Tensor formulation of 3-d mimetic finite differences and applications to elliptic problems∗

The mimetic discretization of a boundary value problem (BVP) seeks to reproduce the same underlying properties that are satisfied by the continuous solution. In particular, the Castillo-Grone mimetic finite difference gradient and divergence fulfill a discrete version of the integration-by-parts theorem on 1-D staggered grids. For the approximation to this integral principle, a boundary flux operator is introduced that also intervenes with the discretization of the given BVP. In this work, we present a tensor formulation of these three mimetic operators on three-dimensional rectangular grids. These operators are used in the formulation of new mimetic schemes for second-order elliptic equations under general Robin boundary conditions. We formally discuss the consistency of these numerical schemes in the case of second-order discretizations and also bound the eigenvalue spectrum of the corresponding linear system. This analysis guarantees the non-singularity of the associated system matrix for a wide range of model parameters and proves the convergence of the proposed mimetic discretizations. In addition, we easily construct fourth-order accurate mimetic operators and extend these discretizations to rectangular grids with a local refinement in any direction. Both of these numerical capabilities are inherited from the original tensor formulation. As a numerical assessment, we solve a boundary-layer test problem with increasing difficulty as a sensitivity parameter is gradually adjusted. Results on uniform grids show optimal convergence rates while the solutions computed after a smooth grid clustering exhibit a significant gain in accuracy for the same number of grid cells.