A two level algorithm for an obstacle problem

Due to the inequality feature of the obstacle problem, the standard quadratic finite element method for solving the problem can only achieve an error bound of the form O(N−3/4+ϵ),N being the total number of degrees of freedom, and ϵ > 0 arbitrary. To achieve a better error bound, the key lies in how to capture the free boundary accurately. In this paper, we propose a two level algorithm for solving the obstacle problem. The first part of the algorithm is through the use of the linear elements on a quasi-uniform mesh. Then information on the approximate free boundary from the linear element solution is used in the construction of a quadratic finite element method. Under some assumptions, it is shown that the numerical solution from the two level algorithm is expected to have a nearly optimal error bound of O(N−1+ϵ), ϵ > 0 arbitrary. Such an expected convergence order is observed numerically in numerical examples.