Description and implementation of an algebraic multigrid preconditioner for H1-conforming finite element schemes

This paper presents detailed aspects regarding the implementation of the Finite Element Method (FEM) to solve a Poisson’s equation with homogeneous boundary conditions. The aim of this paper is to clarify details of this implementation, such as the construction of algorithms, implementation of numerical experiments, and their results. For such purpose, the continuous problem is described, and a classical FEM approach is used to solve it. In addition, a multilevel technique is implemented for an efficient resolution of the corresponding linear system, describing and including some diagrams to explain the process and presenting the implementation codes in MATLAB^®. Finally, codes are validated using several numerical experiments. Results show an adequate behavior of the preconditioner since the number of iterations of the PCG method does not increase, even when the mesh size is reduced.