Spectral properties and conservation laws in Mimetic Finite Difference methods for PDEs

The Mimetic Finite Difference (MFD) methods for PDEs mimic crucial properties of mathematical systems: duality and self-adjointness of differential operators, conservation laws and properties of the solution on general polytopal meshes. In this article the structure and the spectral properties of the linear systems derived by the spatial discretization of diffusion problem are analysed. In addition, the numerical approximation of parabolic equations is discussed where the MFD approach is used in the space discretization while implicit ϑϑ-method and explicit Runge–Kutta–Chebyshev schemes are used in time discretization. Moreover, we will show how the numerical solution preserves certain conservation laws of the theoretical solution.