Analysis and optimization of inner products for mimetic finite difference methods on a triangular grid

Abstract

The support operator method designs mimetic finite difference schemes by first constructing a discrete divergence operator based on the divergence theorem, and then defining the discrete gradient operator as the adjoint operator of the divergence based on the Gauss theorem connecting the divergence and gradient operators, which remains valid also in the discrete case. When evaluating the discrete gradient operator, one needs to define discrete inner products of two discrete vector fields. The local discrete inner product on a given triangle is defined by a 3×3 symmetric positive definite matrix M defined by its six independent elements–parameters. Using the Gauss theorem over our triangle, we evaluate the discrete gradient in the triangle. We require the discrete gradient to be exact for linear functions, which gives us a system of linear equations for elements of the matrix M. This system, together with inequalities which guarantee positive definiteness of the matrix M, results in a one parameter family of inner products which give exact gradients for linear functions. The traditional inner product is a member of this family. The positive free parameter can be used to improve another property of the discrete method. We show that accuracy of the method for quadratic functions improves with decreasing this parameter, however, at the same time, the condition number of the matrix M, which is the local matrix of the linear system for computing the discrete gradient, increases to infinity when the parameter goes to zero, so one needs to choose a compromise between accuracy and solvability of the local system. Our analysis has been performed by computer algebra tools which proved to be essential.