Numerical methods for solving elliptic problems

• Grant number: 16/25855-7
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: April 01, 2017
• End date: January 31, 2018
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Analice Costacurta Brandi
• Grantee: Leticia Braga Berlandi
• Host Institution: Faculdade de Ciências e Tecnologia (FCT). Universidade Estadual Paulista (UNESP). Campus de Presidente Prudente. Presidente Prudente , SP, Brazil

Abstract

The elliptic equations are partial differential equations and they are related to equilibrium problems that do not depend in general on time. The best known of these equations are the Poisson equation and the Laplace equation, whose applications are the most varied. The first represents the movement of a viscous incompressible fluid at low velocity and the second is employed to describe electromagnetic potential, for example. The need to obtain approximate solutions to problems of this kind has led to the study and application of numerical methods. The resolution of an elliptic equation by numerical methods is given discretizing the derivatives by the finite difference method, for example, resulting in a linear equations system typically large and sparse, which requires iterative methods to solve them. In this context, this research project aims to study, implement and compare the solution methods of linear equations system applied in the two-dimensional Poisson equation for different auxiliary conditions. The obtained results will be compared to numerical results and analytical solutions existing in the literature, in order to analyze the convergence of the studied methods, and especially, the computational time spent in the numerical simulations. (AU)