We introduce virtual multi-level iterative methods (VML) which attempt to remove low frequency errors by conducting some special smoothing (residual norm minimization) procedure with respect to the coarse grids. However, there is no coarse grid formed explicitly, no inter-grid transfer operator is needed, and even the smoothing procedure can be done almost locally. These properties are attractive to parallel computers. VML with different relaxation schemes and different smoothing techniques constitutes a class of VML iterative methods. They may be used to accelerate general (single-level) iterative methods or be used with the standard (real) multigrid method to alleviate the inherent lack of parallelism. Numerical experiments with some relaxation and smoothing techniques are used to show how the VML iterative methods work.
Key words: Virtual multi-level iterative method, multigrid method, residual norm minimization, relaxation.