Spectral-finite element approach to three-dimensional electromagnetic induction in a spherical earth

Abstract

We present a spectral-finite element approach to the forward problem of 3-D global-scale electromagnetic induction in a heterogeneous conducting sphere excited by an external source current. It represents an alternative to a variety of numerical methods for 3-D global-scale electromagnetic induction modelling developed recently (the perturbation expansion approach and the finite element and finite difference schemes). Two possible formulations of electromagnetic induction boundary-value problem are introduced. The boundary data used in the Dirichlet boundary-value problem consist of the horizontal components of the total magnetic induction measured on the Earth's surface, whereas the mixed boundary-value problem makes use of the scalar spherical harmonic expansion coefficients of the normal component of total magnetic induction in a near-space atmosphere. The latter problem is then reformulated in a weak sense and parametrized by vector spherical harmonics in the angular direction, whereas piecewise linear finite elements span the radial direction. The solution is searched for using the Galerkin method, which leads to solving a system of linear algebraic equations. We employ the biconjugate gradient method with preconditioning to solve the Galerkin system numerically. Particular care is devoted to the construction of a preconditioner that stabilizes the solution and speeds up the convergence of iterations. The spectral-finite element method and associated numerical code have been tested for 2-D (azimuthally symmetric) and 3-D (off-axis) eccentrically nested spheres models, and good agreement has been obtained.