Thesis Summary:

The main objective of this doctoral thesis is the mathematical and numerical analysis of the application of virtual element methods in general polygonal meshes to solve diverse eigenvalue problems and bending of moderately thick plates, with the aim of making an original contribution and enriching our understanding of virtual element methods. In particular, the Steklov eigenvalue and acoustic vibration problems are considered. In the first case an a posteriori error estimator and an adaptive process based on the estimator are proposed. The thesis also proposes and explores a virtual element method to solve the problem of plate bending modeled by Reissner-Mindlin equations. In particular, a virtual element method is proposed for a formulation written in terms of shear strain and deflection variables, we shown that it does not suffer from locking. In relation to the Steklov eigenvalue problem, the study is focused on developing a virtual element method appropriate for the study of the numerical approximation of the eigenvalues of the problem. Discretization by virtual elements, as presented in [L. Beir˜ao da Veiga et al., Basic principles of virtual element methods, Math. Models Methods Appl. Sci., 23 (2013), pp. 199–214] is proposed to carry out this analysis. Under the standard assumptions on the computational domain, the resulting scheme provides a correct approximation of the spectrum and the error estimations are of optimal order for the eigenfunctions and eigenvalues. Better order estimations are also shown for calculating the eigenfunctions errors in the free boundary, which with some Steklov problems (for example, calculation of sloshing modes) are a quantity of great interest (the free surface of a fluid). All the estimations obtained in the theoretical analysis are corroborated with numerical examples. The second part of the thesis proposes and develops the mathematical and numerical analysis by virtual elements of a residual-type a posteriori error estimator to approximate the Steklov eigenvalue problem in two dimensions. Given that normal fluxes of the virtual solution presented in the first part cannot be calculated explicitly, they are replaced with an appropriate projection, As a consequence of this substitution, new terms appear in the estimator that represent the virtual inconsistency of the virtual element methods. In this way a residual-type a posteriori error estimator is obtained that is completely computable, given that it depends solely on quantities available based on the virtual solution. It is established that the estimator is equivalent to the error except for higher-order terms. Finally, the error estimator is used to guide the refinement of adaptive meshes in a series of test problems with different regularities of the exact solution. The third part of the thesis addresses the mathematical and numerical analysis of the approximation by virtual elements of the acoustic vibration problem. A variational formulation of the spectral problem in terms of fluid displacements is considered. Based on [L. Beir˜ao da Veiga et al., Mixed virtual element methods for general second order elliptic problems on polygonal meshes, ESAIM Math. Model. Numer. Anal., DOI: http://dx.doi.org/10.1051/m2an/2015067 (2016)], virtual discretization of H(div) with a null rotor is proposed. Under standard assumptions of the domain, the resulting scheme provides a correct approximation of the spectrum and the error estimations are optimal for the eigenfunctions and eigenvalues. With this end, we prove approximation properties of the proposed virtual element method. Finally, numerical experiments are presented to corroborate the theoretical results obtained. Finally, a virtual element method is studied for the Reissner-Mindlin plate bending problem, beginning with a variational formulation written in terms of shear strain and deflection variables. A conforming discrete formulation is proposed in [H1(Ω)]2 and H2 (Ω) for shear strain and deflection, respectively. A distinct characteristic of this approach is the direct approximation of shear strain. The rotations are obtained with a simple post-process treatment from shear strains and deflections. The estimation errors prove to be optimal for all the variables involved (in the natural norms of the adopted formulation), with constants independent of the thickness of the plate. Finally, numerical experiments are presented to evaluate the performance of the method, showing that it is convergent and locking-free as the theory predicts.

ISI Publications from the Thesis

Lourenco BEIRAO-DA-VEIGA, David MORA, Gonzalo RIVERA: Virtual elements for a shear-deflection formulation of Reissner-Mindlin plates. Mathematics of Computation, vol. 88, 315, pp. 149-178, (2019).

David MORA, Gonzalo RIVERA, Iván VELáSQUEZ: A virtual element method for the vibration problem of Kirchhoff plates. ESAIM: Mathematical Modelling and Numerical Analysis, vol. 52, 4, pp. 1437-1456, (2018).

David MORA, Gonzalo RIVERA, Rodolfo RODRíGUEZ: A posteriori error estimates for a virtual elements method for the Steklov eigenvalue problem. Computers & Mathematics with Applications, vol. 74, 9, pp. 2172-2190, (2017).

Lourenco BEIRAO-DA-VEIGA, David MORA, Gonzalo RIVERA, Rodolfo RODRíGUEZ: A virtual element method for the acoustic vibration problem. Numerische Mathematik, vol. 136, 3, pp. 725-763, (2017).

David MORA, Gonzalo RIVERA, Rodolfo RODRíGUEZ: A virtual element method for the Steklov eigenvalue problem. Mathematical Models and Methods in Applied Sciences, vol. 25, 8, pp. 1421-1445, (2015).