Keywords: nonsmooth mechanics, contact, friction, hp-FEM, nonconforming approximation, Gauss-Lobatto quadrature.

This paper is concerned with the hp-version of the finite element method (hp-FEM) to treat a variational inequality in a vectorial Sobolev space that models bilateral frictional contact in linear elastostatics. Thus recent work is extended [1] for the boundary element method to a larger class of nonlinear variational problems that are treatable by the finite element method.

By the pioneering work of Babuška and co-workers, the exponentially fast convergence of the hp-FEM for linear elliptic problems is well-known. Recently Maischak and Stephan [2,3], respectively Dörsek and Melenk [4,5] showed the superior convergence properties of adaptive hp boundary element methods (hp-BEM), respectively adaptive hp finite element methods in numerical experiments also for unilateral, nonsmooth problems compared to the standard h-version.

Such an approximation of higher order leads to a nonconforming discretization scheme. Without any regularity assumptions, the convergence of the hp-FEM Galerkin solution in the energy norm is proved. To this end the Glowinski convergence for the friction-type functional is investigated. The key to the 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. Thus the convergence analysis complements prior work of Maischak and Stephan [3] on the hp-BEM for frictionless unilateral contact and more recent work of Dörsek and Melenk [4] on a mixed hp-FEM for frictional bilateral contact.

Finally the convergence analysis can be further extended to other nonlinear variational problems from nonsmooth mechanics. In particular the Bingham fluid problem is considered and a mixed hp-FEM discretization scheme is proposed with analogous convergence properties.

1

J. Gwinner, "On the p-version approximation in the boundary element method for a variational inequality of the second kind modelling unilateral contact and given friction", Appl. Numer. Math., 59, 2774-2784, 2009. doi:10.1016/j.apnum.2008.12.027

2

M. Maischak, E.P. Stephan, "Adaptive hp-versions of BEM for Signorini problems", Appl. Numer. Math., 54, 425-449, 2005. doi:10.1016/j.apnum.2004.09.012

3

M. Maischak, E.P. Stephan, "Adaptive hp-versions of boundary element methods for elastic contact problems", Comput. Mech., 39, 597-607, 2007. doi:10.1007/s00466-006-0109-y

4

P. Dörsek, J.M. Melenk, "Adaptive hp-FEM for the contact problem with Tresca friction in linear elasticity: The primal - dual formulation and a posteriori error estimation", Appl. Numer. Math., 60, 689-704, 2010. doi:10.1016/j.apnum.2010.03.011

5

P. Dörsek, J.M. Melenk, "Adaptive hp-FEM for the contact problem with Tresca friction in linear elasticity: The primal formulation", in "Proceedings of ICOSAHOM 2009", Lecture Notes in Computational Science and Engineering, 76, Springer, Berlin, 1-17, 2011.

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £65 +P&P)