This work presents direct, iterative and multigrid methods for solving systems of equations obtained by means of the finite element method applied to linear elliptic problems, considering two and three dimensional elastic examples. Adaptive analysis aspects are also discussed by taking an error estimator and some stress recovery procedures. All the algorithms are implemented using the object-oriented model with the C++ language.
Comparisons between direct and iterative methods concerned to the number of operations and memory space are presented. The superiority of the iterative techniques accelerated by multigrid strategies can be detected, mainly on three dimensional applications.
The multigrid methods have been used at most with nested meshes. In such cases the treatment of engineering problems with complex boundaries becomes difficult. In this thesis, the meshes are non-structured and non-nested and they are generated by frontal and Delaunay techniques. By using adaptive procedures, it is possible to get an optimal sequence of meshes for solving a problem within a specified admissible error.
The numerical procedures were linked with some other developed programs, using the object paradigm in C++. Two environments with tools for defining geometrical boundaries, automatic meshes generation, errors estimation, meshes refinement and visualization of results are also presented, considering two dimensional linear elastic problems.