1. Administrivia: May 20, 2013 Course project progress reports due Wednesday. Reading in Multigrid Tutorial: Chapters 3-4: Multigrid cycles and implementation Chapter 5: Multigrid convergence (next time) Chapter 8: Algebraic multigrid (optional)

2. Complexity of linear solvers 2D3D Sparse Cholesky:O(n 1.5 )O(n 2 ) CG, exact arithmetic: O(n 2 ) CG, no precond: O(n 1.5 )O(n 1.33 ) CG, modified IC: O(n 1.25 )O(n 1.17 ) CG, support trees: O(n 1.20 )O(n 1.31 ) CG, Spielman/Teng: O(n 1+ε ) Multigrid:O(n) n 1/2 n 1/3 Time to solve model problem (Poisson’s equation) on regular mesh

3. Complexity of direct methods n 1/2 n 1/3 2D3D Space (fill): O(n log n)O(n 4/3 ) Time (flops): O(n 3/2 )O(n 2 ) Time and space to solve any problem on any well- shaped finite element mesh

4. Multigrid Find an approximate solution on a coarser mesh, and then improve it on a finer mesh Reduce smooth fine-mesh error by using coarse mesh again Use idea recursively on hierarchy of meshes Solves the model problem in linear time! Often useful when hierarchy of meshes can be built This is just the intuition – lots of theory and technology

5. 5 Irregular mesh: Tapered Tube (Multigrid)