Abstract
Nonlocal boundary value problems for elliptic equations arise in various mathematical models and applications, where classical local boundary conditions are insufficient to describe the underlying phenomena. Developing accurate and stable numerical methods for such problems is therefore important, particularly in the presence of operator-valued and dependent coefficients. In this work, first- and second-order accurate two-step difference schemes are proposed for the approximate solution of a nonlocal boundary value problem with Samarskii-Ionkin condition I for an elliptic equation in a Banach space with a strongly positive operator. A coercive stability estimate for the solutions of the proposed difference schemes for elliptic equations with dependent coefficients is established. The abstract results are then applied to construct stable difference schemes for the corresponding elliptic nonlocal boundary value problem. The reliability of the theoretical results is demonstrated through corresponding numerical experiments. In addition, a new algorithm is developed for the numerical solution of the resulting difference schemes.