Finite element approximation of the pure Neumann problem using the iterative penalty method

When we use penalty method to solve the pure Neumann problem, the associated stiffness matrix we get is ill-conditioned, which leads to the unstable computation. In this paper, we design an iterative penalty method for the problem and prove some error estimates. In our algorithm, we can use a not very small penalty parameter to avoid the unstable computation. Numerical examples are given to show the algorithm is very effective and powerful.