  The obstacle problem is a class of free boundary problems which finds
applications in many disciplines such as porous media, financial mathematics
and optimal control. In this paper, we propose two operator-splitting methods
to solve the linear and nonlinear obstacle problems. The proposed methods have
three ingredients: (i) Utilize an indicator function to formularize the
constrained problem as an unconstrained problem, and associate it to an initial
value problem. The obstacle problem is then converted to solving for the steady
state solution of an initial value problem. (ii) An operator-splitting strategy
to time discretize the initial value problem. After splitting, a heat equation
with obstacles is solved and other subproblems either have explicit solutions
or can be solved efficiently. (iii) A new constrained alternating direction
explicit method, a fully explicit method, to solve heat equations with
obstacles. The proposed methods are easy to implement, do not require to solve
any linear systems and are more efficient than existing numerical methods while
keeping similar accuracy. Extensions of the proposed methods to related free
boundary problems are also discussed.
