Thesis Summary:

The aim of this thesis is to develop high order mixed finite element discretizations for the numerical solution of partial differential equations arising from continuum mechanics, focusing on scenarios in which our methods contribute to improve the accuracy of the finite element approximation, namely, the treatment of curved domains and the presence of singularities or high gradients of the solution. First, we propose a high order mixed finite element method for steady-state diffusion problems with Dirichlet boundary condition on a curved domain. Our approach is based on approximating the domain by a polyhedral computational subdomain where a high order Galerkin method is considered to compute the solution, and on a transferring technique to approximate the Dirichlet data on the computational boundary. Under suitable hypotheses on the distance between the curved and computational boundaries, and the finite dimensional subspaces, we prove the well-posedness of the resulting Galerkin scheme, and derive the corresponding error estimates, as well. Next, we extend the previous ideas to the Stokes equations in which the pseudostress tensor and the fluid velocity are the only unknowns, whereas the fluid pressure is computed via a postprocessing technique. For the case where the computational boundary is constructed by interpolating the real boundary by a piecewise linear function, we also develop a reliable and quasi-efficient residual-based a posteriori error estimator. Its definition employs a more accurate approximate velocity to achieve the same rate of convergence of the method when the solution is smooth enough. Finally, we present an error analysis of a conforming finite element discretization for a four-field formulation for the stationary Biot's consolidation model in poroelasticity. Assuming standard hypotheses on the discrete spaces, we first prove well-posedness and optimal a priori error estimates of the associated Galerkin scheme. Next, we develop a reliable and efficient residual-based a posteriori error estimator. We show that both the reliability and efficiency estimates are independent of the modulus of dilatation, even in the incompressible limit. For all the problems described above, we provide numerical examples validating the theory.

ISI Publications from the Thesis

Ricardo OYARZúA, Sander RHEBERGEN, Manuel SOLANO, Paulo ZUñIGA: Error analysis of a conforming and locking-free four-field formulation for the stationary Biot’s model. ESAIM: Mathematical Modelling and Numerical Analysis, vol 55, pp. S475-S506, (2021).

Ricardo OYARZúA, Manuel SOLANO, Paulo ZUñIGA: A priori and a posteriori error analyses of a high order unfitted mixed-FEM for Stokes flow. Computer Methods in Applied Mechanics and Engineering, vol. 360, 1, Art. Num. 112780, (2020).

Ricardo OYARZúA, Manuel SOLANO, Paulo ZUñIGA: A high order mixed-FEM for diffusion problems on curved domains. Journal of Scientific Computing, vol. 79, 1, pp. 49-78, (2019).