Thesis Summary:

In this thesis an a posteriori error analysis of the fully mixed formulations of the evolutionary Stokes and Stokes-Darcy problems is performed. To do this, we first analyze the semi-discrete scheme, which consists of fixing a time and discretizing the space by the method of mixed finite elements, and then the temporal interval is discretized using the regressive Euler method . This last is known as the completely discrete scheme. For the purpose of obtaining the desired estimates, for the two approaches described in the previous paragraph, the technique of the elliptical reconstructions is used, in such a way that the error is divided into two contributions, Which allows to resort to the residues of the associated mixed problem. One of these contributions is an evolutionary problem in a mixed form, so that the estimates are deduced using a standard argument for parabolic problems in terms of the norm of energy and integration in time. This estimate is in terms of the initial conditions and the time derivative of the second component of the incognito. In addition, in the completely discrete scheme, terms that depend on the temporal approximation and the mesh changes over time occur. On the other hand, the second contribution of error is a stationary problem, and therefore the estimates are deduced from what is done in the a posteriori analysis of the associated elliptic problems.