for the Navier-Stokes and k-ω turbulence equations. Freezing and limiting strategies are applied to improve the robustness and convergence of the multigrid method. The eddy viscosity and strongly nonlinear production terms of turbulence are frozen in the coarser grids by passing down the values without update of them. The turbulence equations together with the Navier-Stokes equations, however, are consecutively solved on the coarser grids in a loosely coupled fashion. A simple limit for k is also introduced to circumvent slow-down of convergence. Numerical results for the unseparated and separated transonic airfoil flows show that all computations converge well without any robustness problem and the computing time is reduced to a factor of about 3 by the present multigrid method.

### 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.