Spectral-finite element approach to three-dimensional viscoelastic relaxation in a spherical earth

We present a spectral–finite element approach to the forward modelling of the visco-elastic response of a spherical earth with a 3-D viscosity structure to a surface mass load. It represents an alternative to a variety of numerical methods for 2-D and 3-D postglacial rebound modelling used recently (the finite element method, the perturbation method, the semi-analytical approach and the spectral–finite difference method). For a fixed time, the problem is reformulated in a weak sense and parametrized by tensor surface spherical harmonics in the angular direction, whereas piecewise linear finite elements span the radial direction. The solution is obtained with the Galerkin method, which leads to solving a system of linear algebraic equations. The time dependence of the problem is treated directly in the time domain (not in the Laplace domain) as a time evolution problem. The time derivative in the constitutive equation for a Maxwell viscoelastic body is approximated by the explicit Euler time-differencing scheme, which leads to time splitting of the stress tensor. The spectral–finite element method and the associated numerical code have been tested for 2-D (azimuthally symmetric) eccentrically nested spheres models, and good agreement has been obtained.