Numerical Approximations of Problems That Arise in Elasticity

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.