Blankenbach, B., Busse, F., Christensen, U., Cserepes, L., Gunkel, D., Hansen, U., Harder, H., Jarvis, G., Koch, M., Marquart, G., Moore, D., Olson, P., Schmeling, H., and Schnaubelt, T.: A benchmark comparison for mantle convection codes, Geophys. J. Int., 98, 23–38,
https://doi.org/10.1111/j.1365-246X.1989.tb05511.x, 1989.
a
Böhm, F., Bauer, D., Kohl, N., Alappat, C. L., Thönnes, D., Mohr, M., Köstler, H., and Rüde, U.: Code Generation and Performance Engineering for Matrix-Free Finite Element Methods on Hybrid Tetrahedral Grids, SIAM J. Sci. Comput., 47, B131–B159,
https://doi.org/10.1137/24M1653756, 2025.
a,
b,
c,
d,
e,
f
Boussinesq, J.: Theorie analytique de la chaleur vol. 2, Gauthier-Villars, Paris,
https://archive.org/details/thorieanalytiqu00bousgoog (last access: 27 January 2026), 1903. a
Brown, H., Colli, L., and Bunge, H.-P.: Asthenospheric flow through the Izanagi-Pacific slab window and its influence on dynamic topography and intraplate volcanism in East Asia, Front. Earth Sci., 10,
https://doi.org/10.3389/feart.2022.889907, 2022.
a,
b
Burkhart, A., Kohl, N., Wohlmuth, B., and Zawallich, J.: A robust matrix-free approach for large-scale non-isothermal high-contrast viscosity Stokes flow on blended domains with applications to geophysics, GEM – Int. J. Geomath., 17,
https://doi.org/10.1007/s13137-025-00280-5, 2026.
a
Burstedde, C., Stadler, G., Alisic, L., Wilcox, L. C., Tan, E., Gurnis, M., and Ghattas, O.: Large-scale adaptive mantle convection simulation, Geophys. J. Int., 192, 889–906,
https://doi.org/10.1093/gji/ggs070, 2013.
a
Choblet, G., Čadek, O., Couturier, F., and Dumoulin, C.: ŒDIPUS: a new tool to study the dynamics of planetary interiors, Geophys. J. Int., 170, 9–30,
https://doi.org/10.1111/j.1365-246x.2007.03419.x, 2007.
a
Clevenger, T. C. and Heister, T.: Comparison between algebraic and matrix‐free geometric multigrid for a Stokes problem on adaptive meshes with variable viscosity, Numer. Lin. Algeb. Appl., 28,
https://doi.org/10.1002/nla.2375, 2021.
a,
b
Colli, L., Ghelichkhan, S., Bunge, H.-P., and Oeser, J.: Retrodictions of Mid Paleogene mantle flow and dynamic topography in the Atlantic region from compressible high resolution adjoint mantle convection models: Sensitivity to deep mantle viscosity and tomographic input model, Gondwana Res., 53, 252–272,
https://doi.org/10.1016/j.gr.2017.04.027, 2018.
a,
b,
c
Cordier, P. and Goryaeva, A.: Multiscale Modeling of the Mantle Rheology, 156 pp., ISBN 978-2-9564368-1-2,
https://lilloa.hal.science/hal-01881743v2/ (last access: 27 January 2026), 2018. a
Davies, D. R., Davies, J. H., Bollada, P. C., Hassan, O., Morgan, K., and Nithiarasu, P.: A hierarchical mesh refinement technique for global 3-D spherical mantle convection modelling, Geosci. Model Dev., 6, 1095–1107,
https://doi.org/10.5194/gmd-6-1095-2013, 2013.
a,
b
Davies, D. R., Kramer, S. C., Ghelichkhan, S., and Gibson, A.: Towards automatic finite-element methods for geodynamics via Firedrake, Geosci. Model Dev., 15, 5127–5166,
https://doi.org/10.5194/gmd-15-5127-2022, 2022.
a,
b,
c,
d,
e
Engelman, M. S., Sani, R. L., and Gresho, P. M.: The implementation of normal and/or tangential boundary conditions in finite element codes for incompressible fluid flow, Int. J. Numer. Meth. Fluids, 2, 225–238,
https://doi.org/10.1002/fld.1650020302, 1982.
a
Euen, G. T., Liu, S., Gassmöller, R., Heister, T., and King, S. D.: A comparison of 3-D spherical shell thermal convection results at low to moderate Rayleigh number using ASPECT (version 2.2.0) and CitcomS (version 3.3.1), Geosci. Model Dev., 16, 3221–3239,
https://doi.org/10.5194/gmd-16-3221-2023, 2023.
a,
b,
c,
d,
e,
f,
g,
h,
i,
j
Gaspar, F. J., Notay, Y., Oosterlee, C. W., and Rodrigo, C.: A Simple and Efficient Segregated Smoother for the Discrete Stokes Equations, SIAM J. Sci. Comput., 36, A1187–A1206,
https://doi.org/10.1137/130920630, 2014.
