Large-scale 3D geoelectromagnetic modeling using parallel adaptive high-order finite element method

Grayver, Alexander V.; Kolev, Tzanio V.

doi:10.1190/geo2015-0013.1

Large-scale 3D geoelectromagnetic modeling using parallel adaptive high-order finite element method

Journal Article · Sun Nov 01 00:00:00 EDT 2015 · Geophysics

DOI:https://doi.org/10.1190/geo2015-0013.1· OSTI ID:1440727

Grayver, Alexander V. ^[1]; Kolev, Tzanio V. ^[2]

ETH Zurich, Zurich (Switzerland)
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

Here, we have investigated the use of the adaptive high-order finite-element method (FEM) for geoelectromagnetic modeling. Because high-order FEM is challenging from the numerical and computational points of view, most published finite-element studies in geoelectromagnetics use the lowest order formulation. Solution of the resulting large system of linear equations poses the main practical challenge. We have developed a fully parallel and distributed robust and scalable linear solver based on the optimal block-diagonal and auxiliary space preconditioners. The solver was found to be efficient for high finite element orders, unstructured and nonconforming locally refined meshes, a wide range of frequencies, large conductivity contrasts, and number of degrees of freedom (DoFs). Furthermore, the presented linear solver is in essence algebraic; i.e., it acts on the matrix-vector level and thus requires no information about the discretization, boundary conditions, or physical source used, making it readily efficient for a wide range of electromagnetic modeling problems. To get accurate solutions at reduced computational cost, we have also implemented goal-oriented adaptive mesh refinement. The numerical tests indicated that if highly accurate modeling results were required, the high-order FEM in combination with the goal-oriented local mesh refinement required less computational time and DoFs than the lowest order adaptive FEM.

View Accepted Manuscript (DOE)

Research Organization:: Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA)

Grant/Contract Number:: AC52-07NA27344

OSTI ID:: 1440727

Report Number(s):: LLNL-JRNL--665742; 787283

Journal Information:: Geophysics, Journal Name: Geophysics Journal Issue: 6 Vol. 80; ISSN 0016-8033

Publisher:: Society of Exploration GeophysicistsCopyright Statement

Country of Publication:: United States

Language:: English

References (46)

