Estimates of deviations from exact solutions of some nonlinear problems in continuum mechanics

Prof. Dr. Sergey Repin

Nov. 29, 2011, 3:30 p.m. S2 416

In the talk, we discuss estimates measuring the difference between exact solutions of boundary value problems and arbitrary functions from the corresponding (energy) space. The estimates must be computable, consistent and possess necessary continuity properties. In the context of PDE theory, deriving such type estimates present one of the general problems, which unlike, e.g., regularity theory is focused on studying neighborhoods of exact solutions. Being applied to numerical approximations these estimates imply a unified way of a posteriori error estimation. They can be also used for the analysis of modeling errors and errors caused by incomplete knowledge on the problem data. The talk contains a short introduction devoted to historical background, overview of the results obtained in the last decade for elliptic variational inequalities, modeling errors of dimension reduction models, and some very recent results related to models with linear growth energy (as, e.g., Hencky plasticity). Literature: S. Repin. A posteriori error estimates for PDE's, deGruyter, Berlin, 2008. M. Fuchs and S. Repin. A Posteriori Error Estimates for the Approximations of the Stresses in the Hencky Plasticity Problem, Numer. Funct. Analysis and Optimization, 32(2011), 6, 610-640. S. Repin and S. Sauter. Estimates of the modeling error for the Kirchhoff-Love plate model. C. R. Math. Acad. Sci. Paris 348 (2010), no. 17-18, 1039–1043.