On Higher Order Approximation for Nonlinear Variational Problems in Nonsmooth Mechanics

This paper is concerned with the hp-version of the finite element method (hp-FEM) to treat a variational inequality that models frictional contact in linear elastostatics. Such an approximation of higher order leads to a nonconforming discretization scheme. We employ Gauss-Lobatto quadrature for the approximation of the nonsmooth frictiontype functional and take the resulting quadrature error into account in the error analysis. We prove convergence of the hp-FEM Galerkin solution in the energy norm. To this end we investigate Glowinski convergence for the friction-type functional. The key of our norm convergence result for the hp−FEM is the used Gauss-Lobatto integration rule with its high exactness order and its positive weights together with a duality argument in the sense of convex analysis. Then we discuss how our convergence analysis can be further extended to other nonlinear variational problems from nonsmooth mechanics. In particular we treat the Bingham fluid problem and propose a mixed hp-FEM discretization scheme with analogous convergence properties.