Thesis Summary:

In the present work the mixed finite element method is used to solve two types of problems in a harmonic regime: the Helmholtz problem arising from the acoustic wave equation and the elastodynamic equation. Both problems have in common that, in performing the respective dual-dual formulation, one of the bilinear forms does not meet the hypotheses of the Babuska-Brezzi theory, specifically one of the terms has an incorrect sign that causes the loss of coercivity. This problem is solved by properly rewriting the bilinear forms and using a result on the compact inclusion of Sobolev spaces, which allows to use the Fredholm alternative to analyze the existence and uniqueness of both the continuous schemes and the associated discrete schemes. In the first chapter we present the theoretical tools that allow to assure the convergence and stability of the Galerkin schemes associated to compact perturbations of invertible operators. In the second chapter the Helmholtz equation with Dirichlet contour conditions is approached, using as an approximation Raviart-Thomas spaces of zero order. Then, the respective a-posteriori error analysis is performed by an estimator based on the global inf-sup condition and a convenient auxiliary function. Finally, in the third chapter we study a problem arising from the elastodynamics with Dirichlet boundary conditions, using part of the PEERS elements as approximation spaces.