Mathematical Sciences: Finite Element Superconvergence in Computational Mechanics

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9626193 Zhang The investigator studies: (1) Superconvergence points for some typical equations in computational mechanics on different mesh patterns and for various finite element bases; (2) Theoretical justification for superconvergence phenomena for newly developed, highly effective patch recovery techniques in the engineering community; (3) Superconvergence and its recovery for equations with small parameters that model some typical mechanical problems, such as beams, plates, shells, and the Stokes problem; (4) Superconvergence recovery technique for equations with singular solutions, and interior or local estimates for recovery techniques in the presence of boundary singularities. Extensive numerical tests are performed to evaluate the effectness of the proposed recovery techniques for the Poisson equation, the nearly incompressible elasticity equation, the Reissner-Mindlin plate model, and the Stokes equation for different finite element bases. The computational experimentation consider various mesh geometries and some domains with reentrant corners, such as the L-shaped domain and the cracked domain. The primary objective of this project is the study of derivative superconvergence and its recovery in finite element computations. Such study is of fundamental importance in computational mechanics, because engineers are interested in more effective ways to estimate the stress, strain energy, etc. At present, most of the commercial codes used by the engineering community are required to have adaptive capabilities, which allow computations to be carried out in the most efficient way. A knowledge of superconvergence is essential for this adaptivity design through a posteriori error estimates. This project provides a firm theoretical basis for developing reliable and computationally inexpensive superconvergent recovery techniques. The results of this research affect large-scale scientific computing on problems of practical interest arising in structural me chanics.