This diploma thesis presents the mathematical theory and numerical analysis of the contact problem with Tresca friction in plane elasticity. We give an overview of the mathematical formulation of this problem as a variational inequality of the second kind, and prove the existence and uniqueness of the corresponding displacement field using methods of convex analysis. Furthermore, we introduce a primal-dual formulation, where the nonlinear friction functional is replaced by using a Lagrange multiplier function on the contact boundary.
Next, we analyse how the given problem can be appropriately discretised.
It is well known that p-finite element methods can yield exponential convergence, but only if the exact solution is smooth on all elements of the employed mesh. As the displacement field is expected to be nonsmooth near those parts of the contact boundary where the boundary conditions change from sticking to sliding, in addition to corners and transitions between Dirichlet and Neumann boundaries, however, this assumption is not justified for the presented problem. Therefore, in the numerical analysis, we focus on hp-methods. These methods combine fine grids at points where the solution is irregular with high polynomial degrees on elements where it is smooth. We prove a general convergence result for hp-finite element approximations on meshes with arbitrary element size and polynomial degree distributions. Furthermore, given sufficient regularity, we obtain convergence rates using a novel hp-mortar projection operator, which uses a discontinuous Lagrange multiplier space on the boundary.
As the information on the regularity of the exact solution, which is necessary to construct an appropriate mesh, is not available a priori, we apply an error indicator of residual type, generalised to our context, to determine those elements where the local error appears high and which thus should be refined in an adaptive computation. For these elements, we then estimate the local regularity of the solution using the rate of decay of the Legendre series coefficients of the given numerical approximation. Based on this, we decide whether to subdivide the element or increase the polynomial degree.
We finally show numerical results which confirm our analysis. The adaptive methods are able to resolve the irregularities of the solution properly, and give rates of convergence that are significantly higher than those of uniform mesh refinements. In particular, the hp-method empirically yields exponential convergence.