Thesis Summary:

In this thesis work we introduce and analyze a virtual element method (VEM) for a variational formulation of the Poisson problem, and a mixed virtual element method (MVEM) for a mixed variational formulation of the Darcy problem and Stokes problem, respectively. The novelty here constitutes the analysis of a variational formulation of the Stokes problem, since the other two problems were already analyzed in other works. However, further details on the construction of the method and the proofs of the associated results of such problems are provided. Therefore, we consider a non-standard mixed approach for the Stokes problem in which the velocity, the pressure and the pseudostress tensor are the main unknowns. However, the pressure shall be eliminated from the original equations, thus yielding an equivalent formulation in which the velocity and the pseudostress tensor are the only unknowns. We then define the virtual finite element subspaces to be employed, introduce the associated interpolation operators, and provide the respective approximation properties. In particular, the latter includes the estimation of the interpolation error for the pseudostress variable measured in the H(div)-norm. Next, and in order to define calculable discrete bilinear forms, we propose a new local projector onto a suitable space of polynomials, which takes into account the main features of the continuous solution and allows the explicit integration of the terms involving the deviatoric tensors. The uniform boundedness of the resulting family of local projectors and its approximation properties are also established. In addition, we show that the global discrete bilinear forms satisfy all the hypotheses required by the Babuska-Brezzi theory. In this way, we conclude the well-posedness of the actual Galerkin scheme and derive the associated a priori error estimates for the virtual solution as well as for the fully computable projection of it. Finally, several numerical examples illustrating the good performance of the method and confirming the theoretical rates of convergence are presented.

ISI Publications from the Thesis

Ernesto CáCERES, Gabriel N. GATICA, Filander A. SEQUEIRA: A mixed virtual element method for quasi-Newtonian Stokes flows. SIAM Journal on Numerical Analysis, vol. 56, 1, pp. 317-343, (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).

Ernesto CáCERES, Gabriel N. GATICA: A mixed virtual element method for the pseudostress-velocity formulation of the Stokes problem. IMA Journal of Numerical Analysis, vol. 37, 1, pp. 296-331, (2017).