<p>A cell-centered nonlinear correction finite volume (NCFV) scheme preserving the discrete extremum principle is presented to solve diffusion equations with discontinuous and anisotropic coefficient. Its main idea is that we employ a nonlinear technique to modify the structure of the conservative flux in the monotone scheme (Sheng and Yuan 2016), thereby establishing the first bridge from monotone-preserving schemes to extremum-preserving schemes. When constructing the two-point flux, auxiliary unknowns including cell-vertex and cell-edge unknowns are introduced to compute the two nonlinear coefficients of the flux. The positivity of these nonlinear coefficients allows the elimination of all approximation restrictions on the auxiliary unknowns. In other words, our new scheme unconditionally satisfies the discrete extremum principle and can be applied to diffusion problems with discontinuous coefficient on arbitrarily distorted meshes. Furthermore, we conduct a theoretical analysis that includes the priori estimate under a coercivity assumption, the effectiveness of extremum-preserving property and the existence of solution. Numerical results demonstrate that our scheme effectively handles diffusion equations with discontinuous and anisotropic coefficient‌, rigorously satisfies the discrete extremum principle‌, and achieves superior accuracy in specific cases compared to existing schemes‌.</p>