Approximation of boundary conditions for mimetic finite-difference methods

Abstract

The numerical solution of partial differential equations solved with finite-difference approximations that mimic the symmetry properties of the continuum differential operators and satisfy discrete versions of the appropriate integral identities are more likely to produce physically faithful results. Furthermore, those properties are often needed when using the energy method to prove convergence and stability of a particular difference approximation. Unless special care is taken, mimetic difference approximations derived for the interior grid points will fail to preserve the symmetries and identities between the gradient, curl, and divergence operators at the computational boundary. In this paper, we describe how to incorporate boundary conditions into finite-difference methods so the resulting approximations mimic the identities for the differential operators of vector and tensor calculus. The approach is valid for a wide class of partial differential equations of mathematical physics and will be described for Poisson's equation with Dirichlet, Neumann, and Robin boundary conditions. We prove that the resulting difference approximation is symmetric and positive definite for each of these boundary conditions.