Convergence of cubic-spline approach to the solution of a system of boundary-value problems

We use cubic spline to derive some consistency relations which are then used to develop a numerical method for the solution of a system of fourth-order boundary-value problems associated with obstacle, unilateral, and contact problems. It is known that a class of variational inequalities related to contact problems in elastostatics can be characterized by a sequence of variational inequations, which are solved using some numerical method. Boundary formula of order O(h8)O(h8) are formulated. The most common approach for convergence analysis are using monotonicity of the coefficient matrix. But here we study a new approach and give the convergence of prescribed method, so that the matrix associated with the system of linear equations that arises, is not required to be monotone. Numerical examples are given to show the applicability and efficiency of our method.