Error estimates of an efficient linearization scheme for a nonlinear elliptic problem with a nonlocal boundary condition

We consider a nonlinear second order elliptic boundary value problem (BVP) in a bounded domain $\Omega\subset {\mathbb R}^N$ with a nonlocal boundary condition. A Dirichlet BC containing an unknown additive constant, accompanied with a nonlocal (integral) Neumann side condition is prescribed at some boundary part Γn. The rest of the boundary is equipped with Dirichlet or nonlinear Robin type BC. The solution is found via linearization. We design a robust and efficient approximation scheme. Error estimates for the linearization algorithm are derived in L2(Ω),H1(Ω) and L∞(Ω) spaces