A spectral element solution to generate wellconditioned system matrices at any desired frequency

Abstract

Electromagnetic methods have been applied in many advanced applications, such as circuit design, geophysical exploration, medical imaging, and environmental perception. Since it can represent a high-fidelity virtual experiment quite close to the real world, these methods are more reliable than approximate methods. However, as working frequency gets smaller and smaller, the discretized linear system become poorly conditioned and thus difficult to solve. Typical examples include Airborne Transient Electromagnetics (ATEM) systems and Control-Source Electromagnetics (CSEM) Systems, either in time domain or in frequency domain. The purpose of this study is to develop an effective method for solving any desired frequency system especially including these low-frequency electromagnetic models. Here we propose a spectral element method (SEM) with the divergence free constraint and apply it to solve subsurface electromagnetic exploration problem. Because Gauss’ law is now explicitly enforce, making system matrix well-conditioned even at extremely low frequency, we have capability to solve the linear system which cannot be solved correctly before. Several numerical experiments show that the proposed method is accuracy and efficiency.