Projection and self-adaptive projection methods for the Signorini problem with the BEM

Abstract

We propose two new projection methods for solving the Signorini problem and study the relationship between them. Since the Signorini boundary condition is equivalent to a projection fixed point problem, the methods formulate the Signorini boundary condition into a sequence of Robin boundary conditions and only require solving a variational problem with simple boundary conditions at each iterative step. The convergence speed of the first method depends on the parameter ρ heavily, and it is difficult to choose a proper ρ for individual problems. To improve the efficiency of the method, we propose an adaptive projection method which adjusts the parameter ρ automatically per iteration. We establish the convergence analysis of the methods. Because the Signorini boundary condition is given in an iterative form by the potential and its derivative on the boundary of the domain, we can easily provide the boundary element approximation of the problem. Finally, we present some numerical simulations which illustrate the performance of the methods.