Finite Element Methods for Maxwell's Equations Demkowicz, Leszek Encyclopedia of Computational Mechanics Second Edition https://doi.org/10.1002/9781119176817.ecm2023	book	December 2017
Scaling Hypre’s Multigrid Solvers to 100,000 Cores Baker, Allison H.; Falgout, Robert D.; Kolev, Tzanio V. High-Performance Scientific Computing https://doi.org/10.1007/978-1-4471-2437-5_13	book	January 2012
Models and Methods of Magnetotellurics Berdichevsky, Mark N.; Dmitriev, Vladimir I. https://doi.org/10.1007/978-3-540-77814-1	book	January 2008
The h-p version of the finite element method: Part 1: The basic approximation results Guo, B.; Babuška, I. Computational Mechanics, Vol. 1, Issue 1 https://doi.org/10.1007/BF00298636	journal	March 1986
The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids Xu, J. Computing, Vol. 56, Issue 3 https://doi.org/10.1007/BF02238513	journal	September 1996
The h-p version of the finite element method: Part 2: General results and applications Guo, B.; Babuška, I. Computational Mechanics, Vol. 1, Issue 3 https://doi.org/10.1007/bf00272624	journal	September 1986
Adaptive Finite Element Methods with convergence rates Binev, Peter; Dahmen, Wolfgang; DeVore, Ron Numerische Mathematik, Vol. 97, Issue 2 https://doi.org/10.1007/s00211-003-0492-7	journal	April 2004
Fast simplicial quadrature-based finite element operators using Bernstein polynomials Kirby, Robert C.; Thinh, Kieu Tri Numerische Mathematik, Vol. 121, Issue 2 https://doi.org/10.1007/s00211-011-0431-y	journal	November 2011
Moving curved mesh adaptation for higher-order finite element simulations Luo, Xiao-Juan; Shephard, Mark S.; Lee, Lie-Quan Engineering with Computers, Vol. 27, Issue 1 https://doi.org/10.1007/s00366-010-0179-5	journal	February 2010
Numerical Modelling in Geo-Electromagnetics: Advances and Challenges Börner, Ralph-Uwe Surveys in Geophysics, Vol. 31, Issue 2 https://doi.org/10.1007/s10712-009-9087-x	journal	October 2009
Marine Electromagnetic Studies of Seafloor Resources and Tectonics Key, Kerry Surveys in Geophysics, Vol. 33, Issue 1 https://doi.org/10.1007/s10712-011-9139-x	journal	June 2011
Exploring for Geothermal Resources with Electromagnetic Methods Muñoz, Gerard Surveys in Geophysics, Vol. 35, Issue 1 https://doi.org/10.1007/s10712-013-9236-0	journal	May 2013
BoomerAMG: A parallel algebraic multigrid solver and preconditioner Henson, Van Emden; Yang, Ulrike Meier Applied Numerical Mathematics, Vol. 41, Issue 1 https://doi.org/10.1016/S0168-9274(01)00115-5	journal	April 2002
Methods for modelling electromagnetic fields Results from COMMEMI—the international project on the comparison of modelling methods for electromagnetic induction Zhdanov, M. S.; Varentsov, I. M.; Weaver, J. T. Journal of Applied Geophysics, Vol. 37, Issue 3-4 https://doi.org/10.1016/S0926-9851(97)00013-X	journal	October 1997
A residual-based a posteriori error estimator for the hp-finite element method for Maxwellʼs equations Bürg, M. Applied Numerical Mathematics, Vol. 62, Issue 8 https://doi.org/10.1016/j.apnum.2012.02.007	journal	August 2012
Convergence of an automatic hp-adaptive finite element strategy for Maxwellʼs equations Bürg, M. Applied Numerical Mathematics, Vol. 72 https://doi.org/10.1016/j.apnum.2013.04.008	journal	October 2013
Three-dimensional finite-element modelling of magnetotelluric data with a divergence correction Farquharson, Colin G.; Miensopust, Marion P. Journal of Applied Geophysics, Vol. 75, Issue 4 https://doi.org/10.1016/j.jappgeo.2011.09.025	journal	December 2011
Three-dimensional h-adaptivity for the multigroup neutron diffusion equations Wang, Yaqi; Bangerth, Wolfgang; Ragusa, Jean Progress in Nuclear Energy, Vol. 51, Issue 3 https://doi.org/10.1016/j.pnucene.2008.11.005	journal	April 2009
A posteriori error estimation in finite element analysis Ainsworth, Mark; Oden, J. Tinsley Computer Methods in Applied Mechanics and Engineering, Vol. 142, Issue 1-2 https://doi.org/10.1016/s0045-7825(96)01107-3	journal	March 1997
The Magnetotelluric Method Chave, Alan D.; Jones, Alan G. https://doi.org/10.1017/CBO9781139020138	book	April 2012
Convergence and optimality of adaptive edge finite element methods for time-harmonic Maxwell equations Zhong, Liuqiang; Chen, Long; Shu, Shi Mathematics of Computation, Vol. 81, Issue 278 https://doi.org/10.1090/S0025-5718-2011-02544-5	journal	December 2011
Finite Element Methods for Maxwell's Equations Monk, Peter Oxford University Press https://doi.org/10.1093/acprof:oso/9780198508885.001.0001	book	January 2003
Efficient pre-conditioned iterative solution strategies for the electromagnetic diffusion in the Earth: finite-element frequency-domain approach Um, Evan Schankee; Commer, Michael; Newman, Gregory A. Geophysical Journal International, Vol. 193, Issue 3 https://doi.org/10.1093/gji/ggt071	journal	March 2013
A goal-oriented adaptive finite-element approach for plane wave 3-D electromagnetic modelling Ren, Zhengyong; Kalscheuer, Thomas; Greenhalgh, Stewart Geophysical Journal International, Vol. 194, Issue 2 https://doi.org/10.1093/gji/ggt154	journal	May 2013
