Solução numérica da equação de Poisson em malhas estruturadas bidimensionais e tridimensionais

Abstract

In this work, we numerically solve second-order elliptic partial differential equations such as the Laplace and Poisson equations, using the finite difference method in two-dimensional and threedimensional structured meshes. To solve the system of linear equations arising from the finite difference discretization, we use the iterative methods of Gauss-Seidel and SOR. Furthermore, we build manufactured solutions for some Poisson equations and compare the exact and numerical solutions, and test optimal values for the relaxation parameter in the SOR method. We also apply the theory studied in the numerical solution of stationary or equilibrium problems and employ Matlab and Tecplot 360 to visualize the numerical solution. We conclude that the convergence of the SOR method is slow in problems with Neumann boundary conditions and in problems with singularities.