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Error indicators for incompressible Darcy flow problems using enhanced velocity mixed finite element method. (English) Zbl 1437.76019
Summary: Local mesh adaptivity serves as a practical tool in numerical simulations to accurately capture features of interest while reducing computational time and memory requirements. In this work, we suggest a refinement strategy based on pressure and flux error estimates for numerical simulation of an incompressible, single phase flow and transport process in the subsurface porous media. We derive a posteriori error estimates for an Enhanced Velocity Mixed Finite Element Method (EVMFEM) as a spatial domain decomposition approach. We note that the flux errors play an important role in coupled flow and transport systems later demonstrated using numerical experiments. A comparison between explicit (residual based) error estimators and an implicit error estimator; based upon the post-processing proposed by T. Arbogast and Z. Chen [Math. Comput. 64, No. 211, 943–972 (1995; Zbl 0829.65127)], shows that the latter performs better. A residual-based error estimator for pressure was found to be both computationally efficient while sufficiently indicating the large error subdomains. Numerical studies are also presented that confirm our theoretical derivations while demonstrating the advantages of post-processing in detecting velocity errors.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76S05 Flows in porous media; filtration; seepage
65N15 Error bounds for boundary value problems involving PDEs
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[1] Babuška, Ivo; Rheinboldt, Werner C., A-posteriori error estimates for the finite element method, Internat. J. Numer. Methods Engrg., 12, 10, 1597-1615 (1978) · Zbl 0396.65068
[2] Ainsworth, Mark; Oden, J. Tinsley, A Posteriori Error Estimation in Finite Element Analysis, Vol. 37 (2011), John Wiley & Sons · Zbl 0895.76040
[3] Verfürth, Rüdiger, A Posteriori Error Estimation Techniques for Finite Element Methods (2013), OUP Oxford · Zbl 1279.65127
[4] Estep, Donald J.; Larson, Mats G.; Williams, Roy D., Estimating the Error of Numerical Solutions of Systems of Reaction-Diffusion Equations, Vol. 696 (2000), American Mathematical Soc. · Zbl 0998.65096
[5] Larson, Mats G.; Målqvist, Axel, A posteriori error estimates for mixed finite element approximations of elliptic problems, Numer. Math., 108, 3, 487-500 (2008) · Zbl 1136.65101
[6] Vohralík, Martin, Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods, Math. Comp., 79, 272, 2001-2032 (2010) · Zbl 1201.65200
[7] Wohlmuth, Barbara I., A residual based error estimator for mortar finite element discretizations, Numer. Math., 84, 1, 143-171 (1999) · Zbl 0962.65090
[8] Vohralík, Martin, A posteriori error estimates for lowest-order mixed finite element discretizations of convection-diffusion-reaction equations, SIAM J. Numer. Anal., 45, 4, 1570-1599 (2007) · Zbl 1151.65084
[9] Zienkiewicz, Olgierd C.; Zhu, Jian Z., A simple error estimator and adaptive procedure for practical engineerng analysis, Internat. J. Numer. Methods Engrg., 24, 2, 337-357 (1987) · Zbl 0602.73063
[10] Carstensen, Carsten; Hu, Jun, A unifying theory of a posteriori error control for nonconforming finite element methods, Numer. Math., 107, 3, 473-502 (2007) · Zbl 1127.65083
[11] Ern, Alexandre; Vohralík, Martin, Polynomial-degree-robust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations, SIAM J. Numer. Anal., 53, 2, 1058-1081 (2015) · Zbl 1312.76026
[12] Ainsworth, Mark, Robust a posteriori error estimation for nonconforming finite element approximation, SIAM J. Numer. Anal., 42, 6, 2320-2341 (2005) · Zbl 1085.65102
[13] Pencheva, Gergina V.; Vohralík, Martin; Wheeler, Mary F.; Wildey, Tim, Robust a posteriori error control and adaptivity for multiscale, multinumerics, and mortar coupling, SIAM J. Numer. Anal., 51, 1, 526-554 (2013) · Zbl 1267.65165
[14] Sun, Shuyu; Wheeler, Mary F., L2 (h1) norm a posteriori error estimation for discontinuous Galerkin approximations of reactive transport problems, J. Sci. Comput., 22, 1-3, 501-530 (2005) · Zbl 1066.76037
[15] Wheeler, Mary F.; Yotov, Ivan, A posteriori error estimates for the mortar mixed finite element method, SIAM J. Numer. Anal., 43, 3, 1021-1042 (2005) · Zbl 1094.65114
[16] Arbogast, Todd; Estep, Donald; Sheehan, Brendan; Tavener, Simon, A posteriori error estimates for mixed finite element and finite volume methods for problems coupled through a boundary with nonmatching grids, IMA J. Numer. Anal., 34, 4, 1625-1653 (2014) · Zbl 1303.65093
