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Suppose we have a finest level M(1) with N by N points (in 2D)
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Then the k'th coarsest approximation M(k) to this has N/2k by N/2k points
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One way to think about Multigrid is that M(k+1) can form a preconditioner to M(k) so that one can replace natural matrix A(k) by A-1(k+1)A(k)
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A-1(k+1)A(k) is a nicer matrix than A(k) and iterative solvers converge faster as long wavelength terms have been removed
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Basic issue is that A(k) and A(k+1) are of different size so we need prolongation and restriction to map solutions at level k and k+1
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We apply this idea recursively
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