Robust and scalable 3-D geo-electromagnetic modelling approach using the finite element method Grayver, Alexander V.; Bürg, Markus Geophysical Journal International, Vol. 198, Issue 1 https://doi.org/10.1093/gji/ggu119	journal	April 2014
High-Order Symplectic Integration Methods for Finite Element Solutions to Time Dependent Maxwell Equations Rieben, R.; White, D.; Rodrigue, G. IEEE Transactions on Antennas and Propagation, Vol. 52, Issue 8 https://doi.org/10.1109/TAP.2004.832356	journal	August 2004
A parallel goal-oriented adaptive finite element method for 2.5-D electromagnetic modelling: Adaptive finite element modelling for 2.5-D EM Key, Kerry; Ovall, Jeffrey Geophysical Journal International, Vol. 186, Issue 1 https://doi.org/10.1111/j.1365-246X.2011.05025.x	journal	May 2011
Three-dimensional adaptive higher order finite element simulation for geo-electromagnetics-a marine CSEM example: 3-D FE simulation for geo-electromagnetics Schwarzbach, Christoph; Börner, Ralph-Uwe; Spitzer, Klaus Geophysical Journal International, Vol. 187, Issue 1 https://doi.org/10.1111/j.1365-246X.2011.05127.x	journal	August 2011
A multigrid solver for 3D electromagnetic diffusion: Multigrid for 3D EM Mulder, W. A. Geophysical Prospecting, Vol. 54, Issue 5 https://doi.org/10.1111/j.1365-2478.2006.00558.x	journal	September 2006
Toward an h-Independent Algebraic Multigrid Method for Maxwell's Equations Hu, Jonathan J.; Tuminaro, Raymond S.; Bochev, Pavel B. SIAM Journal on Scientific Computing, Vol. 27, Issue 5 https://doi.org/10.1137/040608118	journal	January 2006
Nodal Auxiliary Space Preconditioning in H(curl) and H(div) Spaces Hiptmair, Ralf; Xu, Jinchao SIAM Journal on Numerical Analysis, Vol. 45, Issue 6 https://doi.org/10.1137/060660588	journal	January 2007
A Convergent Adaptive Algorithm for Poisson’s Equation Dörfler, Willy SIAM Journal on Numerical Analysis, Vol. 33, Issue 3 https://doi.org/10.1137/0733054	journal	June 1996
An Adaptive Finite Element Method for the Eddy Current Model with Circuit/Field Couplings Chen, Junqing; Chen, Zhiming; Cui, Tao SIAM Journal on Scientific Computing, Vol. 32, Issue 2 https://doi.org/10.1137/080713112	journal	January 2010
Iterative Methods for Sparse Linear Systems Saad, Yousef https://doi.org/10.1137/1.9780898718003	book	January 2003
Computational Methods in Geophysical Electromagnetics Haber, Eldad Mathematics in Industry https://doi.org/10.1137/1.9781611973808	book	January 2014
`p4est` : Scalable Algorithms for Parallel Adaptive Mesh Refinement on Forests of Octrees Burstedde, Carsten; Wilcox, Lucas C.; Ghattas, Omar SIAM Journal on Scientific Computing, Vol. 33, Issue 3 https://doi.org/10.1137/100791634	journal	January 2011
Bernstein–Bézier Finite Elements of Arbitrary Order and Optimal Assembly Procedures Ainsworth, Mark; Andriamaro, Gaelle; Davydov, Oleg SIAM Journal on Scientific Computing, Vol. 33, Issue 6 https://doi.org/10.1137/11082539X	journal	January 2011
Conditioning of Hierarchic p -Version Nédélec Elements on Meshes of Curvilinear Quadrilaterals and Hexahedra Ainsworth, Mark; Coyle, Joe SIAM Journal on Numerical Analysis, Vol. 41, Issue 2 https://doi.org/10.1137/S003614290239590X	journal	January 2003
Data structures and requirements for hp finite element software Bangerth, W.; Kayser-Herold, O. ACM Transactions on Mathematical Software, Vol. 36, Issue 1 https://doi.org/10.1145/1486525.1486529	journal	March 2009
Algorithms and data structures for massively parallel generic adaptive finite element codes Bangerth, Wolfgang; Burstedde, Carsten; Heister, Timo ACM Transactions on Mathematical Software, Vol. 38, Issue 2 https://doi.org/10.1145/2049673.2049678	journal	December 2011
Low-frequency electromagnetic fields in applied geophysics: Waves or diffusion? Løseth, Lars O.; Pedersen, Hans M.; Ursin, Bjørn GEOPHYSICS, Vol. 71, Issue 4 https://doi.org/10.1190/1.2208275	journal	July 2006
A finite-element-based domain-decomposition approach for plane wave 3D electromagnetic modeling Ren, Zhengyong; Kalscheuer, Thomas; Greenhalgh, Stewart GEOPHYSICS, Vol. 79, Issue 6 https://doi.org/10.1190/geo2013-0376.1	journal	November 2014
Adaptive Finite Element Methods with Convergence Rates Binev, Peter; Dahmen, Wolfgang; DeVore, Ron https://doi.org/10.21236/ada640658	report	January 2003
Robust and scalable 3D geo-electromagnetic modeling approach using the finite element method Grayver, Alexander V.; Bürg, Markus ETH Zurich https://doi.org/10.3929/ethz-b-000109208	text	January 2014
A Multigrid Solver for 3D Electromagnetic Diffusion Mulder, W. A. 67th EAGE Conference & Exhibition https://doi.org/10.3997/2214-4609-pdb.1.p193	conference	January 2005
Parallel Auxiliary Space AMG for H(Curl) Problems Vassilevski, Tzanio V. Kolev Panayot S. Journal of Computational Mathematics, Vol. 27, Issue 5 https://doi.org/10.4208/jcm.2009.27.5.013	journal	June 2009

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