A mimetic interpolation-free cell-centered finite volume scheme for the 2D and 3D heterogeneous anisotropic diffusion equations

We propose a mimetic interpolation -free cell -centered finite volume scheme for diffusion equations on arbitrary two-dimensional (2D) and three dimensional (3D) unstructured, possibly non star -shaped or nonplanar meshes. This scheme borrows ideas from the well known mimetic finite difference methods. This interpolation -free means that we do not need to introduce auxiliary unknowns in the process of constructing the scheme, and the final scheme has only cell -centered unknowns. After carefully handling continuity of solutions and normal fluxes, we construct piecewise linear approximations on each cell. This makes our scheme applicable to arbitrary discontinuities and arbitrary mesh topologies. Interpolation -free makes the program easy to implement and the 2D and 3D programs have a unified framework. It is worth mentioning that the traditional star -shaped mesh assumption has been abolished, and the new scheme has more relaxed restrictions on meshes. Some representative and even challenging numerical examples verify that the scheme is robust and almost has the optimal convergence order on some benchmark polygonal and polyhedral meshes.