Métodos de elementos finitos híbridos estabilizados para problemas de convecção-difusão

• Autor(a) principal: Ordonio Hoyos, Denis Daniel
• Data de Publicação: 2021
• Tipo de documento: Tese
• Idioma: por
• Título da fonte: Biblioteca Digital de Teses e Dissertações do LNCC
• Texto Completo: https://tede.lncc.br/handle/tede/346
• topic: Método dos elementos finitos; Método de Galerkin; Hibridização; Estabilização; Convecção-difusão; CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA::ANALISE

Stabilized hybrid finite element methods are developed for solving stationary and transient linear convection-diffusion equations predominantly convective. We start from a primal hybrid formulation, proposed by Oikawa (2014), where stability of the convective part is achieved by adding an upwind term, taking advantage of the stabilization mechanisms typical of Discontinuous Galerkin (DG) methods, as done in Cockburn, Gopalakrishnan e Lazarov (2009a). From this formulation we make a numerical study of high-order approxi- mations based on the stabilized primal hybrid formulation proposed in Arruda, Loula e Almeida (2013) for the stationary diffusion problem. We propose a hybrid mixed method where a least squares type stabilization term is added similarly to the unconditionally stable mixed finite element method for the Darcy flow proposed and analyzed in Correa e Loula (2008), which includes a coefficient depending on the mesh parameter h. This mixed hybrid formulation gives rise to local problems, involving the degrees of freedom of the scalar variable and its flow, which are solved at the element level and can be eliminated in favor of the Lagrange multiplier, identified as the trace of the scalar variable on the edges of the elements, as occurs in primal hybrid methods. In this way, a global system is assembled involving only the degrees of freedom associated with the Lagrange multipliers. The variable of interest can be obtained by a local post-processing in each element. We also develop a stabilized primal hybrid formulation for transient convection-diffusion problems e and show its equivalence with DG methods obtained numerically by condensing the multiplier degrees of freedom, which are more adequate for lower order approximations. For this transient problem we discretize the time derivative using the Crank-Nicolson method, where stabilization occurs at the instant n ` 1{2 to preserve the second order in time. To illustrate the potential of the proposed formulations, numerical simulations are performed considering problems in dominant convection regimes.