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Finite element three-dimensional Stokes ice sheet dynamics model with enhanced local mass conservation. (English) Zbl 1351.86030

Summary: Parallel finite element nonlinear Stokes models have been successfully used for three-dimensional ice-sheet and glacier simulations due to their accuracy and efficiency, and their capability for easily handling highly irregular domains and different types of boundary conditions. In particular, the well-known Taylor-Hood element pair (continuous piecewise quadratic elements for velocity and continuous piecewise linear elements for pressure) results in highly accuracy velocity and pressure approximations. However, the Taylor-Hood element suffers from poor mass conservation which can lead to significant numerical mass balance errors for long-time simulations. In this paper, we develop and investigate a new finite element Stokes ice sheet dynamics model that enforces local element-wise mass conservation by enriching the pressure finite element space by adding the discontinuous piecewise constant pressure space to the Taylor-Hood pressure space. Through various numerical tests based on manufactured solutions, benchmark test problems, and the realistic Greenland ice-sheet, we demonstrate that, for ice-sheet modeling, the enriched Taylor-Hood finite element model remains highly accurate and efficient, and is physically more reliable and robust compared to the classic Taylor-Hood finite element model.

MSC:

86A40 Glaciology
76M10 Finite element methods applied to problems in fluid mechanics
76T99 Multiphase and multicomponent flows
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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[1] Alley, R. B.; Joughin, I., Modeling ice-sheet flow, Science, 336, 551-552 (2012)
[2] Amestoy, P. R.; Duff, I. S.; L’Excellent, J.-Y., Multifrontal parallel distributed symmetric and unsymmetric solvers, Comput. Methods Appl. Mech. Eng., 184, 501-520 (2000) · Zbl 0956.65017
[3] Blatter, H.; Clarke, G.; Colinge, J., Stress and velocity fields in glaciers: Part II. Sliding and basal stress distribution, J. Glaciol., 44, 457-466 (1998)
[4] Boffi, D., Stability of higher order triangular Hood-Taylor methods for the stationary Stokes equations, Math. Models Methods Appl. Sci., 4, 223-235 (1994) · Zbl 0804.76051
[5] Boffi, D., Three-dimensional finite element methods for the Stokes problem, SIAM J. Numer. Anal., 34, 664-670 (1997) · Zbl 0874.76032
[6] Boffi, D.; Cavallini, N.; Gardini, F.; Gastaldi, L., Local mass conservation of Stokes finite elements, J. Sci. Comput., 52, 383-400 (2012) · Zbl 1264.74259
[7] Burstedde, C.; Ghattas, O.; Stadler, G.; Tu, T.; Wilcox, L. C., Parallel scalable adjoint-based adaptive solution of variable-viscosity Stokes flow problems, Comput. Methods Appl. Mech. Eng., 198, 1691-1700 (2009) · Zbl 1227.76027
[8] Gagliardini, O.; Zwinger, T., The ISMIP-HOM benchmark experiments performed using the finite element code Elmer, Cryosphere, 2, 67-76 (2008)
[9] Gagliardini, O.; Zwinger, T.; Gillet-Chaulet, F.; Durand, G.; Favier, F.; de Fleurian, B.; Greve, R.; Malinen, M.; Martin, C.; Raback, P.; Ruokolainen, J.; Sacchettini, M.; Schafer, M.; Seddik, H.; Thies, J., Capabilities and performance of Elmer/Ice, a new-generation ice sheet model, Geosci. Model Dev., 6, 1299-1318 (2013)
[10] Gresho, P. M.; Lee, R. L.; Chan, S. T.; Leone, J. M., A new finite element for Boussinesq fluids, (Proc. Third Int. Conf. on Finite Elements in Flow Problems (1980), Wiley: Wiley New York), 204-215 · Zbl 0447.76026
[11] Griffiths, D. F., The effect of pressure approximation on finite element calculations of compressible flows, (Morton, K. W.; Baines, M. J., Numerical Methods for Fluid Dynamics (1982), Academic Press: Academic Press San Diego), 359-374
