Iterative scheme for an elliptic non-local free boundary problem

System of variational inequalities with interconnected obstacles

“…Related stationary models were considered in [27,28]. Subsequently, a multidimensional stationary version of the model was introduced in [29] where an iterative scheme was suggested for its numerical solution. Here, we consider the problem discussed in [29] in a more general framework and reformulate it in terms of a switching problem.…”

Numerical Treatment to a Non-local Parabolic Free Boundary Problem Arising in Financial Bubbles

“…is the iterative algorithm given by (5), and u = lim k→∞ u k , then u is a unique continuous viscosity solution of (1).…”

“…To prove that u 2 (t, x) ≥ u 1 (t, x) let us examine equation (5). When k = 1 the obstacle is Ψ 2 := u 1 (t, −x) + ψ(t, x) and for the case k = 0 the obstacle is Ψ 1 := u 0 (t, −x) + ψ(t, x).…”

“…Let |ρx| ≤ σ 2 ∆x for every x ∈ Ω, and w(t, x) be a continuous viscosity solution to the parabolic bubble problem (4) determined on [0, T ] × Ω. Define u (k) (t, x) to be an increasing iterative sequence (5). If we set by u Proof.…”

“…Other stationary models were studied in [3,4]. The multidimensional stationary problem was considered in [5] where existence and uniqueness of the viscosity solution were proved.…”