Mimetic finite difference operators and higher order quadratures

Abstract

Mimetic finite difference operators textbf{D}, textbf{G} are discrete analogs of the continuous divergence (div) and gradient (grad) operators. In the discrete sense, these discrete operators satisfy the same properties as those of their continuum counterparts. In particular, they satisfy a discrete extended Gauss’ divergence theorem. This paper investigates the higher-order quadratures associated with the fourth- and sixth- order mimetic finite difference operators, and show that they are indeed numerical quadratures and satisfy the divergence theorem. In addition, extensions to curvilinear coordinates are treated. Examples in one and two dimensions to illustrate numerical results are presented that confirm the validity of the theoretical findings.