Eficiência dos Métodos Multigrid Algébricos para a Solução da Equação do Fluxo Livre Estacionário em Domínio Georreferenciado

ABSTRACT

In this paper, the algebraic multigrid method based on smoothed aggregation, the classical method of Ruge-Stúben and the method GMRES preconditioned by algebraic multigrid were used to solve the steady-state free flow equation in a georeferenced domain. The availability of computer codes allowed us to evaluate the approximation of finite elements from the perspective of algebraic multigrid methods and their combination, as preconditioning, with the GMRES method. The maximum differences between solutions by different methods, the time required to obtain the solutions of the associated linear systems, in each of the Picard iterations, the residuals of each of the iterative methods and the residuals of each of the Picard iterations are presented and discussed. As a result of the analysis, the preconditioned methods are more efficient in the sense of less computational time combined with the stability of the number of iterations. The analysis of the residuals of Picard iterations allows comparing the evolution of the different methods of solving linear systems. The detailing of the residuals of the iterative methods in each step of Picard’s iterations allowed a more comprehensive view and an analysis of convergence. In detail, the method based on smoothed aggregation needs a significantly smaller number of iterations when compared to the classical method of Ruge Stüben in the first iterations of Picard. The preconditioning reduces the number of iterations in relation to the initial iterations and there is a persistence of the reduction in the number of iterations of the aggregation-based method in relation to the classical method.

Keywords:
finite elements; FEniCs; Python; interative methods