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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
J.E. Castillo, G.F. Miranda, Mimetic Discretization Methods, 1st edn. (CRC Press, Boca Raton, 2013)
J.E. Castillo, J.M. Hyman, M.J. Shashkov, S. Steinberg, The sensitivity and accuracy of fourth order finite difference schemes on nonuniform grids in one dimension. Comput. Math. Appl. 30(8), 41–55 (1995)
E. Sanchez, C. Paolini, P. Blomgren, J. Castillo, Algorithms for higher-order mimetic operators, in Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014: Selected papers from the ICOSAHOM conference, June 23–27, 2014, Salt Lake City, Utah, USA (Springer International Publishing, Cham, 2015), pp. 425–434
E.J. Sanchez, Mimetic finite differences and parallel computing to simulate carbon dioxide subsurface mass transport, PhD thesis, Claremont Graduate University and San Diego State University, 2015
G.B. Dantzig, Linear programming and extensions, Technical report, August 1963
J.E. Castillo, J.M. Hyman, M. Shashkov, S. Steinberg, Fourth- and sixth-order conservative finite difference approximations of the divergence and gradient. Appl. Numer. Math. 37, 171–187 (2001)
E.J. Sanchez, C.P. Paolini, J.E. Castillo, The mimetic methods toolkit: an object-oriented api for mimetic finite differences. J. Comput. Appl. Math. 270, 308–322 (2014). Fourth International Conference on Finite Element Methods in Engineering and Sciences (FEMTEC 2013)
E.J. Sanchez, Mimetic Methods Toolkit (MTK) (2012), http://www.csrc.sdsu.edu/mtk/,
J.B. Runyan, A novel higher order finite difference time domain method based on the Castillo-Grone mimetic curl operator with application concerning the time-dependent Maxwell equations, Master’s thesis, San Diego State University, San Diego, CA, 2011
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Sanchez, E.J., Miranda, G.F., Cela, J.M., Castillo, J.E. (2017). Supercritical-Order Mimetic Operators on Higher-Dimensional Staggered Grids. In: Bittencourt, M., Dumont, N., Hesthaven, J. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016. Lecture Notes in Computational Science and Engineering, vol 119. Springer, Cham. https://doi.org/10.1007/978-3-319-65870-4_48
Download citation
DOI: https://doi.org/10.1007/978-3-319-65870-4_48
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-65869-8
Online ISBN: 978-3-319-65870-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)