Numerical analysis of a free boundary problem with non-local obstacles

The paper deals with the obstacle-like minimization problem in the cylindrical domain Ω=D×(−l,l)J(u)=∫_Ω|∇u|²dx+2∫_Dmax{v(x^′),0}dx^′, where x=(x^′,x_n), and v(x^′)=∫_−l^lu(x^′,x_n)dx_n. The corresponding Euler–Lagrange equation is Δu(x^′,x_n)=χ_{v>0}(x^′)+−∂_xnu(x^′,−l)+∂_xnu(x^′,l)χ_{v=0}(x^′). Due to the non-local nature of the obstacle, the comparison principle does not hold for the minimizers u(x), which makes the problem challenging both analytically and numerically. The standard optimization techniques such as Newton or quasi-Newton's methods require approximations of the Jacobians that are four dimensional tensors and are prohibitively expensive both in storage and computational time due to the nature of the three dimensional problem. In this paper, a new algorithm that can compute the global minimum is introduced. Non-trivial exact solutions have been constructed; and second order accuracy has been confirmed. Another important contribution is the numerical testing of the comparison principle for functions v(x^′), as conjectured by M. Chipot and the second author in Chipot and Mikayelyan (2022).