On 1D diffusion problems with a gradient-dependent diffusion coefficient

The mimetic finite difference (MFD) methods mimic important properties of physical and mathematical models. As a result, conservation laws, solution symmetries, and the fundamental identities of the vector and tensor calculus are held for discrete models. The MFD methods retain these attractive properties for full tensor coefficients and arbitrary polygonal meshes which may include non-convex and degenerate elements. The existing MFD methods for solving diffusion-type problems are second-order accurate for the conservative variable (temperature, pressure, energy, etc.) and only first-order accurate for its flux. We developed new high-order MFD methods which are second-order accurate for both scalar and vector variables. The second-order convergence rates are demonstrated with a few numerical examples on randomly perturbed quadrilateral and polygonal meshes.