Numerical Enclosure-Methods for the Weak Solution of Elliptic Problems with Unilateral Constraints

In this paper we discuss elliptic boundary value problems of second order with unilateral constraints and give a complete numerical solution scheme related to the lately developed Verification-Methods or E-Methods which are the basis for computations to prove automatically the existence, uniqueness, and an enclosure of the solution. Here we apply the penalty method to prove the existence and uniqueness of the solution of the variational problem and to get finite element approximations which we prove to converge to the solution of the analytic problem. We give the associated bounds for error estimates and then develop direct and iterative E-methods for verified computations.