[17] Arbogast, Todd; Estep, Donald; Sheehan, Brendan; Tavener, Simon, A posteriori error estimates for mixed finite element and finite volume methods for parabolic problems coupled through a boundary, SIAM/ASA J. Uncertain. Quantif., 3, 1, 169-198 (2015) · Zbl 1322.65093
[18] Sun, Shuyu; Wheeler, Mary F., Discontinuous Galerkin methods for simulating bioreactive transport of viruses in porous media, Adv. Water Resour., 30, 6, 1696-1710 (2007)
[19] Riviere, Béatrice; Wheeler, Mary F., A posteriori error estimates for a discontinuous Galerkin method applied to elliptic problems. log number: R74, Comput. Math. Appl., 46, 1, 141-163 (2003) · Zbl 1059.65098
[20] Keith, Brendan; Astaneh, Ali Vaziri; Demkowicz, Leszek F., Goal-oriented adaptive mesh refinement for discontinuous Petrov-Galerkin methods, SIAM J. Numer. Anal., 57, 4, 1649-1676 (2019) · Zbl 1422.65391
[21] Fuentes, Federico; Demkowicz, Leszek; Wilder, Aleta, Using a DPG method to validate DMA experimental calibration of viscoelastic materials, Comput. Methods Appl. Mech. Engrg., 325, 748-765 (2017)
[22] Amanbek, Yerlan; Singh, Gurpreet; Wheeler, Mary F.; van Duijn, Hans, Adaptive numerical homogenization for upscaling single phase flow and transport, J. Comput. Phys., 387, 117-133 (2019)
[23] Thomas, Sunil G.; Wheeler, Mary F., Enhanced velocity mixed finite element methods for modeling coupled flow and transport on non-matching multiblock grids, Comput. Geosci., 15, 4, 605-625 (2011) · Zbl 1348.76101
[24] Singh, Gurpreet; Leung, Wingtat; Wheeler, Mary F., Multiscale methods for model order reduction of non-linear multiphase flow problems, Comput. Geosci., 23, 2, 305-323 (2019) · Zbl 1414.76088
[25] Wheeler, John A.; Wheeler, Mary F.; Yotov, Ivan, Enhanced velocity mixed finite element methods for flow in multiblock domains, Comput. Geosci., 6, 3-4, 315-332 (2002) · Zbl 1023.76023
[26] Singh, Gurpreet; Amanbek, Yerlan; Wheeler, Mary F., Adaptive homogenization for upscaling heterogeneous porous medium, (SPE Annual Technical Conference and Exhibition (2017), Society of Petroleum Engineers)
[27] Amanbek, Yerlan, A New Adaptive Modeling of Flow and Transport in Porous Media Using an Enhanced Velocity Scheme (2018), (Ph.D. thesis)
[28] Mandli, Kyle T.; Dawson, Clint N., Adaptive mesh refinement for storm surge, Ocean Model., 75, 36-50 (2014)
[29] Amanbek, Yerlan; Wheeler, Mary F., A priori error analysis for transient problems using enhanced velocity approach in the discrete-time setting, J. Comput. Appl. Math., 361, 459-471 (2019) · Zbl 07085902
[30] Burstedde, Carsten; Stadler, Georg; Alisic, Laura; Wilcox, Lucas C.; Tan, Eh; Gurnis, Michael; Ghattas, Omar, Large-scale adaptive mantle convection simulation, Geophys. J. Int., 192, 3, 889-906 (2013)
[31] Jenny, Patrick; Lee, Seong H.; Tchelepi, Hamdi A., Adaptive multiscale finite-volume method for multiphase flow and transport in porous media, Multiscale Model. Simul., 3, 1, 50-64 (2005) · Zbl 1160.76372
[32] Ewing, Richard; Efendiev, Yalchin; Ginting, Victor; Wang, Hong, Upscaling of transport equations for multiphase and multicomponent flows, (Domain Decomposition Methods in Science and Engineering XVII (2008), Springer), 193-200 · Zbl 1359.76292
[33] Arbogast, Todd; Chen, Zhangxin, On the implementation of mixed methods as nonconforming methods for second-order elliptic problems, Math. Comp., 64, 211, 943-972 (1995) · Zbl 0829.65127
[34] Arbogast, Todd; Pencheva, Gergina; Wheeler, Mary F.; Yotov, Ivan, A multiscale mortar mixed finite element method, Multiscale Model. Simul., 6, 1, 319-346 (2007) · Zbl 1322.76039
[35] Gilbarg, David; Trudinger, Neil S., Elliptic Partial Differential Equations of Second Order (2015), Springer · Zbl 1042.35002
[36] Ciarlet, Philippe G., The finite element method for elliptic problems, Classics Appl. Math., 40, 1-511 (2002)
[37] Carstensen, Carsten, A posteriori error estimate for the mixed finite element method, Math. Comput. Amer. Math. Soc., 66, 218, 465-476 (1997) · Zbl 0864.65068
[38] Amanbek, Yerlan; Singh, Gurpreet; Wheeler, Mary F., Recovery of the interface velocity for the incompressible flow in enhanced velocity mixed finite element method, (International Conference on Computational Science (2019), Springer), 510-523
[39] Karakashian, Ohannes A.; Pascal, Frederic, A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems, SIAM J. Numer. Anal., 41, 6, 2374-2399 (2003) · Zbl 1058.65120
[40] Burman, Erik; Ern, Alexandre, Continuous interior penalty hp-finite element methods for advection and advection-diffusion equations, Math. Comp., 76, 259, 1119-1140 (2007) · Zbl 1118.65118
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