SPECIAL SECTION

Spectral Upscaling for Graph Laplacian Problems with Application to Reservoir Simulation

Abstract

We consider coarsening procedures for graph Laplacian problems written in a mixed saddle-point form. In that form, in addition to the original (vertex) degrees of freedom (dofs), we also have edge degrees of freedom. We extend previously developed aggregation-based coarsening procedures applied to both sets of dofs [P. S. Vassilevski and L. T. Zikatanov, Numer. Linear Algebra Appl., 21 (2014), pp. 297--315] to now allow more than one coarse vertex dof per aggregate. Those dofs are selected as certain eigenvectors of local graph Laplacians associated with each aggregate. Additionally, we coarsen the edge dofs by using traces of the discrete gradients of the already constructed coarse vertex dofs. These traces are defined on the interface edges that connect any two adjacent aggregates. The overall procedure is a modification of the spectral upscaling procedure developed in [P. Jenny, S. H. Lee, and H. A. Tchelepi, J. Comput. Phys., 187 (2003), pp. 47--67] for the mixed finite element discretization of diffusion type PDEs which has the important property of maintaining inf-sup stability on coarse levels and having provable approximation properties. We consider applications to partitioning a general graph and to a finite volume discretization interpreted as a graph Laplacian, developing consistent and accurate coarse-scale models of a fine-scale problem.

Keywords

  1. graph Laplacian
  2. finite volume methods
  3. numerical upscaling
  4. algebraic multigrid
  5. reservoir simulation

