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

A cascadic multigrid algorithm is substantiated for a grid problem obtained by discretization of a second-order elliptic equation with second-order finite elements on triangles. The efficiency of the algorithm is proved. In particular, it is shown that the number of arithmetic operations required to achieve the order of accuracy of an approximate solution equal to that of the discretization error depends linearly on the number of unknowns. The rate of convergence is found to be higher than one for linear finite elements despite achieving a higher order of accuracy.