Solving Large Sparse Linear Systems

Abstract

In the previous chapters we have seen how finite element discretization gives rise to linear systems, which must be solved in order to obtain the discrete solution. The size of these linear systems is generally large, as it is direct proportional to the number of nodes in the mesh. Indeed, it is not unusual to have millions of nodes in large meshes. This puts high demands on the linear algebra algorithms and software that is used to solve the linear systems in terms of computational complexity (i.e., number of floating point operations), memory requirements, and time consumption. To cope with these problems it is necessary and important to exploit the fact that these linear systems are sparse, which means that they have very few non-zero entries as compared to their size. This is due to the fact that the finite element basis functions have very limited support and only interact with their nearest neighbours. In this chapter we review some of the most common direct and iterative methods for solving large sparse linear systems. We emphasize that the aim is not to present and analyze these methods rigorously in any way, but only to give an overview of them and their connection to finite elements. 6.1 Linear Systems We consider the problem of solving the linear system Ax D b (6.1) where A is a given n n matrix, b is a given n 1 vector, and x the sought n 1 solution vector. The assumption is that n is large, say, 10 6 , and that A is sparse. A sparse matrix is somewhat vaguely defined as one with very few non-zero entries A ij. A prime example of such a matrix is the stiffness matrix resulting from finite element discretization of the Laplace operator. If A is invertible, which is the case when the underlying differential equation is well posed, the solution x to (6.1) can formally be found by first computing the inverse A 1 of A, and then multiplying