Multigrid finite element method

To solve a number of important physical boundary value problems (which solutions of the equations be-ing equivalent to finding the minimum of correspond-ing functional) the multigrid finite element method (MFEM) which is realized on the basis of ratios and algorithms of the method of final elements (MFE) in the form of Ritz method with application of multigrid final elements (MFEM) was proposed. To construct a -grid finite element (FE), the -nested grids were used. A finite grid is generated by basic body partition taking into account its irregular shape and physical features of the boundary value problem (e.g. the in-homogeneous structure of elastic body). Other grids were used to reduce MFE dimension, and with increasing MFE dimension decreases. The essence of MFE is as follows: at a basic partition of grid FE, consisting of known single-grid FE, the functional of boundary value problem was determined as a function of many variables, being the values of the required function at the nodes of a fine grid. On the n n 1n 2n F other grids some approximating functions were used for the decrease of the dimension of func-tion, allowing one to develop small dimensional MFE. The developing -grid FE is carried out according to a single matrix procedure There are some essential differences between MFEM and FEM. First, in regard to MFEM, some arbitrarily fine base body partitions can be applied, which makes it possible to take into account their irregular shape heterogeneous and microheterogeneous structure (without increasing the dimensions of the multigrid discrete models). As to FEM, it is impossible to use any arbitrarily fine parti-tions of the bodies, as the computer resources are limited, i.e. MFEM is more efficient than FEM. Sec-ondly, the implementation of MFEM based on the essential models of bodies takes less computer memory and span time than that of FEM for essential models, i.e. MFEM is more time and memory-saving than FEM. Thirdly, in MFEM some elastic homoge-neous and inhomogeneous MFE are applied, using the nested grids to construct, significantly expanding the scope of MFEM. Therefore, MFEM can be as-sumed to be a generalization of FEM, i.e. FEM is a special case of MFEM. The procedures of developing MFE of various shapes were presented. The top as-sessment of errors of approximate decisions is of-fered.