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Nested Dissection of a Regular Finite Element Mesh

Alan George
SIAM Journal on Numerical Analysis
Vol. 10, No. 2 (Apr., 1973), pp. 345-363
Stable URL: http://www.jstor.org/stable/2156361
Page Count: 19
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Nested Dissection of a Regular Finite Element Mesh
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Abstract

Let M be a mesh consisting of n2 squares called elements, formed by subdividing the unit square (0, 1) × (0, 1) into n2 small squares of side 1/h, and having a node at each of the (n + 1)2 grid points. With M we associate the N × N symmetric positive definite system Ax = b, where N = (n + 1)2, each xi is associated with a node of M, and Aij ≠ 0 if and only if xi and xj are associated with nodes of the same element. If we solve the equations via the standard symmetric factorization of A, then O(n4) arithmetic operations are required if the usual row by row (banded) numbering scheme is used, and the storage required is O(n3). In this paper we present an unusual numbering of the mesh (unknowns) and show that if we avoid operating on zeros, the LDLT factorization of A can be computed using the same standard algorithm in O(n3) arithmetic operations. Furthermore, the storage required is only O(n2 log2 n). Finally, we prove that all orderings of the mesh must yield an operation count of at least O(n3), provided we use the standard factorization algorithm.

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