<p>This paper presents a new approach for modeling problems with local discontinuities. The method relies on the finite volume and extended finite element methods. The proposed approach builds on the foundations of the recently introduced Dual Mesh Control Domain Method (DMCDM), which elegantly connects the finite volume and finite element methods. In DMCDM, a primary mesh is used to discretize the domain, and finite element-type interpolation functions are used to build the approximation of the primary state variables. Additionally, a finite volume mesh, referred to as the dual mesh, is used to define the control domains in which integral statements of the governing equations are formulated. The main idea of the proposed approach is to enhance the approximation space of the primary mesh to account for discontinuities by relying on the extended finite element method (XFEM). This is achieved by employing the intrinsic version of XFEM to construct new shape functions that account for the discontinuities in the domain without introducing additional unknowns. The modified shape functions are derived locally near the discontinuity using the moving least squares (MLS) approach. As a result, special shape functions are obtained near the discontinuities, while standard FE shape functions are used elsewhere. While the proposed approach is general, in this study, it is developed for the Poisson equation to demonstrate its performance in the presence of weak discontinuities in the domain. The study shows that the proposed method improves the accuracy and convergence of the finite volume method (FVM) for problems with a local discontinuity in the case of non-conforming meshes.</p>