Multigrid iterative algorithms in the finite element method using numerical integration
In this paper a grid problem obtained by discretization of a second-order elliptic equation with the piecewise linear elements on triangles is considered with the help of numerical integration. The full multigrid cascadic algorithm with a W-cycle constitutes the simplest variant of multigrid methods applied on a sequence of embedded grids. Discrete problems on the coarser grids are constructed such that the matrices of systems of equations on applied on the neighboring grids are related via interpolation and restriction operators. It has been proved that the number of arithmetic operations per one unknown required to determine an approximate solution (with an error of the same order as that of the discrete problem with numerical integration) is independent of the number of unknowns and the number of grids for both algorithms.