Supercritical-Order Mimetic Operators on Higher-Dimensional Staggered Grids

Abstract

We focus on the construction of 2- and 3D mimetic gradient, divergence, curl, and Laplacian operators. We base this work on the method by Castillo and Grone, which constructs mimetic gradient and divergence operators via a discrete instance of Gauss’ divergence theorem. This method can not construct tenth-order gradient nor eighth-order divergence operators (nor higher) because the computed weights discretizing the corresponding weighted inner products are not all positive for these cases. Thus, we define the tenth order and the eighth order thresholds as critical orders of accuracy for the gradient and divergence operators, respectively. In previous works, we introduced the Castillo–Blomgren–Sanchez algorithm. This algorithm constructs supercritical-order mimetic operators. The contribution of this work is the extension to higher dimensions of the operators constructed by this algorithm. This includes detailing the mathematics of this extension. We also detail the construction of a mimetic curl operator via a linear combination of the divergence of auxiliary Gaussian fluxes. This avoids any interpolation from classic discretization approaches based on Stokes’ theorem. We validate our operators by solving higher-dimensional elliptic problems.