The 3-D spectral-element method (SE) based on Gauss–Lobatto–Legendre (GLL) polynomials is a very efficient and accurate solver for high-frequency computational electromagnetic (EM) modeling. Although the SE method is based on a weighted residual technique similar to the finite-element method (FE), it utilizes polynomial interpolation functions rather than linear interpolation functions. The GLL polynomials have the characteristic of exponential convergence with the order of polynomial that helps improve the modeling accuracy.
We apply the SE method in the frequency range (900 Hz to 56 kHz) to airborne EM modeling. Starting from the vector Maxwell's equations, we use the SE method to establish spatial-discrete forms of 3-D vector Helmholtz equations and adopt GLL integration to calculate the matrix elements. The magnetic field is calculated using Faraday's law. To check the accuracy of the SE algorithm, we compare our results with semi-analytical solutions for a homogeneous half-space and a layered earth model. We further analyze the influence of grids, expansion of outside boundary of model domain, and the order of interpolation polynomial on EM modeling accuracy and reveal that increasing the SE polynomial order can especially improve the accuracy for complex models. Finally, we demonstrate the feasibility of our SE algorithm for modeling an airborne EM system by numerical 3-D experiments.