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

We present a spectral-finite element approach to the forward problem of 3-D global-scale electromagnetic induction in a heterogeneous conducting spher e 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 bound ary data used in the Dirichlet boundary-value problem consist of the horizo ntal components of the total magnetic induction measured on the Earth's sur face, whereas the mixed boundary-value problem makes use of the scalar sphe rical harmonic expansion coefficients of the normal component of total magn etic induction in a near-space atmosphere. The latter problem is then refor mulated in a weak sense and parametrized by vector spherical harmonics in t he angular direction, whereas piecewise linear finite elements span the rad ial direction. The solution is searched for using the Galerkin method, whic h leads to solving a system of linear algebraic equations. We employ the bi conjugate gradient method with preconditioning to solve the Galerkin system numerically. Particular care is devoted to the construction of a precondit ioner that stabilizes the solution and speeds up the convergence of iterati ons. The spectral-finite element method and associated numerical code have been tested for 2-D (azimuthally symmetric) and 3-D (off-axis) eccentricall y nested spheres models, and good agreement has been obtained.