An Algorithmic Finite Element Approach for a Class of Non-linear Quasi Variational Inequalities

In this paper, we introduce a new method to analyze the convergence of the standard finite element method for a class of elliptic quasi-variational inequalities (QVIs) with non-linear source terms. It consists of approximating the solution of the non-linear QVI by a sequence of solutions of linear QVIs. Under a realistic assumption on the non-linearity, we first prove that the resulting iterative scheme is geometrically convergent to the solution of the QVI. Afterwards, we establish an error estimate in the maximum norm between the continuous iterative scheme and its finite element counterpart. Finally, combining this latter estimate with the geometrical convergence of the iterative scheme, we also derive an error estimate between the solution of the QVI and its finite element counterpart.