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A matrix arising from the discretization of a 2D or 3D mesh is a really bad example of a "band matrix". See the discussion under "sparse matrix".
Attempting to treat a matrix arising from a 2D square mesh as a "band matrix", for example, will lead to a provably non-optimal factorization (in terms of fill-in and flop count).
--- Tim Davis, Univ. of Florida.
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