The mimetic finite difference method for elliptic and parabolic problems with a staggered discretization of diffusion coefficient
Numerical schemes for nonlinear parabolic equations based on the harmonic averaging of cell-centered diffusion coefficients break down when some of these coefficients go to zero or their ratio grows. To tackle this problem, we propose new mimetic finite difference schemes that use a staggered discretization of the diffusion coefficient. The primary mimetic operator approximates div(k?) div ( k ? ) ; the derived (dual) mimetic operator approximates ??(?) ? ? ( ? ) . The new mimetic schemes preserve symmetry and positive-definiteness of the continuum problem which allows us to use algebraic solvers with optimal complexity. We perform detailed numerical analysis of the new schemes for linear elliptic problems and a specially designed linear parabolic problem that has solution dynamics typical for nonlinear problems. We show that the new schemes are competitive with the state-of-the-art schemes for steady-state problems but provide much more accurate solution dynamics for the transient problem.
mimetic finite differences elliptic and parabolic problems compatible discretizations unstructured polygonal meshes