On the additive version of the algebraic multilevel iteration method for anisotropic elliptic problems

In this paper a recently proposed additive version of the algebraic multile vel iteration method for iterative solution of elliptic boundary value prob lems is studied. The method constructs a nearly optimal order parameter-fre e preconditioner, which is robust with respect to anisotropy and discontinu ity of the problem coefficients. It uses a new strategy for approximating t he blocks corresponding to "new" basis functions on each discretization lev el. To cope with the difficulties arising from the anisotropy, the problem on the coarsest mesh is solved using a bordering technique with a special c hoice of bordering vectors. The aim is to find a parameter-free "black-box" robust solver. The results are derived in the framework of a hierarchical basis, linear fi nite element discretization of an elliptic problem on arbitrary triangular meshes, and a hierarchical basis, bilinear finite element discretization on Cartesian meshes. A comparison of the method with some other iterative solution techniques is presented. Robustness and high efficiency of the proposed algorithm are de monstrated on several model-type problems.