A cascadic multigrid algorithm in the finite element method for the three-dimensional Dirichlet problem in a curvilinear boundary domain

The paper deals with the three-dimensional Dirichlet problem for a second order elliptic equation in a domain with a smooth curvilinear boundary. To construct a finite element scheme, the embedded subspaces of basic functions are used without strict embeddedness of a sequence of spatial triangulations. It is proved that the discretization error is of the same order as in the case of the standard piecewise linear elements on a polyhedron. For solving the obtained system of linear algebraic equations on a sequence of grids, the cascadic organization of two iterative processes is applied providing a simple version of a multigrid method without any preconditioning or restriction to coarser grids. The cascadic algorithm starts on the coarsest grid where the grid problem is directly solved. On finer grids, approximate solutions are obtained by an iterative process, where interpolation of an approximate solution from the previous coarser grid is taken as an initial guess. It is proved that the convergence rate of this algorithm does not depend on the number of unknown values as well as on the number of grids.