Volume 45, pp. 457-475, 2016.

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

J. Blanco, O. Rojas, C. Chacón, J. M. Guevara-Jordan, and J. Castillo

Abstract

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.

Full Text (PDF) [1.8 MB]

Key words

mimetic finite differences, tensor products, locally refined grids, elliptic equations

AMS subject classifications

65H17, 65N06, 40A30

Links to the cited ETNA articles

[3]Vol. 34 (2008-2009), pp. 152-162 E. D. Batista and J. E. Castillo: Mimetic schemes on non-uniform structured meshes

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