For problems involving aseismic ground deformation in seismically active areas, numerical approaches based on the finite element scheme with discontinuity have been developed. We emphasise the utility of such numerical techniques in tackling geodynamics problems. In a viscoelastic half space representing the lithosphere-asthenosphere system, a long strike-slip fault is considered. Under the influence of tectonic forces caused by mantle convection, the fault moves abruptly. The ensuing boundary value issues were solved using a numerical technique based on a finite element scheme with proper boundary conditions that was developed specifically for the task. The numerical algorithms described here can be tweaked to solve more general deformation issues where analytical methods become too complex. Appropriate modifications can also be incorporated in our model in case when the tectonic forces are not anti-symmetric in nature as well as fault plane is not vertical.