a
Gassmöller, R., Dannberg, J., Bangerth, W., Heister, T., and Myhill, R.: On formulations of compressible mantle convection, Geophys. J. Int., 221, 1264–1280,
https://doi.org/10.1093/gji/ggaa078, 2020.
a
Gmeiner, B., Huber, M., John, L., Rüde, U., and Wohlmuth, B.: A quantitative performance study for Stokes solvers at the extreme scale, J. Comput. Sci., 17, 509–521,
https://doi.org/10.1016/j.jocs.2016.06.006, 2016.
a,
b
Gresho, P. M., Lee, R. L., Sani, R. L., Maslanik, M. K., and Eaton, B. E.: The consistent Galerkin FEM for computing derived boundary quantities in thermal and or fluids problems, Int. J. Numer. Meth. Fluids, 7, 371–394,
https://doi.org/10.1002/fld.1650070406, 1987.
a
Ham, D. A., Kelly, P. H., Mitchell, L., Cotter, C., Kirby, R. C., Sagiyama, K., Bouziani, N., Vorderwuelbecke, S., Gregory, T., Betteridge, J., Shapero, D. R., Nixon-Hill, R., Ward, C., Farrell, P. E., Brubeck, P. D., Marsden, I., Gibson, T. H., Homolya, M., Sun, T., McRae, A. T., Luporini, F., Gregory, A., Lange, M., Funke, S. W., Rathgeber, F., Bercea, G.-T., and Markall, G. R.: Firedrake User Manual, in: 1st Edn., Imperial College London and University of Oxford and Baylor University and University of Washington,
https://doi.org/10.25561/104839, 2023.
a
Heister, T., Dannberg, J., Gassmöller, R., and Bangerth, W.: High accuracy mantle convection simulation through modern numerical methods – II: realistic models and problems, Geophys. J. Int., 210, 833–851,
https://doi.org/10.1093/gji/ggx195, 2017.
a,
b
Horbach, A., Mohr, M., and Bunge, H.-P.: A semi-analytic accuracy benchmark for Stokes flow in 3-D spherical mantle convection codes, GEM – Int. J. Geomath., 11,
https://doi.org/10.1007/s13137-019-0137-3, 2019.
a
Ilangovan, P., D'Ascoli, E., Kohl, N., and Mohr, M.: Numerical Studies on Coupled Stokes-Transport Systems for Mantle Convection, Springer Nature, Switzerland, 288–302, ISBN 9783031637599,
https://doi.org/10.1007/978-3-031-63759-9_33, 2024.
a,
b,
c
Ilangovan, P., Kohl, N., and Mohr, M.: Geodynamic Benchmark Simulations with HyTeG: Supplementary Data for Article “Highly Scalable Geodynamic Simulations with HyTeG”, Leibniz Supercomputing Centre (LRZ) [data set],
https://doi.org/10.25927/zvnf9-cvc38, 2025.
a
Jarvis, G. T.: Effects of curvature on two‐dimensional models of mantle convection: Cylindrical polar coordinates, J. Geophys. Res.-Solid, 98, 4477–4485,
https://doi.org/10.1029/92jb02117, 1993.
a
King, S. D., Lee, C., van Keken, P. E., Leng, W., Zhong, S., Tan, E., Tosi, N., and Kameyama, M. C.: A community benchmark for 2-D Cartesian compressible convection in the Earth’s mantle, Geophys. J. Int., 180, 73–87,
https://doi.org/10.1111/j.1365-246x.2009.04413.x, 2010.
a,
b,
c,
d,
e
Kohl, N. and Rüde, U.: Textbook Efficiency: Massively Parallel Matrix-Free Multigrid for the Stokes System, SIAM J. Sci. Comput., 44, C124–C155,
https://doi.org/10.1137/20M1376005, 2022.
a,
b,
c,
d,
e
Kohl, N., Thönnes, D., Drzisga, D., Bartuschat, D., and Rüde, U.: The
HyTeG finite-element software framework for scalable multigrid solvers, Int. J. Parallel Emerg. Distrib. Syst., 34, 477–496,
https://doi.org/10.1080/17445760.2018.1506453, 2019.
a,
b
Kohl, N., Mohr, M., Eibl, S., and Rüde, U.: A Massively Parallel Eulerian-Lagrangian Method for Advection-Dominated Transport in Viscous Fluids, SIAM J. Sci. Comput., 44, C260–C285,
https://doi.org/10.1137/21M1402510, 2022.
a,
b
Kohl, N., Bauer, D., Böhm, F., and Rüde, U.: Fundamental data structures for matrix-free finite elements on hybrid tetrahedral grids, Int. J. Parallel Emerg. Distrib. Syst., 39, 51–74,
https://doi.org/10.1080/17445760.2023.2266875, 2024.
