An approach to the numerical verification of solutions for variational inequalities using Schauder fixed point theory

In this paper, we describe a numerical method to verify the existence of solutions for a unilateral boundary value problems for second order equation governed by the variational inequalities. It is based on Nakao's method by using finite element approximation and its explicit error estimates for the problem. Using the Riesz representation theory in Hilbert space, we first transform the iterative procedure of variational inequalities into a fixed point form. Then, using Schauder fixed point theory, we construct a high efficiency numerical verification method that through numerical computation generates a bounded, closed, convex set which includes the approximate solution. Finally, a numerical example is illustrated.