Monotone finite volume schemes for diffusion equations on polygonal meshes

Weconstruct a nonlinear finite volume (FV) scheme for diffusion equationon star-shapedpolygonalmeshes andprove that the scheme ismonotone, 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 distortedmeshes 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. 2008 Elsevier Inc. All rights reserved. MSC: 65M06; 65M12; 65M55