The Role of Incomplete LU-Factorization in Multigrid Methods

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Since their introduction as preconditioners for conjugate gradient methods (ICCG method, [37]) and the widely appreciated demonstration of the efficiency of these preconditioners in [27], incomplete LU (ILU) factorizations have found widespread use in the context of conjugate gradient and related methods. Not long after its introduction, ILU was also applied as smoother in a multigrid method and comparison was made with a conjugate gradient method, preconditioned with the same ILU factorization, for a Navier-Stokes application ([55]). More extensive comparisons of conjugate gradients and multigrid as linear systems solvers are reported in [45] and [12]. Viewed as linear systems solvers, conjugate gradients and multigrid can be regarded as two alternate ways to accelerate convergence of basic iterative methods. Generally speaking, for medium-sized linear problems conjugate gradients and multigrid are about equally effective, but for sufficiently large problems multigrid is faster, because of its linear computational complexity. Unlike multigrid, conjugate gradient methods are limited to linear problems. On the other hand, conjugate gradient methods are much easier to program, especially when the computational grid is unstructured.