Approximate And Incomplete Factorizations

. In this chapter, we give a brief overview of a particular class of preconditioners known as incomplete factorizations. They can be thought of as approximating the exact LU factorization of a given matrix A (e.g. computed via Gaussian elimination) by disallowing certain fill-ins. As opposed to other PDE-based preconditioners such as multigrid and domain decomposition, this class of preconditioners are primarily algebraic in nature and can in principle be applied to any sparse matrices. When applied to PDE problems, they are usually not optimal in the sense that the condition number of the preconditioned system will grow as the mesh size h is reduced, although usually at a slower rate than for the unpreconditioned system. On the other hand, they are often quite robust with respect to other more algebraic features of the problem such as rough and anisotropic coefficients and strong convection terms. We will describe the basic ILU and (modified) MILU preconditioners. Then we will review ..