Low Complexity Fast Direct Solution for Multiscale Problems with Nested Equivalence Source Approximation

We propose a multi-level low complexity fast direct solution of electric field integral equations (EFIEs). Far coupling submatrices are compressed by a kernel-independent method with nested equivalence source approximation (NESA). In this work, equivalence sources are constructed at upper levels while skeletons are selected via the adaptive cross approximation (ACA) at the bottom level to further accelerate the computation. To factorize and inverse the system matrix, an elimination matrix is introduced to each group to reduce the far blocks dimension. Then, an LU factorization is performed to reduce the near blocks dimension. This process is repeated recursively from the bottom level to the top. The inverse of the system matrix is then represented as a multiplication of a series of matrices arising from the factorization process. Numerical results verify the accuracy and the linear complexity of our proposed method.