MSC codes

  1. 65N55
  2. 65N08
  3. 65F15

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References

1.
A. Brandt, Multiscale solvers and systematic upscaling in computational physics, Comput. Phys. Commun., 169 (2005), pp. 438--441, https://doi.org/10.1016/j.cpc.2005.03.097.
2.
J. Brannick, Y. Chen, J. Kraus, and L. Zikatanov, Algebraic multilevel preconditioners for the graph Laplacian based on matching in graphs, SIAM J. Numer. Anal., 51 (2013), pp. 1805--1827, https://doi.org/10.1137/120876083.
3.
M. A. Christie and M. J. Blunt, Tenth SPE comparative solution project: A comparison of upscaling techniques, SPE Reservoir Simulation Symposium, Society of Petroleum Engineers, 2001, pp. 308--317, https://doi.org/10.2118/66599-MS.
4.
P. D'Ambra and P. S. Vassilevski, Adaptive AMG with coarsening based on compatible weighted matching, Comput. Vis. Sci., 16 (2013), pp. 59--76, https://doi.org/10.1007/s00791-014-0224-9.
5.
C. L. Farmer, Upscaling: A review, Internat. J. Numer. Methods Fluids, 40 (2002), pp. 63--78, https://doi.org/10.1002/fld.267.
6.
V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer Ser. Comput. Math. 5, Springer-Verlag, Berlin, 1986, https://doi.org/10.1007/978-3-642-61623-5.
7.
X. Hu, P. S. Vassilevski, and J. Xu, A two-grid SA-AMG convergence bound that improves when increasing the polynomial degree, Numer. Linear Algebra Appl., 23 (2016), pp. 746--771, https://doi.org/10.1002/nla.2053.
8.
J. D. Jansen, R. M. Fonseca, S. Kahrobaei, M. M. Siraj, G. M. Van Essen, and P. M. J. Van den Hof, The egg model -- A geological ensemble for reservoir simulation, Geoscience Data Journal, 1 (2014), pp. 192--195, https://doi.org/10.1002/gdj3.21.
9.
J. D. Jansen, G. M. Van Essen, S. Kahrobaei, M. Siraj, P. M. J. Van den Hof, and R. M. Fonseca, The egg model - data files, 2013, https://doi.org/10.4121/uuid:916c86cd-3558-4672-829a-105c62985ab2.
10.
P. Jenny, S. H. Lee, and H. A. Tchelepi, Multi-scale finite volume method for elliptic problems in subsurface flow simulation, J. Comput. Phys., 187 (2003), pp. 47--67, https://doi.org/10.1016/S0021-9991(03)00075-5.
11.
P. Jenny, S. H. Lee, and H. A. Tchelepi, Adaptive fully implicit multi-scale finite-volume method for multi-phase flow and transport in heterogeneous porous media, J. Comput. Phys., 217 (2006), pp. 627--641, https://doi.org/10.1016/j.jcp.2006.01.028.
12.
D. Z. Kalchev, C. S. Lee, U. Villa, Y. Efendiev, and P. S. Vassilevski, Upscaling of mixed finite element discretization problems by the spectral AMGe method, SIAM J. Sci. Comput., 38 (2016), pp. A2912--2933, https://doi.org/10.1137/15M1036683.
13.
G. Karypis and V. Kumar, A fast and high quality multilevel scheme for partitioning irregular graphs, SIAM J. Sci. Comput., 20 (1998), pp. 359--392, https://doi.org/10.1137/S1064827595287997.
14.
I. V. Lashuk and P. S. Vassilevski, Element agglomeration coarse Raviart–-Thomas spaces with improved approximation properties, Numer. Linear Algebra Appl., 19 (2012), pp. 414--426, https://doi.org/10.1002/nla.1819.
15.
I. V. Lashuk and P. S. Vassilevski, The construction of coarse de Rham complexes with improved approximation properties, Comput. Methods Appl. Math., 14 (2014), pp. 257--303, https://doi.org/10.1515/cmam-2014-0004.
16.
S. H. Lee, C. Wolfsteiner, and H. A. Tchelepi, Multiscale finite-volume formulation of multiphase flow in porous media: Black oil formulation of compressible, three-phase flow with gravity, Comput. Geosci., 12 (2008), pp. 351--366, https:/doi.org/10.1007/s10596-007-9069-3.
17.
O. Livne and A. Brandt, Lean algebraic multigrid (LAMG): Fast graph Laplacian linear solver, SIAM J. Sci. Comput., 34 (2012), pp. B499–--B522, https://doi.org/10.1137/110843563.
18.
I. Lunati and P. Jenny, Multiscale finite-volume method for compressible multiphase flow in porous media, J. Comput. Phys., 216 (2006), pp. 616--636, https://doi.org/10.1016/j.jcp.2006.01.001.
19.
K. A. Mardal and R. Winther, Preconditioning discretizations of systems of partial differential equations, Numer. Linear Algebra Appl., 18 (2011), pp. 1--40, https://doi.org/10.1002/nla.716.
20.
A. Napov and Y. Notay, An efficient multigrid method for graph Laplacian systems, Electron. Trans. Numer. Anal., 45 (2016), pp. 201--218.
21.
Y. Notay, An aggregation-based algebraic multigrid method, Electron. Trans. Numer. Anal., 37 (2010), pp. 123--146.
22.
J. E. Pasciak and P. S. Vassilevski, Exact de Rham sequences of spaces defined on macro-elements in two and three spatial dimensions, SIAM J. Sci. Comput., 30 (2008), pp. 2427--2446, https://doi.org/10.1137/070698178.
23.
H. A. Tchelepi, P. Jenny, S. H. Lee, and C. Wolfsteiner, Adaptive multiscale finite-volume framework for reservoir simulation, SPE J., 12 (2007), pp. 188--195.
24.
P. Vaněk, J. Mandel, and M. Brezina, Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems, Computing, 56 (1996), pp. 179--196, https://doi.org/10.1007/BF02238511.
25.
P. S. Vassilevski, Sparse matrix element topology with application to AMG(e) and preconditioning, Numer. Linear Algebra Appl., 9 (2002), pp. 429--444, https://doi.org/10.1002/nla.300.
26.
P. S. Vassilevski, Multilevel Block-Factorization Preconditioners: Matrix-based Analysis and Algorithms for Solving Finite Element Equations, Springer-Verlag, New York, 2008, https://doi.org/10.1007/978-0-387-71564-3.
27.
P. S. Vassilevski, Coarse spaces by algebraic multigrid: Multigrid convergence and upscaling error estimates, Adv. Adapt. Data Anal., 3 (2011), pp. 229--249, https://doi.org/10.1142/S1793536911000830.
28.
P. S. Vassilevski and L. T. Zikatanov, Commuting projections on graphs, Numer. Linear Algebra Appl., 21 (2014), pp. 297--315, https://doi.org/10.1002/nla.1872.

Information & Authors

Information

Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: S323 - S346
ISSN (online): 1095-7197

History

Submitted: 31 May 2016
Accepted: 1 February 2017
Published online: 26 October 2017

Keywords

  1. graph Laplacian
  2. finite volume methods
  3. numerical upscaling
  4. algebraic multigrid
  5. reservoir simulation

MSC codes

  1. 65N55
  2. 65N08
  3. 65F15

Authors

Affiliations

Panayot S. Vassilevski

Funding Information

U.S. Department of Energy https://doi.org/10.13039/100000015 : DE-AC52-07NA27344

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