AMG-Khandelwal & Visaria

AMG-Khandelwal & Visaria - ME 608 Final Report...

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Khandelwal & Visaria 1 ME 608 Final Report Purdue University Spring, 2006 Algebraic Multigrid Solver for Structured Meshes Somesh Khandelwal Department of Mechanical Engineering Purdue University 585 Purdue Mall W. Lafayette IN 47907 Milan Visaria Department of Mechanical Engineering Purdue University 585 Purdue Mall W. Lafayette IN 47907 ABSTRACT A general, fixed cycle, algebraic multigrid solver based on the additive correction strategy was made. The performance of the solver was demonstrated on a variety of problems. The effect of boundary conditions, anisotropy, and grid aspect ratio on the solver was analyzed. INTRODUCTION The fluid flow, heat transfer and solid mechanics problems can be described by governing equations expressed as partial differential equations (PDE). By using any of the available discretization schemes, these PDE’s can be expressed as a set of algebraic equations for the whole domain. For small number of equations, the direct methods are the best way to solve this set of equations. But problems of practical interest often require the solution of more than 1000 to 10000 equations or so. For these problems the iterative solvers such as Jacobi and Gauss-Siedel are more attractive. Point Gauss- Siedel solvers are particularly important for unstructured grids where there is no equivalent of the line-iterative methods that are commonly used on structured grids. These iterative solvers have a local stencil and cannot feel the effects of neighbors which are computationally far away. Thus although the Gauss-Seidel scheme rapidly removes local (high- frequency) errors in the solution, global (low-frequency) errors are reduced at a rate inversely related to the grid size. Thus, for a large number of nodes, the solver stalls and the residual reduction rate becomes prohibitively low. These global errors however become local errors if the number of neighbors decrease, or in other words if the grid size is increased. This property can be exploited to increase the rate of convergence of an iterative solver by iterations on successively coarser meshes. Multigrid (MG) methods in numerical analysis are a group of algorithms for solving differential equations using a hierarchy of coarse level solutions. The typical application for multigrid is in the numerical solution of elliptic partial differential equations in two or more dimensions. MG can be applied in combination with any of the common discretization techniques. In these cases, multigrid is among the fastest solution techniques known today. In contrast to other methods, multigrid is general in that it can treat arbitrary regions and boundary conditions. It does not depend on the separability of the equations or other special properties of the equation. MG is also directly applicable to more complicated non-symmetric and nonlinear systems of equations, like the Lamé system of elasticity or the Navier-Stokes equations. Multigrid can be generalized in many different ways. It can be applied naturally
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