An Efficient ILU Preconditioning for Highly Sparse Matrices Constructed using the FDFD Method

Finding a good preconditioner to solve a given sparse linear system is often considered a difficult but important task. One of the simplest ways of defining a preconditioner is to perform an incomplete LU decomposition (ILU) of the original matrix. The ILU factorization is considered to be an easy and inexpensive preconditioner to use. However, it fails to provide a solution of a sparse linear system generated from general three-dimensional problems with three unknowns. In this paper, a modified ILU preconditioner is proposed to provide an efficient preconditioner for highly sparse matrices, especially for matrices constructed using the finite-difference frequency-domain (FDFD) method for three-dimensional applications. It has been proven that the proposed ILU preconditioner provides a valid solution when the classical ILU preconditioner fails. The efficiency of the proposed ILU preconditioner is also demonstrated by the small memory requirements relative to traditional ILU preconditioners.