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

The support operators method of discretizing partial differential equations produces discrete analogs of continuum initial boundary value problems tha t exactly satisfy discrete conservation laws analogous to those satisfied b y the continuum system. Thus, the stability of the method is assured, but c urrently there is no theory that predicts the accuracy of the method on non uniform grids. In this paper, we numerically investigate how the accuracy, particularly the accuracy of the fluxes, depends on the definition of the i nner product for discrete vector fields. We introduce two different discret e inner products, the standard inner product that we have used previously a nd a new more accurate inner product. The definitions of these inner produc ts are based on interpolation of the fluxes of vector fields. The derivatio n of the new inner product is closely related to the use of the Piola trans form in mixed finite elements. Computing the formulas for the new accurate inner product requires a nontrivial use of computer algebra. From the resul ts of our numerical experiments, we can conclude that using more accurate i nner product produces a method with the same order of convergence as the st andard inner product, but the constant in error estimate is about three tim es 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. (C) 2001 Elsevier Science Ltd. All rights reserved .