Geodesic
Tensor formulation of 3-D mimetic finite differences and applications to elliptic problems
Electronic transactions on numerical analysis, Tome 45 (2016), pp. 457-475
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.
Classification : 65H17, 65N06, 40A30
Keywords: mimetic finite differences, tensor products, locally refined grids, elliptic equations 
@article{ETNA_2016__45__a2,
     author = {Blanco,  J. and Rojas,  O. and Chac\'on,  C. and Guevara-Jordan,  J.M. and Castillo,  J.},
     title = {Tensor formulation of {3-D} mimetic finite differences and applications to elliptic problems},
     journal = {Electronic transactions on numerical analysis},
     pages = {457--475},
     year = {2016},
     volume = {45},
     zbl = {1355.65141},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2016__45__a2/}
}
TY  - JOUR
AU  - Blanco,  J.
AU  - Rojas,  O.
AU  - Chacón,  C.
AU  - Guevara-Jordan,  J.M.
AU  - Castillo,  J.
TI  - Tensor formulation of 3-D mimetic finite differences and applications to elliptic problems
JO  - Electronic transactions on numerical analysis
PY  - 2016
SP  - 457
EP  - 475
VL  - 45
UR  - http://geodesic.mathdoc.fr/item/ETNA_2016__45__a2/
LA  - en
ID  - ETNA_2016__45__a2
ER  - 
%0 Journal Article
%A Blanco,  J.
%A Rojas,  O.
%A Chacón,  C.
%A Guevara-Jordan,  J.M.
%A Castillo,  J.
%T Tensor formulation of 3-D mimetic finite differences and applications to elliptic problems
%J Electronic transactions on numerical analysis
%D 2016
%P 457-475
%V 45
%U http://geodesic.mathdoc.fr/item/ETNA_2016__45__a2/
%G en
%F ETNA_2016__45__a2
Blanco,  J.; Rojas,  O.; Chacón,  C.; Guevara-Jordan,  J.M.; Castillo,  J. Tensor formulation of 3-D mimetic finite differences and applications to elliptic problems. Electronic transactions on numerical analysis, Tome 45 (2016), pp. 457-475. http://geodesic.mathdoc.fr/item/ETNA_2016__45__a2/