A theoretical study on the rank´s dependence with electric size of the inverse finite element matrix for large-scale electrodynamic analysis

The rank of the inverse finite-element matrix is theoretically studied for 1-D, 2-D, and 3-D electrodynamic problems. We find that the rank of the inverse finite-element matrix is a constant irrespective of electric size for 1-D electrodynamic problems. For 2-D electrodynamic problems, the rank grows very slowly with electric size as square root of the logarithm of the electric size of the problem. For 3-D electrodynamic problems, the rank scales linearly with the electric size. The findings of this work are both theoretically proved and numerically verified. They are applicable to problems with inhomogeneous materials and arbitrarily shaped structures.