Barzilai-borwein Method for a Nonlocal Elliptic Problem

The object of interest in the present paper is a nonlocal nonlinear problem for a general second order elliptic operator. The problem under consideration represents a model of nonlocal reaction diffusion process. Furthermore, applications in computational biology are also available. The strong problem is reduced to a discrete minimization problem. The approximate problem is obtained by Lagrangian finite element discretizations. Due to its simplicity and efficiency, the Barzilai and Borwein gradient method is used for finding positive solutions with respect to the inhomogeneous strong Allee effect growth pattern. The corresponding fast and stable iterative algorithm converges monotonically with respect to the objective functional. A rigorous proof of the monotone convergence theorem is presented. Computer implementations of the method support the considered theory.