Convergence of the multigrid cascadic algorithm for second order finite elements in a domain with a smooth boundary

In this paper, the cascadic multigrid algorithm for a grid problem obtained by discretization of a second order elliptic equation with second order finite elements on triangles is substantiated. The efficiency of the algorithm is proved. This means that the number of arithmetic operations required to achieve the order of accuracy of an approximate solution equal to that of the discretization error linearly depends on the number of unknowns. The rate of convergence is found to be higher than that for linear finite elements in spite of a higher order of accuracy.