Thesis Summary:

This thesis addresses two main topics. First, we apply and analyse a high order hybridizable discontinuous Galerkin (HDG) method to two interesting problems in the context of fluid mechanics, which are Stokes problem and Oseen equations. The novelty of this part is that curved domains are considered instead of polyhedral domains. The analysis is based on approximating the curved domain by a polyhedral computational subdomain where an HDG solution can be computed. To obtain a high order approximation of the Dirichlet boundary data in the computational domain, we employ a transferring technique based on integrating the approximation of the gradient of the velocity. In addition, we first seek for a discrete pressure having zero-mean in the computational domain and then the zero-mean condition in the entire domain is recovered by a post-process that involves an extrapolation of the discrete pressure. Since the problems we are interested on are modelled by similar sets of equations (there is only one additional term in Oseen equations), the treatment of the boundary condition and pressure is the same for both problems. We prove that the method provides optimal order of convergence for the approximations of the pressure, the velocity and its gradient, that is, order $h^{k+1}$ if the local discrete spaces are constructed using polynomials of degree $k$ and the meshsize is $h$. Numerical experiments validating the method are presented. Secondly, we also present a dispersion analysis of HDG methods. Considering the Helmholtz system, we quantify the discrepancies between the exact and discrete wavenumbers. In particular, we obtain an analytic expansion for the wavenumber error for the lowest order Single Face HDG (SFH) method. The novel result of this part is that the expansion shows that the SFH method exhibits convergence rates of the wavenumber errors comparable to that of the mixed hybrid Raviart–Thomas (HRT) method. In addition, the same behavior for the higher order cases is observed in numerical experiments.

ISI Publications from the Thesis

Manuel SOLANO, Felipe VARGAS: An unfitted HDG method for Oseen equations. Journal of Computational and Applied Mathematics, vol. 399, Art. Num. 113721, (2022).

Manuel SOLANO, Felipe VARGAS: A high order HDG method for Stokes flow in curved domains. Journal of Scientific Computing, vol. 79, 3, pp 1505-1533, (2019).

Jay GOPALAKRISHNAN, Manuel SOLANO, Felipe VARGAS: Dispersion analysis of HDG methods. Journal of Scientific Computing, vol. 77, 3, pp. 1703-1735, (2018).