Thesis Summary:

The aim of this thesis is to contribute to the development of hybridizable discontinuous Galerkin (HDG) methods to solve partial differential equations arising from fluid mechanics. First, we propose a novel technique to solve elliptic problems involving a non-polygonal interface/boundary. It is based on a high order HDG method where the mesh does not exactly fit the domain. We first study the case of a curved-boundary value problem with mixed boundary conditions. On the computational boundary, the Dirichlet data is approximated by using a transferring technique and the treatment of the Neumann data is based on extrapolating the approximation of the gradient. We then extend these ideas to curved interfaces. We provide numerical results suggesting that optimal high order convergence is achieved if the computational domain is constructed by interpolating the boundary/interface using piecewise linear segments. Next, we introduce and analyze an HDG method for the gradient-velocity-pressure formulation of the Brinkman problem. We present an a priori error analysis of the method, showing optimal order of convergence of the error. We also introduce an a posteriori error estimator, of the residual type, which helps us to improve the quality of the numerical solution. We establish reliability and local efficiency of our estimator for the natural norms, with constants written explicitly in terms of the physical parameters and independent of the meshsize. In particular, our results are also valid for the Stokes problem. Finally, we provide numerical experiments showing the quality of our adaptive scheme. At last, we propose a residual type a posteriori error estimator for an HDG method applied to the Oseen problem with unknowns as in the Brinkman problem. Similarly to the Brinkman problem, we prove reliability and local efficiency keeping track on the parameters dependence. For both Brinkman and Oseen problems we use the approximation properties of the Oswald interpolation operator and, for the last one, we employ a weighted function technique to control the L²-error of the velocity. Numerical experiments in three dimensions illustrate the quality of our adaptive scheme.

ISI Publications from the Thesis

Rodolfo ARAYA, Manuel SOLANO, Patrick VEGA: A posteriori error analysis of an HDG method for the Oseen problem. Applied Numerical Mathematics, vol. 146, pp. 291-308, (2019).

Rodolfo ARAYA, Manuel SOLANO, Patrick VEGA: Analysis of an adaptive HDG method for the Brinkman problem. IMA Journal of Numerical Analysis, vol. 39, 3, pp. 1502-1528, (2019).

Weifeng QIU, Manuel SOLANO, Patrick VEGA: A high order HDG method for curved-interface problems via approximations from straight triangulations. Journal of Scientific Computing, vol. 69, 3, pp. 1384-1407, (2016).