Thesis Summary:

In this thesis we introduce and analyze further applications of the Virtual Element Method (VEM), which can be interpreted as an evolution of the Mimetic Finite Differences method offering high order approximation spaces on computational meshes consisting of polygonal/polyhedral elements. In particular, we focus our analysis on mixed discretizations (mixed-VEM) of some nonlinear problems in fluids mechanics. Also, we are interested in developing the tools required to implement adaptive algorithms that are able to take advantage of the flexibility offered by such general meshes. Firstly, we propose and analyze a mixed-VEM for a dual-mixed formulation of a nonlinear Brinkman model of porous media flow. We introduce the main features for the corresponding discrete scheme, which employs an explicit piecewise-polynomial subspace and a virtual element subspace to approximate the gradient of velocity and the pseudostress, respectively. The velocity and the pressure are computed via simple postprocessing formulae. In turn, the associated computable discrete nonlinear operator is defined in terms of the L^2-orthogonal projector onto a suitable space of polynomials, which allows the explicit integration of the terms involving deviatoric tensors that appear in the original setting. The well-posedness of the discrete scheme and the associated a priori error estimates for the virtual element solution, as well as for the fully computable projection of it, are derived. Several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are presented. Next, a mixed-VEM for a pseudostress-velocity formulation of the Navier-Stokes equations is proposed and analyzed. We describe the main VEM ingredients that are required for our discrete analysis, which, besides projectors commonly utilized for related models, include, as the main novelty, the simultaneous use of virtual element subspaces for H^1 and H(div) in order to approximate the velocity and the pseudostress, respectively. The pressure is computed via a postprocessing formula. Then, the discrete bilinear and trilinear forms involved, their main properties and the associated mixed virtual scheme are defined, and the corresponding solvability analysis is performed using appropriate fixed-point arguments. Moreover, Strang-type estimates are applied to derive the a priori error estimates for the two components of the virtual element solution as well as for the fully computable projections of them and the postprocessed pressure. Furthermore, some numerical examples confirming the theoretical error bounds and illustrating the performance of the proposed discrete scheme are reported. Additionally, we extend our study and propose a mixed-VEM for the Boussinesq problem. Here, the main unknows are given by the pseudostress tensor, the velocity and the temperature, and as before, the pressure is computed via postprocessing. The discrete problem is treated employing a mixed-VEM for the scheme associated with the fluid equations in such a way that the pseudostress and the velocity are approximated on virtual element subspaces of H(div) and H^1, respectively, whereas another VEM is proposed to approximate the temperature on a virtual element subspace of H^1. The corresponding solvability analysis is performed by using fixed-point strategies. Further, Strang-type estimates are applied to derive the a priori error estimates for the components of the virtual element solution as well as for the fully computable projections of them and the postprocessed pressure. Finally, we present an a posteriori error analysis for the mixed-VEM applied to second order elliptic equations in divergence form and with mixed boundary conditions. The resulting error estimator is of residual-type. It only depends on quantities directly available from the VEM solution and applies on very general polygonal meshes. Properties of interpolation operators, Helmholtz decompositions, inverse inequalities and localization techniques based on bubble functions are used for the analysis. Then, via the inclusion of a fully local postprocessing of the mixed-VEM solution, we show that the estimator provides a reliable and efficient control on the broken H(div)-norm error between the exact and the postprocessed flux. In the same way, we propose an a posteriori error analysis of a mixed-VEM discretization for the nonlinear Brinkman model described above. For the analysis we make use again of the aforementioned techniques. For both problems, we provide numerical experiments showing the quality of ours adaptive schemes.

ISI Publications from the Thesis

Gabriel N. GATICA, Mauricio MUNAR, Filander A. SEQUEIRA: A mixed virtual element method for the Boussinesq problem on polygonal meshes. Journal of Computational Mathematics, vol. 39, 3, pp. 392-427, (2021).

Mauricio MUNAR, Filander A. SEQUEIRA: A posteriori error analysis of a mixed virtual element method for a nonlinear Brinkman model of porous media flow. Computers & Mathematics with Applications, vol. 80, 5, pp. 1240-1259, (2020).

Gabriel N. GATICA, Mauricio MUNAR, Filander A. SEQUEIRA: A mixed virtual element method for the Navier-Stokes equations. Mathematical Models and Methods in Applied Sciences, vol. 28, 14, pp. 2719-2762, (2018).

Gabriel N. GATICA, Mauricio MUNAR, Filander A. SEQUEIRA: A mixed virtual element method for a nonlinear Brinkman model of porous media flow. Calcolo, vol. 55, 2, article:21, (2018).