Solving the obstacle problem : Finite element simulation with a study on exact solutions

In this paper, we focus on the elliptic and the parabolic obstacle problem. We will discuss the elliptic obstacle problem in both one and two dimensions. On the other hand, we will consider the parabolic obstacle problem in one dimension only. We use the finite element method and quadratic programming in MATLAB to simulate solutions. We provide the corresponding numerical implementation and show that there exist some elegant exact solutions to the elliptic obstacle problem in specific cases. The method we use could be a step towards finding more exact solutions of obstacle problems.