Numerical Approximations of Problems That Arise in Elasticity

<p>Two problems arising from elasticity are investigated in this report. The first one involves the nonstandard mixed finite element formulations of linear elasticity problems for which we demonstrate a necessary and sufficient condition for a subspace where existence and uniqueness of solutions are guaranteed. In a numerical setting, a stable boundary finite element is constructed that improves the approximation of boundary conditions. A numerical example is conducted to show its efficacy. The second problem is a mathematical model that simulates ground motion during an earthquake where dislocation occurs in a thin fault region. We illustrate that, under appropriate scaling, solutions of this problem can be approximated by solutions of a limit problem where the fault region reduces to a surface. In a numerical context, the reduced model eliminates the need to resolve the large deformation in the fault region. A numerical example is presented to exhibit the effectiveness of this strategy.</p>