A parallel preconditioner for 2D elliptic boundary value problems

This work presents the implementation on a Linux Cluster of a parallel preconditioner for the solution of the linear system resulting from the finite element discretization of a 2D second order elliptic boundary value problem. The numerical method, proposed by Bramble, Pasciak and Schatz, is developed using Domain Decomposition techniques, which are based on the splitting of the computational domain into subregions of smaller size, enforcing suitable compatibility conditions. The Fortran code is implemented using PETSc: a suite of data structures and routines devoted to the scientific parallel computing and based on the MPI standard for all message-passing communications. The main interest of the paper is to investigate how the architectural aspects of the cluster influence the performance of the considered algorithm. We provide an analysis of the execution times as well as of the scalability, using as test case the classical Poisson equation with Dirichlet boundary conditions.