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

Abstract

A new hierarchical renumbering technique for sparse matrices arising from the application of the Finite Element Method (FEM) to three-dimensional Maxwell's equations is presented. It allows the complete Cholesky decomposition of the matrix, which leads to a direct solver of 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.