The mixed method with two Lagrange multiplier formulations for the Signorini problem

Abstract

Introducing stress as a new unknown, we consider the Signorini problem in the mixed form. The well-posedness theory of the continuous and discrete mixed variational inequalities has been established by reformulating them into projection problems. Two Lagrange multiplier formulations are introduced utilizing different projections. The equivalency among the Lagrange multiplier formulations and the mixed variational inequality is demonstrated in continuous and discrete sense, respectively. For the discrete mixed variational inequality, we develop the error estimates. Based on two Lagrange multiplier formulations, we design two Active/Inactive set algorithms and investigate the convergence. Several numerical experiments are conducted to verify the theoretical convergence rates of the finite element discretization and the iteration algorithms.