Two multigrid iterative algorithms for a discrete analogue of the biharmonic equation

A standard scheme of the finite element method with the use of bicubic elements on a rectangular quasiuniform grid is considered as applied to the two-dimensional Dirichlet problem for the biharmonic equation in a rectangle. To solve this scheme, two multigrid algorithms are treated on a sequence of embedded rectangular grids: a full multigrid with $V$-cycle and a simpler cascadic algorithm. The presence of angular points of a rectangle results in deficiency of solution smoothness which complicates substantiation of convergence of the algorithm as compared to a smooth case. At the same time, a number of arithmetical operations remains almost optimal for the cascadic algorithm and optimal for $V$-cycles.