a,
b
Kramer, S. C., Davies, D. R., and Wilson, C. R.: Analytical solutions for mantle flow in cylindrical and spherical shells, Geosci. Model Dev., 14, 1899–1919,
https://doi.org/10.5194/gmd-14-1899-2021, 2021.
a,
b,
c,
d
Liu, S. and King, S. D.: A benchmark study of incompressible Stokes flow in a 3-D spherical shell using ASPECT, Geophys. J. Int., 217, 650–667,
https://doi.org/10.1093/gji/ggz036, 2019.
a
May, D., Brown, J., and Le Pourhiet, L.: A scalable, matrix-free multigrid preconditioner for finite element discretizations of heterogeneous Stokes flow, Comput. Meth. Appl. Mech. Eng., 290, 496–523,
https://doi.org/10.1016/j.cma.2015.03.014, 2015.
a
Moresi, L.-N. and Solomatov, V. S.: Numerical investigation of 2D convection with extremely large viscosity variations, Phys. Fluids, 7, 2154–2162,
https://doi.org/10.1063/1.868465, 1995.
a
Oeser, J., Bunge, H.-P., and Mohr, M.: Cluster Design in the Earth Sciences: TETHYS, in: High Performance Computing and Communications – Second International Conference, HPCC 2006, Munich, Germany, vol. 4208 of Lecture Notes in Computer Science, edited by: Gerndt, M. and Kranzlmüller, D., 31–40, Springer,
https://doi.org/10.1007/11847366_4, 2006.
a
Ratcliff, J. T., Schubert, G., and Zebib, A.: Steady tetrahedral and cubic patterns of spherical shell convection with temperature‐dependent viscosity, J. Geophys. Res.-Solid, 101, 25473–25484,
https://doi.org/10.1029/96jb02097, 1996.
a,
b,
c,
d,
e
Rudi, J., Stadler, G., and Ghattas, O.: Weighted BFBT Preconditioner for Stokes Flow Problems with Highly Heterogeneous Viscosity, SIAM J. Sci. Comput., 39, S272–S297,
https://doi.org/10.1137/16m108450x, 2017.
a,
b
Tan, E., Choi, E., Thoutireddy, P., Gurnis, M., and Aivazis, M.: GeoFramework: Coupling multiple models of mantle convection within a computational framework, Geochem. Geophy. Geosy., 7,
https://doi.org/10.1029/2005gc001155, 2006.
a
Tan, E., Leng, W., Zhong, S., and Gurnis, M.: On the location of plumes and lateral movement of thermochemical structures with high bulk modulus in the 3-D compressible mantle: Plumes and chemical structures, Geochem. Geophy. Geosy., 12,
https://doi.org/10.1029/2011gc003665, 2011.
a,
b
Terrel, A. R., Scott, L. R., Knepley, M. G., Kirby, R. C., and Wells, G. N.: Finite elements for incompressible fluids, Springer, Berlin, Heidelberg, 385–397,
https://doi.org/10.1007/978-3-642-23099-8_20, 2012.
a
Tosi, N., Stein, C., Noack, L., Hüttig, C., Maierová, P., Samuel, H., Davies, D. R., Wilson, C. R., Kramer, S. C., Thieulot, C., Glerum, A., Fraters, M., Spakman, W., Rozel, A., and Tackley, P. J.: A community benchmark for viscoplastic thermal convection in a 2-D square box, Geochem. Geophy. Geosy., 16, 2175–2196,
https://doi.org/10.1002/2015GC005807, 2015.
a,
b,
c
Williams, K., Stamps, D. S., Austermann, J., King, S., and Njinju, E.: Effects of using the consistent boundary flux method on dynamic topography estimates, Geophys. J. Int., 238, 1137–1149,
https://doi.org/10.1093/gji/ggae203, 2024.
a
Zhong, S., Gurnis, M., and Hulbert, G.: Accurate determination of surface normal stress in viscous flow from a consistent boundary flux method, Phys. Earth Planet. Inter., 78, 1–8,
https://doi.org/10.1016/0031-9201(93)90078-n, 1993.
a
Zhong, S., Zuber, M. T., Moresi, L., and Gurnis, M.: Role of temperature-dependent viscosity and surface plates in spherical shell models of mantle convection, J. Geophys. Res.-Solid, 105, 11063–11082,
https://doi.org/10.1029/2000jb900003, 2000.
a
Zhong, S., McNamara, A., Tan, E., Moresi, L., and Gurnis, M.: A benchmark study on mantle convection in a 3-D spherical shell using CitcomS, Geochem. Geophy. Geosy., 9,
https://doi.org/10.1029/2008gc002048, 2008.
a,
b,
c,
d,
e,
f,
g,
h,
i,
j
