Two projection methods for the solution of Signorini problems

Abstract

Two iterative methods, based on projection and boundary element methods, are considered for Signorini problems. The regularized problem with the projection boundary condition is first deduced from the Signorini problem. By using the equivalence between the Signorini boundary condition and the projection fixed point problem, our methods formulate the Signorini boundary condition into a sequence of Robin boundary conditions. In the new boundary condition there is a penalty parameter ρ. The convergence speed of the first method is greatly influenced by the value of ρ which is difficult to choose for individual problems. To improve the performance of this method, we present a self-adaptive projection method which adjusts the parameter ρ automatically per iteration based on the iterative data. The main result of this work is to provide the convergence of the methods under mild assumptions. As the iteration process is given by the potential and its derivative on the boundary of the domain, the unknowns of the problem are computed explicitly by using the boundary element method. Both theoretical results and numerical experiments indicate efficiency of the methods proposed.