A cascadic multigrid algorithm in the finite element method for the threedimensional Dirichlet problem

A standard scheme of the finite element method with the use of piecewise-linear elements on tetrahedrons is considered as applied to the three-dimensional elliptic second order Dirichlet problem. In order to solve this scheme, a cascadic arrangement of two iterative algorithms is used on a sequence of embedded threedimensional triangulations that gives a simple version of the multigrid methods without preconditioning and restriction to a coarser grid. The cascadic algorithm starts on the coarsest grid where the grid problem is solved by direct methods. In order to obtain approximate solutions on finer grids, the iteration method is used; the initial guess is taken by interpolation of the approximate solution from the preceeding coarser grid. It has been proved that the convergence rate of this algorithm does not depend on the number of unknowns and the number of grids.