Computational simulations have led to a deeper understanding of the mechanics of earthquakes and wave propagation. In this research, based on Finite Element solutions, I study two unresolved seismological problems: the influence of fault geometry on the rupture duration of an earthquake, and the effect of soft layers on the strong ground motion in the Valley of Mexico. For the purpose of solving dynamic rupture problem, I develop Split Node and Stress Glut-type methods. Using the new formulation of the Split Node Method, I analyze the effect of one and two kinks in a fault with a slip weakening friction, and conclude that the relationship between geometry and traction is an efficient mechanism to generate slip-rates with raise times that are compatible with kinematic inversions. In order to address the second problem, I propose a lithospheric and geotechnical model of Central Mexico, and investigate the amplification in the amplitude and durations of the soil motions due to strong earthquakes in the Pacific Coast of Mexico. Finally, I show some comparison between the synthetic and the recorded seismograms for the 1995 Copala Earthquake.