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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 ( N  4/3 ) 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.

Hierarchical Cholesky decomposition of sparse matrices arising from curl–curl-equation