Mixed hybrid finite element method in elasticity and poroelasticity - Quinelato, Thiago de Oliveira (2017)
This thesis is focused on the development and analysis of finite dimensional approximations of the equations describing linear elasticity and poroelasticity problems. The approximation strategy is based on mixed hybrid finite element formulations of those problems and the construction of the finite dimensional spaces is guided by several desired properties: continuity of the tractions (conservation of linear momentum), symmetry of the stress tensor (conservation of angular momentum), reduced number of global degrees of freedom, and robustness under mesh distortion. The main difficulty is related with the simultaneous fulfillment of the inf-sup condition and the symmetry of the stress tensor. The last requirement is relaxed, being enforced in the weak sense through the introduction of a Lagrange multiplier. The main contribution is the development and analysis of stable finite dimensional spaces for mixed approximation of linear elasticity and poroelasticity problems on arbitrary quadrilateral meshes. These spaces are capable of providing optimal order convergence of the stress field in the H(div)-norm on meshes of arbitrary quadrilaterals, which is proved by numerical analysis and confirmed by experimentation.