Thesis Summary:

This dissertation deals with diverse mathematical and numerical aspects of new mixed finite element methods and hybridized discontinuous Galerkin schemes, based on the introduction of pseudostress auxiliary variables, for analyzing nonlinear and transmission problems governed by systems of partial differential equations arising in continuum mechanics. Firstly, we present the a priori and a posteriori error analyses of a non-standard mixed finite element method for the linear elasticity problem with non-homogeneous Dirichlet boundary conditions, which does not require symmetric tensor spaces in the finite element discretization. Here, physical quantities such as the stress, the strain tensor of small deformations, and the rotation, are computed through a simple postprocessing in terms of the pseudostress variable. Furthermore, we also introduce a second element-by-element postprocessing formula for the stress, which yields an optimally convergent approximation of this unknown with respect to the broken H(div)-norm. A reliable and efficient residual-based a posteriori error estimator for this problem, is also provided. Next, we introduce and analyze an augmented mixed finite element method for the two-dimensional nonlinear Brinkman model of porous media flow with mixed boundary conditions. Here, we employ a dual-mixed formulation in which the main unknowns are given by the gradient of the velocity and the pseudostress. In this way, the original unknowns corresponding to the velocity and pressure are easily recovered through a simple postprocessing. We apply known results from nonlinear functional analysis to prove that the corresponding continuous and discrete schemes are both well-posed. In addition, we derive a reliable and efficient residual-based a posteriori error estimator for this nonlinear system. On the other hand, we apply the hybridizable discontinuous Galerkin (HDG) method for numerically solving a class of nonlinear Stokes models arising in quasi-Newtonian fluids. We use the incompressibility condition to eliminate the pressure, and set the velocity gradient as an auxiliary unknown. Then, we enrich the HDG formulation with two suitable augmented equations, which allows us to apply a nonlinear version of Babuška-Brezzi theory and the classical Banach fixed-point theorem to show that the discrete scheme is well-posed, yielding in turn the derivation of the corresponding a priori error estimates. In addition, a second approach for this problem is also considered. For this new version, the main features of the aforementioned augmented formulation are maintained, but after introducing slight modifications of the finite element subspaces for the pseudostress and velocity, we are able to significantly improve our previous analyses and results. More precisely, on one hand we omit the utilization of any fixed-point argument and related parameters to establish the well-posedness of the discrete scheme, and on the other hand we now prove optimally convergent approximations for all the unknowns. Furthermore, we develop a reliable and efficient residual-based a posteriori error estimator, and propose the associated adaptive algorithm for our HDG approximation of the nonlinear model problem. Additionally, we present an HDG method for the coupling of fluid flow with porous media flow. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. We consider a fully-mixed formulation in which the main unknowns are given by the stress, the vorticity, the velocity, and the trace of the velocity, (all of them in the fluid), along with the velocity, the pressure, and the trace of the pressure in the porous medium. In addition, we enhance the finite dimensional subspace for the stress, in order to obtain optimally convergent approximations for all unknowns, as well as a superconvergent approximation of the trace variables. To do that, as in previous papers dealing with the development of a priori error estimates for HDG methods, we use the projection-based error analysis in order to simplify the corresponding study. Finally, we close this thesis by providing H(div) conforming and discontinuous Galerkin (DG) methods for the incompressible Euler equation in two and three dimensions. More precisely, we consider velocity-pressure formulations in which the main goal is prove the L2-stability of each scheme, along to the local conservative properties for the DG methods. Once we have developed H(div) conforming methods, it guides us in designing DG methods using the post-processing idea employed in previous papers. In addition, we obtain a priori error estimates for both the semi-discrete and fully-discrete methods using Backward Euler time stepping. In all the cases, we consider central and upwind fluxes. For all the situations described above, several numerical experiments illustrating the correct performance of the methods, and confirming the theoretical results, are reported.

ISI Publications from the Thesis

Johnny GUZMAN, Filander A. SEQUEIRA, Chi - Wang SHU: H(div) conforming and DG methods for the incompressible Euler equations. IMA Journal of Numerical Analysis, vol. 37, 4, pp. 1733-1771, (2017).

Gabriel N. GATICA, Filander A. SEQUEIRA: Analysis of the HDG method for the Stokes-Darcy coupling. Numerical Methods for Partial Differential Equations, vol. 33, 3, pp. 885-917, (2017).

Gabriel N. GATICA, Filander A. SEQUEIRA: A priori and a posteriori error analyses of an augmented HDG method for a class of quasi-Newtonian Stokes flows. Journal of Scientific Computing, vol. 69, 3, pp. 1192-1250, (2016).

Gabriel N. GATICA, Luis F. GATICA, Filander A. SEQUEIRA: A priori and a posteriori error analyses of a pseudostress-based mixed formulation for linear elasticity. Computers & Mathematics with Applications, vol. 71, 2, pp. 585-614, (2016).

Gabriel N. GATICA, Filander A. SEQUEIRA: Analysis of an augmented HDG method for a class of quasi-Newtonian Stokes flows. Journal of Scientific Computing, vol. 65, 3, pp. 1270-1308, (2015).

Gabriel N. GATICA, Luis F. GATICA, Filander A. SEQUEIRA: A RT_k - P_k approximation for linear elasticity yielding a broken H(div) convergent postprocessed stress. Applied Mathematics Letters, vol. 49, pp. 133-140, (2015).

Gabriel N. GATICA, Luis F. GATICA, Filander A. SEQUEIRA: Analysis of an augmented pseudostress-based mixed formulation for a nonlinear Brinkman model of porous media flow. Computer Methods in Applied Mechanics and Engineering, vol. 289, 1, pp. 104-130, (2015).

Other Publications (ISI)

Luis F. GATICA, Filander A. SEQUEIRA: A priori and a posteriori error analyses of an HDG method for the Brinkman problem. Computers & Mathematics with Applications, vol. 75, 4, pp. 1191-1212, (2018).

Ernesto CáCERES, Gabriel N. GATICA, Filander A. SEQUEIRA: A mixed virtual element method for the Brinkman problem. Mathematical Models and Methods in Applied Sciences, vol. 27, 4, pp. 707-743, (2017).