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l42 - Lecture 42 Algebraic Multigrid Method 1 Last Time...

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1 Lecture 42: Algebraic Multigrid Method

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2 Last Time … Discussed the basics of the multigrid method » Use coarse mesh only to accelerate convergence » Therefore coarse mesh corrections go to zero at convergence » Can choose coarse-level coefficient matrix for convenience Looked particularly at the geometric multigrid method
3 This Time… We will Consider an alternative to the geometric multigrid method called the algebraic multigrid method Develop coarse-level discrete equations Consider agglomeration strategies

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4 Geometric Multigrid Make series of coarse meshes » Nested (by agglomeration) or non-nested (by mesh generation – not preferred) Discretize directly at each mesh level using finite volume or other discretization technique At each level, we must solve an equation set of type Ax = b » Called relaxation sweep » Can use Gauss-Seidel, Jacobi, or even LBL-TDMA Restrict/prolongate between mesh levels
5 Nested Coarse Meshes Does not guarantee convex polyhedra at coarse levels

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6 Non-Convex Polyhedra Centroid can fall outside cell Recall diffusion flux on unstructured mesh: If centroid falls outside, can lead to negative coefficients Not meaningful in any case because centroid value no longer represents value in cell Must find a way to create coarse-level discrete system without direct discretization
7 Algebraic Multigrid Method (AMG) Basic idea the same as for geometric multigrid Primary difference is that coarse-mesh discrete equations are not created by direct discretization Coarse level algebraic equations created by agglomerating fine-level algebraic equations » Not created by agglomerating meshes and then discretizing No mesh generation of any type at coarse levels

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l42 - Lecture 42 Algebraic Multigrid Method 1 Last Time...

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