Qingguang Guan, Max Gunzburger


Numerical Analysis of Finite Element Method for Nonlocal Obstacle Problems


For classical obstacle problem, the obstacle function ψ should be a smooth function, however, we considered the nonlocal obstacle problem which can allow ψ to be a function with jump discontinuities. The Laplacian operator in classical obstacle problem is replaced by the nonlocal operator, for which the fractional Laplacian is a special case. We considered the problem of minimizing an energy functional with the constraint u ≥ ψ, where u is the solution of the problem. In a proper space, we proved the existence and uniqueness of the solution. A finite element method is applied to solve the minimization problem. The convergence of the numerical solution to the exact solution was proved. Also, we gave some numerical results to verify our theorems.