[12] Gunzburger, M., Finite Element Methods for the Navier-Stokes Equations (1989), Academic: Academic Boston
[13] Ju, L., Conforming centroidal Voronoi Delaunay triangulation for quality mesh generation, Int. J. Numer. Anal. Model., 4, 531-547 (2007) · Zbl 1132.65012
[14] Leng, W.; Ju, L.; Gunzburger, M.; Price, S.; Ringler, T., A parallel high-order accurate finite element Stokes ice sheet model, J. Geophys. Res., 117, F01001 (2012)
[15] Leng, W.; Ju, L.; Gunzburger, M.; Price, S., Manufactured solutions and the verification of three-dimensional Stokes ice-sheet models, Cryosphere, 7, 19-29 (2013)
[16] Leng, W.; Ju, L.; Gunzburger, M.; Price, S., A parallel computational model for three-dimensional, thermo-mechanical Stokes flow simulations of glaciers and ice sheets, Commun. Comput. Phys. (2013), in press
[17] Le Meur, E.; Gagliardini, O.; Zwinger, T.; Ruokolainen, J., Glacier flow modelling: a comparison of the shallow ice approximation and the full-Stokes solution, C. R. Phys., 5, 709-722 (2004)
[18] Nguyen, H.; Burkardt, J.; Gunzburger, M.; Ju, L., Constrained CVT meshes and a comparison of triangular mesh generators, Comput. Geom. Theory Appl., 42, 1-19 (2009) · Zbl 1152.65035
[19] Nye, J., The distribution of stress and velocity in glaciers and ice sheets, Proc. R. Soc. Lond., Ser. A, 239, 113-133 (1957) · Zbl 0077.38201
[20] Okabe, A.; Boots, B.; Sugihara, K.; Chiu, S. N., Spatial Tessellations: Concepts and Applications of Voronoi Diagrams (2000), John Wiley and Sons, Inc. · Zbl 0946.68144
[21] Paterson, W., The Physics of Glaciers (1994), Elsevier Science: Elsevier Science Oxford, UK
[22] Pattyn, F.; Perichon, L.; Aschwanden, A.; Breuer, B.; Smedt, D. B.; Gagliardini, O.; Gudmundsson, G. H.; Hindmarsh, R. C.A.; Hubbard, A.; Johnson, J. V.; Kleiner, T.; Konovalov, Y.; Martin, C.; Payne, A. J.; Pollard, D.; Price, S.; Ruckamp, M.; Saito, F.; Sugiyama, S.; Zwinger, T., Benchmark experiments for higher-order and full-Stokes ice sheet models (ISMIP-HOM), Cryosphere, 2, 95-108 (2008)
[23] Pierre, R., Local mass conservation and \(C^0\)-discretizations of the Stokes problem, Houst. J. Math., 20, 115-127 (1994) · Zbl 0803.65105
[24] Shannon, S.; Payne, A.; Bartholomew, I.; van den Broeke, M. R.; Edwards, T.; Fettweis, X.; Gagliardini, O.; Gillet-Chaulet, F.; Goelzer, H.; Hoffman, M.; Huybrechts, P.; Mair, D.; Nienow, P.; Perego, M.; Price, S.; Smeets, C.; Sole, A.; vande Wal, R.; Zwinger, T., Enhanced basal lubrication and the contribution of the Greenland ice sheet to future sea-level rise, Proc. Natl. Acad. Sci. USA, 110, 14156-14161 (2013)
[25] Qin, J.; Zhang, S., Stability of the finite elements \(9 /(4 c + 1)\) and \(9 / 5 c\) for stationary Stokes equations, Comput. Struct., 84, 70-77 (2005)
[26] Sadd, Y., A flexible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Comput., 14, 461-469 (1993) · Zbl 0780.65022
[27] Schoof, C., Variational methods for glacier flow over plastic till, J. Fluid Mech., 555, 299-320 (2006) · Zbl 1091.76058
[28] Schoof, C., Coulomb friction and other sliding laws in a higher order glacier flow model, Math. Models Methods Appl. Sci., 20, 157-189 (2010) · Zbl 1225.74054
[29] Thatcher, R. W., Locally mass-conserving Taylor-Hood elements for two- and three-dimensional flow, Int. J. Numer. Methods Fluids, 11, 341-353 (1990) · Zbl 0709.76083
[30] Valli, A. Q., Domain Decomposition Methods for Partial Differential Equations (1999), Oxford Science Publications
[31] Zhang, H.; Ju, L.; Gunzburger, M.; Ringler, T.; Price, S., Coupled models and parallel simulations for three dimensional full-Stokes ice sheet modeling, Numer. Math., Theory Methods Appl., 4, 359-381 (2011)
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