The effect of inner products for discrete vector fields on the accuracy of mimetic finite difference methods

The support operators method of discretizing partial differential equations produces discrete analogs of continuum initial boundary value problems that exactly satisfy discrete conservation laws analogous to those satisfied by the continuum system. Thus, the stability of the method is assured, but currently there is no theory that predicts the accuracy of the method on nonuniform grids. In this paper, we numerically investigate how the accuracy, particularly the accuracy of the fluxes, depends on the definition of the inner product for discrete vector fields. We introduce two different discrete inner products, the standard inner product that we have used previously and a new more accurate inner product. The definitions of these inner products are based on interpolation of the fluxes of vector fields. The derivation of the new inner product is closely related to the use of the Piola transform in mixed finite elements. Computing the formulas for the new accurate inner product requires a nontrivial use of computer algebra. From the results of our numerical experiments, we can conclude that using more accurate inner product produces a method with the same order of convergence as the standard inner product, but the constant in error estimate is about three times less. However, the method based on the standard inner product is easier to compute with and less sensitive to grid irregularities, so we recommend its use for rough grids.