# Solution of Nonlinear Stokes Equations Discretized By High-Order Finite Elements on Nonconforming and Anisotropic Meshes, with Application to Ice Sheet Dynamics

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*Department of Defense World Geodetic System 1984: Its definition and relationships with local geodetic systems*, Tech. rep. NGA TR8350.2, U.S. National Geospatial-Intelligence Agency, Springfield, VA, 1984. Revised in 2004.

*A parallel maximal independent set algorithm*, in Proceedings of the 5th Copper Mountain Conference on Iterative Methods, Boulder, CO, 1998.

*The stability of mixed hp-finite element methods for Stokes flow on high aspect ratio elements*, SIAM J. Numer. Anal., 38 (2000), pp. 1721--1761.

*PETSc users manual*, Tech. rep. ANL-95/11, Revision 3.3, Argonne National Laboratory, Lemont, IL, 2012.

*Velocity and stress fields in grounded glaciers: A simple algorithm for including deviatoric stress gradients*, J. Glaciology, 41 (1995), pp. 333--344.

*Efficient nonlinear solvers for nodal high-order finite elements in 3D*, J. Sci. Comput., 45 (2010), pp. 48--63.

*Achieving textbook multigrid efficiency for hydrostatic ice sheet flow*, SIAM J. Sci. Comput., 35 (2013), pp. B359--B375.

*Parallel scalable adjoint-based adaptive solution for variable-viscosity Stokes flows*, Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 1691--1700.

*p4est: Scalable algorithms for parallel adaptive mesh refinement on forests of octrees*, SIAM J. Sci. Comput., 33 (2011), pp. 1103--1133.

*Efficient variable-coefficient finite-volume Stokes solvers*, Commun. Comput. Phys., 16 (2014), pp. 1263--1297.

*Finite-element preconditioning of G-NI spectral methods*, SIAM J. Sci. Comput., 31 (2010), pp. 4422--4451.

*A survey of parallelization techniques for multigrid solvers*, in Parallel Processing for Scientific Computing, M. A. Heroux, P. Raghavan, and H. D. Simon, eds., SIAM, Philadelphia, 2006, pp. 179--201.

*Adaptive mesh, finite volume modeling of marine ice sheets*, J. Comput. Phys., 232 (2012), pp. 529--549.

*A monolithic FEM-multigrid solver for non-isothermal incompressible flow on general meshes*, J. Comput. Phys., 228 (2009), pp. 3869--3881.

*Finite-element preconditioning for pseudospectral solutions of elliptic problems*, SIAM J. Sci. Statist. Comput., 11 (1990), pp. 311--342.

*High-Order Methods for Incompressible Fluid Flow*, Cambridge Monogr. Appl. Comput. Math. 9, Cambridge University Press, Cambridge, UK, 2002.

*Influence of Gauss and Gauss-Lobatto quadrature rules on the accuracy of a quadrilateral finite element method in the time domain*, Numer. Methods Partial Differential Equations, 25 (2009), pp. 526--551.

*Choosing the forcing terms in an inexact Newton method*, SIAM J. Sci. Comput., 17 (1996), pp. 16--32.

*Block preconditioners based on approximate commutators*, SIAM J. Sci. Comput., 27 (2006), pp. 1651--1668.

*Finite Elements and Fast Iterative Solvers with applications in incompressible fluid dynamics*, Oxford University Press, Oxford, UK, 2005.

*The Geoforschungszentrum Potsdam/groupe de Recherche de Geodesie Spatiale satellite-only and combined gravity field models: EIGEN-GL04S1 and EIGEN-GL04C*, J. Geodesy, 82 (2008), pp. 331--346.

*Development of a Stokes flow solver robust to large viscosity jumps using a Schur complement approach with mixed precision arithmetic*, J. Comput. Phys., 230 (2011), pp. 8835--8851.

*The ISMIP-HOM benchmark experiments performed using the finite-element code Elmer*, The Cryosphere, 2 (2008), pp. 67--76.

*A new smoothed aggregation multigrid method for anisotropic problems*, Numer. Linear Algebra Appl., 16 (2009), pp. 19--37.

*Scalable robust solvers for unstructured FE geodynamic modeling applications: Solving the Stokes equation for models with large localized viscosity contrasts*, Geochem. Geophys. Geosyst., 10 (2009), Q09002.

*Numerical Methods for Non-Newtonian Fluids: Special Volume*, Handbook Numer. Anal. 16, North--Holland, Amsterdam, 2011.

*An iterative method for the Stokes-type problem with variable viscosity*, SIAM J. Sci. Comput., 31 (2009), pp. 3959--3978.

*On the inf-sup condition for higher order mixed fem on meshes with hanging nodes*, ESAIM Math. Model. Numer. Anal., 41 (2007), pp. 1--20.

*Algebraic multigrid for higher-order finite elements*, J. Comput. Phys., 204 (2005), pp. 520--532.

