We introduce a new finite volume method for highly anisotropic diffusion operators on triangular cells. The main idea is to calculate the gradient using a nonlinear scheme. For parabolic problems, the resulting global matrix is a strictly diagonally dominant M-Matrix without geometrical constraints on the mesh and restrictive conditions on the anisotropy ratio. We verify the accuracy of the method by comparing our computed solutions with analytical solutions. The efficiency of the algorithm is demonstrated by comparing it with numerical schemes which do not satisfy discrete minimum and maximum principles.