The goal of this chapter is to investigate efficient methods to solve linear systems of the form AU = F where the matrix A and the right-hand side F result from a finite element approximation to a linear model problem satisfying the well-posedness conditions of the BNB Theorem. The first section is concerned with the concept of matrix conditioning. The idea is to evaluate a real number that quantifies the stability of the linear system with respect to perturbations. In particular, we estimate the condition number of the mass matrix and that of the stiffness matrix. The second section deals with reordering techniques for sparse matrices. These techniques are particularly useful when solving sparse linear systems using direct methods. The third section reviews elementary properties of some widely used iterative solution methods: the Conjugate Gradient algorithm for symmetric positive definite systems and, more generally, projection based Krylov-type methods. We investigate the convergence rate of these methods and show how this rate can be improved using preconditioning techniques. The last section presents a brief introduction to the parallel implementation of iterative solution methods. For the sake of brevity, relaxation methods and multigrid methods are not discussed herein. The reader is referred, e.g., to [BrS94, GoV89, LaT93, Ort87, QuV97, Saa96] for further insight.
KeywordsStiffness Matrix Conjugate Gradient Conjugate Gradient Algorithm Sparsity Pattern Adjacency Graph
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