Error Control and Adaptivity in Mixed Finite Element Methods for Contact Problems

Prof. Dr. Andreas Schröder

June 25, 2014, 1:45 p.m. S2 054

The talk presents mixed finite element discretizations of higher-order for obstacle problems and frictional contact problems in linear elasticity as well as elastoplasticity. In particular, the discretization of Lagrange multipliers with standard polynomial approximation spaces and, alternatively, with biorthogonality is discussed. Moreover, residual-based a posteriori error estimates are derived where the residual is defined via the discrete displacement and the discrete Lagrange multipliers. The error estimates consist of the dual norm of the residual plus some computable remainder terms. The dual norm can be estimated by the discretization error of an auxiliary problem that is a variational equation so that well-known error controls for variational equations can be applied. The talk also discusses goal-oriented error estimates based on the dual weighted residual approach (DWR) for the mixed formulation of frictional contact problems. The estimates measure the error in terms of a quantity of interest and are evaluated in the displacement field as well as the Lagrange multipliers. Due to the low-regularity nature of contact problems, only low convergence rates are expected for higher-order discretiziations when uniform mesh refinement is applied. The use of adaptivity based on a posteriori error control may recover optimal algebraic rates and significantly improve the convergence of higher-order schemes based on mixed methods. Indeed, this expectation is confirmed in the numerical experiments. It is observed that the end points of the contact zone as well as those points where gliding switches to sticking are resolved.