Abstract: In this paper, we provide a relatively complete a posteriori error analysis for the regularization method via duality theory for elliptic variational inequalities. The model problems considered in the paper are a friction contact problem and an obstacle problem, respectively. Choosing a different bounded operator form and a functional form, we perform their dual formations and give an $H^1$-norm a posteriori error estimation based on the regularization method which is usually used in solving non-differentiable minimization problems. A posteriori error estimates, with residual type for an obstacle problem in the general framework, is established by using duality theory in convex analysis. At the same time, we make a particular choice of the dual variable that leads to a residual-based error estimate of the model problem and its efficiency. A posteriori error estimates for numerical solutions are the basis for developing efficient adaptive algorithms, whereas a posteriori estimates for modeling errors are useful for analyzing the effects of uncertainties in problem data on the solution.

Key words: a posteriori error estimation, regularization method, elliptic variational inequality; efficiency, dual theory