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 t he visco-elastic response of a spherical earth with a 3-D viscosity structu re to a surface mass load. It represents an alternative to a variety of num erical methods for 2-D and 3-D postglacial rebound modelling used recently (the finite element method, the perturbation method, the semi-analytical ap proach and the spectral-finite difference method). For a fixed time, the pr oblem is reformulated in a weak sense and parametrized by tensor surface sp herical harmonics in the angular direction, whereas piecewise linear finite elements span the radial direction. The solution is obtained with the Gale rkin 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 derivativ e in the constitutive equation for a Maxwell viscoelastic body is approxima ted by the explicit Euler time-differencing scheme, which leads to time spl itting of the stress tensor. The spectral-finite element method and the ass ociated numerical code have been tested for 2-D (azimuthally symmetric) ecc entrically nested spheres models, and good agreement has been obtained.