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A new hierarchical renumbering technique for sparse matrices arising from the application of the Finite Element Method (FEM) to the three-dimensional Maxwell's equations is presented. It allows the complete Cholesky decomposition of the matrix, which leads to a direct solver with O(N4/3) memory requirement. In addition, an approximate factorisation, yielding a preconditioner for the matrix, can be constructed. For this, two algorithms using low-rank approximation are presented, which have almost linear arithmetic complexity and memory requirement. The efficiency of the methods is demonstrated on several numerical examples.
Key Words: sparse matrix,; reordering,; hierarchical matrix,; clustering,; approximate Cholesky decomposition
Published Online: 2007-08-15
Published in Print: 2007-05-23
Copyright 2007, Walter de Gruyter