Гришанов А.Н.  

Method of building approximate solutions using multigrid finite elements

A variant of building sequence of solutions in tasks for calculating elastic body using finite elements method (FEM) with the help of an effective way of defining the stiffness matrix on the basis of multigrid finite elements (MFE) has been considered. MFE allow for successive segmentation of discretization grid with fixed dimension of the main finite elements method (MFE) linear algebraic equations system. The approach under consideration is analogous to the hp-version of the MFE-increase of finite elemental task pane presentation dimension. A posteriori error estimates are conducted on the basis of the method suggested by O.C. Zienkiewicz and J. Z. Zhu, in norm L2.
To give an example, analysis of solutions sequence convergence as well as of error estimates in the numerical calculation of three-layered cylindrical shells stress-strain state under local loading has been presented. It has been demonstrated that using MFE in discretization both in the case of MFE h-version and in the case of MFE hp-version on the basis of the suggested method of calculatinf stiffness matrixes generates converging sequences of approximate solutions in norm L2. Comparison of error estimates and the nature of convergence in two variants of finite-elemental discretization using three-grid finite elements has been conducted. The use of MFE in the suggested method of calculating stiffness matrixes allows one to use small dimension MFE equations systems in building aptoximate solutions which ensures computer resource saving in comparison with discretization in the case of MFE h-version.

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