Graduate Thesis of Felipe Vargas
Program | PhD in Applied Sciences with mention in Mathematical Engineering, Universidad de Concepción | |
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Enrollment Year | 2014 | |
Senior Year | 2019 | |
Thesis Title | Discontinuous Galerkin Methods in Continuum Mechanic | |
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. | ||
Thesis Director(s) | Jay Gopalakrishnan, Manuel Solano | |
Thesis Project Approval Date | 2016, December 21 | |
Thesis Defense Date | 2019, January 11 | |
Professional Monitoring | ||
PDF Thesis | Download Thesis PDF | |
ISI Publications from the ThesisJay GOPALAKRISHNAN, Manuel SOLANO, Felipe VARGAS: Dispersion analysis of HDG methods. Journal of Scientific Computing, vol. 77, 3, pp. 1703-1735, (2018). |
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