An efficient spectral element method for two-dimensional magnetotelluric modeling

We introduce a new efficient spectral element approach to solve the two-dimensional magnetotelluric forward problem based on Gauss-Lobatto-Legendre polynomials. It combines the high accuracy of the spectral technique and the perfect flexibility of the finite element approach, which can significantly improve the calculation accuracy. This method mainly includes two steps: 1) transforming the boundary value problem in the partial differential form into the variational problem in the integral form and 2) solving large symmetric sparse systems based on the combination of incomplete LU factorization and the double conjugate gradient stability algorithm through the spectral element with quadrilateral meshes. We imply the spectral element method on a resistivity half-space model to obtain a simple analytical solution and find that the magnetic field solutions simulated by the spectral element approach matched closely to the exact solutions. The experiment result shows that the spectral element solution has high accuracy with coarse meshes. We further compare the numerical results of the spectral element, finite difference, and finite element approaches on the COMMEMI 2D-1 and smooth models, respectively. The numerical results of the spectral element procedure are highly consistent with the other two techniques. All these comparison results suggest that the spectral element technique can not only give high accuracy for modeling results but also provide more detailed information. In particular, a few nodes are required in this method relative to the finite difference and finite element methods, which can decrease the relative errors. We then deduce that the spectral element method might be an alternative approach to simulate the magnetotelluric responses in two- or three-dimensional structures.