The paper deals with certain adaptive multilevel methods at the confluence of nested multigrid methods and iterative methods based on the cascade principle of [10]. From the multigrid point of view, no correction cycles are needed; from the cascade principle view, a basic iteration method without any preconditioner is used at successive refinement levels. For a prescribed error tolerance on the final level, more iterations must be spent on coarser grids in order to allow for less iterations on finer grids, A first candidate of such a cascadic multigrid method was the recently suggested cascadic conjugate gradient method of [9], in short CCG method, which used the CG method as basic iteration method on each level. In [18] it has been proven, that the CCG method is accurate with optimal complexity for elliptic problems in 2D and quasi-uniform triangulations, The present paper simplifies that theory and extends it to more general basic iteration methods like the traditional multigrid smoothers. Moreover, an adaptive control strategy for the number of iterations on successive refinement levels for possibly highly non-uniform grids is worked out on the basis of a posteriori estimates. Numerical tests confirm the efficiency and robustness of the cascadic multigrid method.
