Thesis Summary:

In this thesis we study mathematical and numerical aspects of adaptive finite element methods with applications to geosciences. First we develop an a posteriori error estimator of hierarchical type for the Local Projection Stabilized (LPS) finite element method, applied to the incompressible Navier-Stokes equations. To approach the error, the technique uses the solution of local problems posed on appropriate finite dimensional spaces of bubble-like functions. Next we propose and analyze a residual a posteriori error estimator for the Multiscale Hybrid-Mixed (MHM) method for the Stokes and Brinkman equations. The error estimator relies on the multi-level structure of the MHM method and considers the two levels of approximation of the method. As a result, the error estimator accounts for a first-level global estimator defined on the skeleton of the partition and second-level contributions from element-wise approximations. The analysis establishes local efficiency and reliability of the complete multiscale estimator. Also, it yields a new face-adaptive strategy on the mesh's skeleton which avoids changing the topology of the global mesh. Specially designed to work on multiscale problems, the present estimator can leverage parallel computers since local error estimators are independent of each other. Finally, we consider a non-linear Stokes problem that models the behavior of a glacier. The nonlinearity of the problem is due to the relationship between the viscosity of the fluid and its velocity, which in this context is given by Glen's law. We propose a MHM scheme to solve the non-linear problem inspired on the MHM scheme for the linear Stokes problem previously studied. For the situations described above, we report several numerical results which assess the performance of the method and confirm theoretical results.

ISI Publications from the Thesis

Rodolfo ARAYA, Ramiro REBOLLEDO, Frederic VALENTIN: On a multiscale a posteriori error estimator for the Stokes and Brinkman equations. IMA Journal of Numerical Analysis, vol. 41, 1, pp. 344-380, (2021).

Rodolfo ARAYA, Ramiro REBOLLEDO: An a posteriori error estimator for a LPS method for Navier-Stokes equations. Applied Numerical Mathematics, vol. 127, pp. 179-195, (2018).