Performance of meshfree methods in approximations with diffuse derivatives

The study and solution of partial differential equations (PDEs) is a central field in the mathematical analysis. In the numerical context the mesh free approximation methods arise as an alternative to the conventional techniques that require some kind of mesh to the realization of the approximations. In cases where it has relatively complex eometries (as domains with discontinuities) traditional methods are difficult to implement. Therefore the objective of the mesh free methods is, in part, to remove these difficulties. In this thesis we focus our attention into the study of the Element Free Galerkin method (EFG) coupled with approximations with diffuse derivative, for the solutions of (PDEs) problems. In the first part of this work, we show the known theoretical results (convergence theorems and convergence orders) with original proves based on existing work and corroborate them numerically for classic PDEs problems. The second part of this thesis consists on the numerical solution and convergence analysis of the EFG method for problems that involves fractional operators and discontinuous geometries. The obtained results are compared with theoretical information and the unexpected events are discussed and justified.

• Autores: Carmona Otálvaro, Jhonatan
• Fecha de publicación: 2016
• Institución: Universidad Nacional de Colombia
• Repositorio: Universidad Nacional de Colombia
• Idioma: spa
• OAI Identifier: oai:repositorio.unal.edu.co:unal/58847
• Acceso en línea: https://repositorio.unal.edu.co/handle/unal/58847; http://bdigital.unal.edu.co/55840/
• Palabra clave: 51 Matemáticas / Mathematics; Métodos libres de malla; Método de Galerkin libre de elementos; Derivada difusa; Operadores fraccionarios; Geometrías discontinuas; Mesh free approximations methods; Element Free Galerkin method; Diffuse derivative; Fractional operators; Discontinuous geometries
• Rights: openAccess
• License: Atribución-NoComercial 4.0 Internacional