A monotone finite volume scheme for diffusion equations on general non-conforming meshes

A nonlinear monotone finite volume scheme on general non-conforming meshes for diffusion equations is introduced, which deals with discontinuous tensor coefficients rigorously. Since the expression of normal flux depends on auxiliary unknowns defined at cell-vertex including hanging nodes, we propose a new method to eliminate vertex-unknown by using primary unknowns at the centers of the cells sharing the vertex. Especially the unknowns defined on hanging nodes are eliminated by flux continuous conditions. The resulting scheme is monotone and preserves positivity of analytical solutions for strongly anisotropic and heterogeneous full tensor coefficient problems. Numerical results show that the convergent order of the monotone scheme by different methods of eliminating vertex unknowns will vary remarkably, and our new method can assure that it has almost second order accuracy and more accurate than some existing methods.