# BEM for Contact Problems

Ernst P. Stephan^{*}

Firstly, we consider Signorini contact for the Laplacian in $R^2$. We use the PoincarĂ©-Steklov operator, which realizes the Dirichlet-to-Neumann map, and represent the negative of the unknown normal derivative on the contact boundary by a Lagrange multiplier. Herewith we derive a mixed formulation which is equivalent to a variational inequality on the contact boundary, where the non-penetration condition is incorporated in the convex set of admissible ansatz and test functions. Both formulations are uniquely solvable. To obtain a higher order method (hp-version) we use Gauss-Lobatto-Lagrange basis functions on a regular mesh on the contact boundary for the primal variable and biorthogonal basis functions of the same degree on the same mesh for the Lagrange multiplier. We present a reliable a posteriori error estimate of residual type for the Galerkin solution of the mixed formulation. The discrete mixed system is solved by the semi-smooth Newton algorithm in combination with a penalized Fischer-Burmeister complementarity function taking care of the contact condition. Numerical experiments are given which support our theoretical results. Secondly, we consider a class of linear elastic problems with nonlinear, nonsmooth boundary conditions modeling adhesion. With the PoincarĂ©-Steklov operator we derive a hemivariational inequality which is solved with boundary elements by a Bundle-Newton algorithm. Numerical benchmarks in 2D and 3D are given.

Mathematics Subject Classification: 65N30 65N38

Keywords: Signorini problems; PoincarĂ©-Steklov operator; biorthogonal basis functions; adhesion; hemivariational inequality; semi-smooth Newton algorithm; Bundle-Newton algorithm

Minisymposion: Nonsmooth and Unilateral Problems - Modelling, Analysis and Optimization Methods