Kvasnytsia H. Construction and analysis of h-adaptive finite element scheme for elasticity problems

The dissertation is devoted to the construction and research of h-adaptive schemes of the finite element method for solving elasticity problems, in particular, singularly perturbed. The classical finite element scheme with the piecewise-quadratic approximations of the displacement vectors on the Delaunay triangulations is supplemented by the construction of element-wise implicit residual Dirichlet and Neumann a posteriori error estimators, which are capable of calculating the lower and upper bounds of the true error. The reliability and efficiency of such estimators for the elasticity problems are proved. There have been proposed criteria for the local refinement of Delaunay triangulations with multiple element subdivision, and on their basis have been developed a reliable and effective strategy of iterative calculation of a convergent sequence of approximations, which guarantees finding approximate solutions of the mentioned problems with the required accuracy. In the original software has been implemented the h-adaptive finite element scheme for two-dimensional elasticity problems. The probability of theoretical conclusions and dissertation provisions has been confirmed by the results of numerical experiments with singularly perturbed problems. The proposed method of constructing a posteriori error estimators has been extended to the problem of time-harmonic vibrations of a viscoelastic body with short-term memory, for which have been proved the correctness of the corresponding variational problem, convergence of finite element schemes and have been constructed local a posteriori error indicators. For the torsion problem has been proposed an adaptation criterion with an exact total error estimate for dual problems in the Saint-Venant’s and Prandtl’s formulations.