**1. Introduction**

Numerical verification methods of solutions for differential equations have been the subject of extensive study in recent years and much progress have been made both mathematically and computationally [1–23]. However, for some problems governed by the elliptic variational inequalities, there are very few approaches. As far as we know, it is hard to find any applicable methods except for those of Nakao and Ryoo [13, 24–46].

The authors have studied for several years the numerical verification method of solutions for elliptic variational inequalities using finite element method and the constructive error estimates combining with Schauder's and Banach's fixed point theorem. Several results in our research are already published in [13, 24–46]. In this chapter, we briefly overview our resent research results including works not yet published.

The outline of this chapter is as follows. In Section 2, the two types of elliptic variational inequalities are considered. In Subsection 2.1, we describe the elliptic variational inequalities and give a fixed point formulation to prove the existence of solutions. In Subsections 2.2 and 2.3, the main tool of the verification method is explained at an abstract level. In Subsection 2.2, we present a simple iteration method for numerical verification of solutions for the elliptic variational inequalities. We construct the concepts of rounding and rounding error for functions and present a computer algorithm to construct the set satisfying the verification conditions. However, it is difficult to apply the method in Subsection 2.2 to a problem in which an associated operator is not retractive in a neighborhood of the solution, because it is based upon a simple iteration method. In Subsection 2.3, we propose

another approach to overcome such a difficulty. This method can be applied to general elliptic variational inequalities without any retraction property of the associated operator. We introduce a Newton-like operator and reformulate the problem using it. Particularly, special emphasis is placed on the way to devise the Newtonlike operator for a kind of non-differentiable map which defines the original problem. We introduce a computational verification condition. In order to show a concrete usage of the tool, in Section 3, we present an application to some problems governed by the elliptic variational inequalities. Many difficulties remain to be overcome in the construction of general techniques applicable to a broader range of problems. However, the authors have no doubt that investigation along this line will lead to a new approach employing numerical methods in the field of existence theory of solutions for various variational inequalities that appear in mathematical analysis.
