An ℋ-matrix parametric cascading based fast direct finite-element solver for large-scale circuit extraction

In the electromagnetic analysis of problems with complex irregular geometries and arbitrary inhomogeneity, the finite-element method (FEM) is a favorable choice compared to other computational electromagnetic methods such as finite-difference-based methods and integral-equation-based methods. Although an FEM based analysis generates a sparse system matrix, its solution is highly computationally challenging if the problem size is large. An ℋ-matrix based direct finite element solver was recently developed to solve large-scale electrodynamic problems. The storage requirements and matrix-vector multiplication have been shown to be of complexity O(kNlogN), and the complexity of direct inverse and LU is shown to be O(k²Nlog²N), where k is a small parameter that is adaptively determined based on accuracy requirements, and N is the matrix size. In, a layered ℋ-inverse based direct solver was developed. It takes advantage of the zero blocks in the original FEM matrix, and reduces the storage complexity to O(MlogM) and the time complexity to O(Nlog²M), where M is the number of unknowns in a single layer, which is, in general, orders of magnitude smaller than N. The methods developed in do not make any theoretical approximation or simplification. They are developed for the analysis of general electromagnetic problems. The engineering problem concerned in this work is the analysis of very large scale integrated (VLSI) circuits.