Estudo e aplicação do método multimalha na resolução de sistemas lineares

Abstract

The search for solutions to linear systems appears in different contexts, such as in the field of optimization, control theory, computational tomography, and the study of fluid dynamics problems. Physical phenomena related to fluid dynamics are often modeled using differential equations, which often lack a known analytical solution. In particular, there is a frequent need to obtain solutions for elliptic partial differential equations, such as the Poisson equation or the Laplace equation. The search for a numerical solution to this type of equation requires solving sparse linear systems of large dimension and not always with well-defined structure, which usually also need to be solved numerically.In this context, the present work focuses on the development and implementation of multigrid methods to investigate their characteristics and evaluate their efficiency in solving systems. This technique is widely used to accelerate the convergence of equation systems by exploiting the advantage that some numerical methods have in effectively smoothing high-frequency errors at different refinement levels. Multigrid methods in a V-cycle composed of 2, 3, or four levels are used to solve sparse linear systems resulting from the discretization of the Laplace and Poisson equations via second-order accurate finite differences. Comparative analyses of runtime and the number of cycles needed were conducted and demonstrate the potential use of the technique as a convergence accelerator for large-scale problems.