Application of the tree-cotree splitting for improving matrix conditioning in the full-wave finite-element analysis of high-speed circuits

In recent years the development of high-speed circuits leads to a demand in accurate and reliable full-wave electromagnetic analysis from high frequencies all the way to DC, and thus brings a challenge to current full-wave techniques. A full-wave solver yields solutions with deteriorated accuracy or even breakdowns at low frequencies due to ill-conditioned system matrices. The same situation also happens when very small elements (compared to the wavelength) exist in the finite-element discretization, which is similar to a low-frequency case [1]. This is actually very common in circuit problems since there are very dense meshes both inside and outside conductors for modeling the conductor loss and the field singularity, respectively. In this case an element may not only be very small, but also have a high aspect ratio, which make a system matrix very ill-conditioned even at not too low frequencies.