Monotone finite volume schemes for diffusion equations on polygonal meshes

We construct a nonlinear finite volume (FV) scheme for diffusion equation on star-shaped polygonal meshes and prove that the scheme is monotone, i.e., it preserves positivity of analytical solutions for strongly anisotropic and heterogeneous full tensor coefficients. Our scheme has only cell-centered unknowns, and it treats material discontinuities rigorously and offers an explicit expression for the normal flux. Numerical results are presented to show how our scheme works for preserving positivity on various distorted meshes for both smooth and non-smooth highly anisotropic solutions. And numerical results show that our scheme appears to be approximate second-order accuracy for the solution and first-order accuracy for the flux.