Multigrid method: Difference between revisions

Revision as of 07:03, 2 August 2007

Revision as of 08:57, 20 September 2007

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Assume that one has a differential equation which can be solved approximately (with a given accuracy) on a grid <math>i</math> with a given grid point

density <math>N_i</math>. Assume furthermore that a solution on any grid <math>N_i</math> may be obtained with a given

effort <math>W_i = \rho K N_i</math> from a solution on a coarser grid <math>i+1</math>. withHere, <math>\rho = N_{j+1} / N_j < 1</math> is the ratio of grid points on ``neighboring'' grids and is assumed to be constant throughout the grid hierarchy, and <math>K</math> is some constant modeling the effort of computing the result for one grid point.

density <math>N_{i+1} = \rho N_i</math> (that is, <math>K</math> is not dependent on <math>i</math>).

Using the [[geometric series]], we then find for the effort involved in finding the solution on