VML: A class of virtual multi-level iterative methods for solving partial differential equations

We introduce virtual multi-level iterative methods (VML) which attempt to remove the 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 constitute a class of VML iterative methods. They may be used to accelerate general (single-level) iterative methods or be used with the standard (real) multi-grid 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.