This paper presents a method that modifies commercial engineering-oriented finite element packages for the modelling of Glacial Isostatic Adjustment (GIA) on a self-gravitating, compressible and spherical Earth with 3-D structures. The approach, called the iterative finite element body and surface force (FEMIBSF) approach, solves the equilibrium equation for deformation using the ABAQUS finite element package and calculates potential perturbation consistently with finite element theory, avoiding the use of spherical harmonics. The key to this approach lies in computing the mean external body forces for each finite element within the Earth and pressure on Earth's surface and core-mantle boundary (CMB). These quantities, which drive the deformation and stress perturbation of GIA but are not included in the equation of motion of commercial finite element packages, are implemented therein. The method also demonstrates how to calculate degree-1 deformation directly in the spatial domain and Earth-load system for GIA models. To validate the FEMIBSF method, loading Love numbers (LLNs) for homogeneous and layered earth models are calculated and compared with three independent GIA methodologies: the normal-mode method, the iterative body force method and the spectral-finite element method. Results show that the FEMIBSF method can accurately reproduce the unstable modes for the homogeneous compressible model and agree reasonably well with the Love number results from other methods. It is found that the accuracy of the FEMIBSF method increases with higher resolution, but a non-conformal mesh should be avoided due to creating the so-called hanging nodes. The role of a potential force at the CMB is also studied and found to only affect the long-wavelength surface potential perturbation and deformation in the viscous time regime. In conclusion, the FEMIBSF method is ready for use in realistic GIA studies, with modelled vertical and horizontal displacement rates in a disc load case showing agreement with other two GIA methods within the uncertainty level of GNSS measurements.

Significant land uplift and horizontal motions have been recorded with Global Navigation Satellite Systems (GNSS) in areas such as Alaska, Iceland and the Northern Antarctic Peninsula (NAP) as a result of Glacial Isostatic Adjustment (GIA) due to ice melt after the Little Ice Age. Here, analysis of horizontal displacement rates can be of extra importance, as they are more sensitive to Earth properties in shallower layers than vertical displacement rates. Proper modelling of horizontal displacement rates with dedicated GIA models requires a spherical Earth with compressible rheology. However, in these small areas, the used GIA models are often incompressible using a Cartesian geometry to ease computation and in some cases allow for lateral viscosity changes or more complex rheology. We investigate the validity of modelled horizontal displacement rates using different approximations, that is using spherical or Cartesian Earth structures, and incompressible, material compressible or compressible rheology. Although the lack of self-gravity and sphericity compensate each other in the vertical, this is less the case for the horizontal. For a disc ice sheet with a radius just over 200 km and a thickness of 1000 m, differences due to sphericity are minimal and the modelled horizontal displacement rates of compressible Cartesian models differ from those simulated by a compressible spherical model by 0.63 mm a-1. Thus, compressible Cartesian GIA models can be applied for modelling horizontal displacement rates of small ice sheets like those in Alaska, Iceland and NAP. Unfortunately, the implementation of compressibility in Abaqus that we use here cannot be extended to spherical models as gravity can not be specified for a spherical body. Other modelling approaches are recommended in such cases.