In this paper, the multilevel ILU (MLILU) decomposition is introduced. During an incomplete Gaussian elimination process new matrix entries are generated such that a special ordering strategy yields distinct levels. On these levels, some smoothing steps are computed. The MLILU decomposition exists and the corresponding iterative scheme converges for all symmetric and positive definite matrices. Convergence rates independent of the number of unknowns are shown numerically for several examples. Many numerical experiments including unsymmetric and anisotropic problems, problems with jumping coefficients as well as realistic problems are presented. They indicate a very robust convergence behavior of the MLILU method.
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