Monotone Finite Volume Scheme For Diffusion Equation On Nonconforming Meshes

In this paper, a nonlinear finite volume scheme is constructed for sta-tionary diffusion equation on nonconforming rectangular meshes and random quadrilateral meshes, and the monotonicity of the scheme is proven.Firstly, the auxiliary meshes, which is called vertex-centered control vol-umes on the primary nonconforming meshes, is introduced. While constructing the scheme by using integral interpolation, we need cell-center unknowns and cell-vertex unknowns. In order to reduce the computational costs, we eliminate the cell-center unknowns through the interpolation method based on Taylor expansion, so there are only cell-vertex unknowns in the resulting finite volume scheme, and the scheme have explicit expression of discrete flux and have local stencil.In particular, a monotone finite volume scheme is built for heterogeneous full tensor coefficient. The gradient is discontinuous on both sides of discontin-uous lines, we reconstruct the scheme by a new method based on the continuity of tangential gradient and normal flux through both sides of the discontinuous lines in order to get the explicit expression of discrete flux.Afterwards, numerical results are presented to show how our scheme work-s for positivity-preserving on nonconforming rectangular meshes and random quadrilateral meshes for both smooth and non-smooth highly anisotropic so-lutions.And numerical results show that the scheme is robust and effective.