*A numerical comparison of approximations to the Stokes equations used in ice sheet and glacier modeling*, J. Geophys. Res., 109 (2004), F01012.

*Theoretical Glaciology*, Mathematical Approaches to Geophysics, D. Reidel, Dordrecht, The Netherlands, 1983.

*Low-cost parallel algorithms for 2:1 octree balance*, in Proceedings of the 26th IEEE International Parallel and Distributed Processing Symposium, IEEE, Washington, DC, 2012. http://dx.doi.org/10.1109/IPDPS.2012.47.

*Scalable and efficient algorithms for the propagation of uncertainty from data through inference to prediction for large-scale problems, with application to flow of the Antarctic ice sheet*, J. Comput. Phys., 296 (2015), pp. 348--368.

*Smoothed aggregation multigrid for a Stokes problem*, Comput. Vis. Sci., 11 (2008), pp. 169--180.

*Analysis and finite element approximation of a nonlinear stationary Stokes problem arising in glaciology*, Adv. Numer. Anal., 2011 (2012), 164581.

*Algorithm for non-hydrostatic dynamics in the regional oceanic modeling system*, Ocean Model., 18 (2007), pp. 143--174.

*Algebraic multigrid for finite element elasticity equations: Determination of nodal dependence via edge-matrices and two-level convergence*, Internat. J. Numer. Methods Engrg., 83 (2010), pp. 642--670.

*Algorithms for geodesics*, J. Geodesy, 87 (2013), pp. 43--55.

*Piecewise bilinear preconditioning of high-order finite element methods*, Electron. Trans. Numer. Anal., 26 (2007), pp. 228--242.

*An improved Antarctic dataset for high resolution numerical ice sheet models (ALBMAP v1)*, Earth Syst. Sci. Data, 2 (2010), pp. 247--260.

*A parallel high-order accurate finite element nonlinear Stokes ice-sheet model and benchmark experiments*, J. Geophys. Res., 117 (2012), F01001.

*An incomplete factorization technique for positive definite linear systems*, Math. Comp., 34 (1980), pp. 473--497.

*Preconditioned iterative methods for Stokes flow problems arising in computational geodynamics*, Phys. Earth Planetary Interiors, 171 (2008), pp. 33--47.

*The generation and compaction of partially molten rock*, J. Petrology, 25 (1984), pp. 713--765.

*Global climate projections*, in Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change, Cambridge University Press, Cambridge, UK, 2007, pp. 747--845.

*Inversion of basal friction in Antarctica using exact and incomplete adjoints of a higher-order model*, J. Geophys. Res. Earth Surface, 118 (2013), pp. 1746--1753.

*Numerical Optimization*, Springer-Verlag, Berlin, Heidelberg, New York, 1999.

*Algebraic multigrid preconditioning of high-order spectral elements for elliptic problems on a simplicial mesh*, SIAM J. Sci. Comput., 29 (2007), pp. 2189--2209.

*Benchmark experiments for higher-order and full-Stokes ice sheet models (ISMIP-HOM)*, The Cryosphere, 2 (2008), pp. 95--108.

*An inexact Gauss-Newton method for inversion of basal sliding and rheology parameters in a nonlinear Stokes ice sheet model*, J. Glaciology, 58 (2012), pp. 889--903.

*A flexible inner-outer preconditioned GMRES algorithm*, SIAM J. Sci. Comput., 14 (1993), pp. 461--469.

*Iterative Methods for Sparse Linear Systems*, 2nd ed., SIAM, Philadelphia, 2003.

*Mantle Convection in the Earth and Planets*, Cambridge University Press, Cambridge, UK, 2001.

*Coupling ice flow models of varying orders of complexity with the Tiling method*, J. Glaciology, 58 (2012), pp. 776--786.

*Triangle: Engineering a 2D quality mesh generator and Delaunay triangulator*, in Applied Computational Geometry: Towards Geometric Engineering, M. C. Lin and D. Manocha, eds., Lecture Notes in Comput. Sci. 1148, Springer-Verlag, Berlin, 1996, pp. 203--222.

*Albany/felix: A parallel, scalable and robust, finite element, first-order Stokes approximation ice sheet solver built for advanced analysis*, Geosci. Model Development, 8 (2015), pp. 1197--1220.

*Mixed hp-finite element approximations on geometric edge and boundary layer meshes in three dimensions*, Numer. Math., 94 (2003), pp. 771--801.

*Multigrid*, Academic Press, London, 2001.

*Convergence of algebraic multigrid based on smoothed aggregation*, Numer. Math., 88 (2001), pp. 559--579.

*Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems*, Computing, 56 (1996), pp. 179--196.

*Glaciers and laws of friction and sliding*, Acta Mechanica, 166 (2003), pp. 185--206.

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**Submitted**: 25 June 2014

**Accepted**: 18 August 2015

**Published online**: 3 November 2015

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