Nonlocal problem for a general second-order elliptic operator

The paper deals with a nonlinear second-order elliptic equation with Dirichlet boundary conditions. The nonlocal term involved in the strong problem essentially increases its complexity and the necessary total computational work. The existence and uniqueness of the weak solution is established. The nonlinear weak formulation is reduced to the minimization of a nonlinear functional. Finite element discretizations by Lagrangian finite elements are applied to obtain an approximate minimization problem. A two-point step size gradient method with an original steplength is used for finding approximate solutions of the problem under consideration. No line search is necessary in the new approach. The present method is computer implemented and tested on different triangulations. The test examples indicate that the method slightly depends on the initial guesses.