Método Galerkin descontínuo misto híbrido para um problema não-linear com efeito térmico

• Autor(a) principal: Oliveira, Eduardo Lima de
• Data de Publicação: 2017
• 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/342
• topic: Análise numérica; Método de Galerkin; CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA::ANALISE

In this work was studied a coupled system of nonlinear equations consisting of two elliptic subproblems associated with potential and temperature. The couplings appear in the electrical conductivity that depends on temperature, heat source and electric charge flow. This class of problems has received attention in the specialized literature due to its applications and the mathematical challenges associatedwith analysis of existence, uniqueness and regularity of solution as well as the development of numerical algorithms and methods. In the analysis of this problem we use formulations, associated with a fixed point algorithm, and approximations by hybridized finite elements based in the Raviart-Thomas (RTH) methods and in the Hybridized Discontinuous Galerkin method (HDG). The hybrid method of Raviart-Thomas hybridized is analyzed restricted to a mixed problem, where the vector variable is searched in a subspace divH(div; Ω), hypothesizing the regularity of the charge flow. The HDG approximation follows the strategy proposed by the authors, which uses a local design based on the structure of the numerical trait, which defines the method. For this, we also use the argument of duality to obtain estimates of errors. We verified that due to nonlinearity, the estimates for the flow and for the scalar variables are coupled and depend on the argument of duality and regularity of problem solution. Results are presented confirming theexpected convergence rates, i.g. optimum rates for flows in H(div; Ω) and for the scalar variables in L 2 (Ω). For the HDG and RTH methods, the multipliers also have predicted convergences, for example, when the polynomials that constitute the approximation spaces have degree k the order of convergence is of order k+1/2.