Journal of Scientific Computing

, Volume 74, Issue 2, pp 895–919 | Cite as

Surface Couplings for Subdomain-Wise Isoviscous Gradient Based Stokes Finite Element Discretizations

  • Markus Huber
  • Ulrich Rüde
  • Christian WalugaEmail author
  • Barbara Wohlmuth


The Stokes system with constant viscosity can be cast into different formulations by exploiting the incompressibility constraint. For instance, the rate of strain tensor in the weak formulation can be replaced by the velocity-gradient yielding a decoupling of the velocity components in the different coordinate directions. Consequently, the discretization of this partly decoupled formulation leads to fewer nonzero entries in the stiffness matrix. This is of particular interest in large scale simulations where a reduced memory bandwidth requirement can help to significantly accelerate the computations. In the case of a piecewise constant viscosity, as it typically arises in multi-phase flows, or when the boundary conditions involve traction, the situation is more complex, and one has to treat the cross derivatives in the original Stokes system with care. A naive application of the standard vectorial Laplacian results in a physically incorrect solution, while formulations based on the rate of strain tensor increase the computational effort globally. Here, we propose a new approach that is consistent with the stress-divergence formulation and preserves the decoupling advantages of the velocity-gradient-divergence formulation in isoviscous subdomains. The modification is equivalent to locally changing the discretization stencils at interfaces or boundaries. Hence, the more expensive discretization is of lower complexity, making the additional computational cost in large scale simulations negligible. We establish consistency and convergence properties and show that in a massively parallel setup, the multigrid solution of the resulting discrete systems is faster than for the classical stress-divergence formulation.


Finite elements Stokes equation Interface problem Multigrid method Matrix-free methods Stabilization Traction boundary conditions Non-isoviscous flow 


  1. 1.
    Amestoy, P.R., Guermouche, A., L’Excellent, J.Y., Pralet, S.: Hybrid scheduling for the parallel solution of linear systems. Parallel Comput. 32(2), 136–156 (2006)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Baker, A.H., Klawonn, A., Kolev, T., Lanser, M., Rheinbach, O., Yang, U.M.: Scalability of classical algebraic multigrid for elasticity to half a million parallel tasks. In: Bungartz, H.J., Neumann, P., Nagel, W.E. (eds.) Software for Exascale Computing - SPPEXA 2013-2015, pp. 113–140. Springer, Cham (2016). doi: 10.1007/978-3-319-40528-5_6 CrossRefGoogle Scholar
  3. 3.
    Bastian, P., Blatt, M., Dedner, A., Engwer, C., Klöfkorn, R., Kornhuber, R., Ohlberger, M., Sander, O.: A generic grid interface for parallel and adaptive scientific computing. Part II: implementation and tests in DUNE. Computing 82(2–3), 121–138 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bauer, S., Bunge, H.P., Drzisga, D., Gmeiner, B., Huber, M., John, L., Mohr, M.R., Rüde, U., Stengel, H., Waluga, C., Weismüller, J., Wellein, G., Wittmann, M., Wohlmuth, B.: Hybrid Parallel Multigrid Methods for Geodynamical Simulations. Springer, Cham (2016). doi: 10.1007/978-3-319-40528-5_10 CrossRefGoogle Scholar
  5. 5.
    Bergen, B., Gradl, T., Rüde, U., Hülsemann, F.: A massively parallel multigrid method for finite elements. Comput. Sci. Eng. 8(6), 56–62 (2006)CrossRefGoogle Scholar
  6. 6.
    Bergen, B., Hülsemann, F.: Hierarchical hybrid grids: data structures and core algorithms for multigrid. Numer. Linear Algebra Appl. 11, 279–291 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bey, J.: Tetrahedral grid refinement. Computing 55(4), 355–378 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  9. 9.
    Boffi, D., Cavallini, N., Gardini, F., Gastaldi, L.: Local mass conservation of Stokes finite elements. J. Sci. Comput. 52(2), 383–400 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Brezzi, F., Pitkäranta, J.: On the stabilization of finite element approximations of the Stokes equations. In: Hackbusch, W. (ed.) Efficient Solutions of Elliptic Systems. Springer, Berlin (1984)Google Scholar
  11. 11.
    Centre, J.S.: JUQUEEN: IBM Blue Gene/Q supercomputer system at the jülich supercomputing centre. J. Large-scale Res Facil. (2015). doi: 10.17815/jlsrf-1-18 Google Scholar
  12. 12.
    Cousins, B.R., Le Borne, S., Linke, A., Rebholz, L.G., Wang, Z.: Efficient linear solvers for incompressible flow simulations using Scott-Vogelius finite elements. Numer. Methods Partial Differ. Equ. 29(4), 1217–1237 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dongarra, J., Beckman, P., Moore, T., Aerts, P., Aloisio, G., Andre, J.C., Barkai, D., Berthou, J.Y., Boku, T., Braunschweig, B., Cappello, F., Chapman, B., Chi, X., Choudhary, A., Dosanjh, S., Dunning, T., Fiore, S., Geist, A., Gropp, B., Harrison, R., Hereld, M., Heroux, M., Hoisie, A., Hotta, K., Jin, Z., Ishikawa, Y., Johnson, F., Kale, S., Kenway, R., Keyes, D., Kramer, B., Labarta, J., Lichnewsky, A., Lippert, T., Lucas, B., Maccabe, B., Matsuoka, S., Messina, P., Michielse, P., Mohr, B., Mueller, M.S., Nagel, W.E., Nakashima, H., Papka, M.E., Reed, D., Sato, M., Seidel, E., Shalf, J., Skinner, D., Snir, M., Sterling, T., Stevens, R., Streitz, F., Sugar, B., Sumimoto, S., Tang, W., Taylor, J., Thakur, R., Trefethen, A., Valero, M., Van Der Steen, A., Vetter, J., Williams, P., Wisniewski, R., Yelick, K.: The international exascale software project roadmap. Int. J. High Perform. Comput. Appl. 25(1), 3–60 (2011)CrossRefGoogle Scholar
  14. 14.
    Falgout, R., Meier-Yang, U.: hypre: a library of high performance preconditioners. Comput. Sci.-ICCS 2002, 632–641 (2002)zbMATHGoogle Scholar
  15. 15.
    Gerya, T.: Introduction to Numerical Geodynamic Modelling. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  16. 16.
    Gmeiner, B., Huber, M., John, L., Rüde, U., Wohlmuth, B.: A quantitative performance study for Stokes solvers at the extreme scale. J. Comput Sci. 17, part 3, 509 – 521 (2016). doi: 10.1016/j.jocs.2016.06.006. Recent Advances in Parallel Techniques for Scientific Computing
  17. 17.
    Gmeiner, B., Rüde, U., Stengel, H., Waluga, C., Wohlmuth, B.: Performance and scalability of hierarchical hybrid multigrid solvers for stokes systems. SIAM J. Sci. Comput. 37(2), C143–C168 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gmeiner, B., Rüde, U., Stengel, H., Waluga, C., Wohlmuth, B.: Towards textbook efficiency for parallel multigrid. Numer. Math. Theory Methods Appl. 8(1), 22–46 (2015). doi: 10.4208/nmtma.2015.w10si MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gmeiner, B., Waluga, C., Wohlmuth, B.: Local mass-corrections for continuous pressure approximations of incompressible flow. SIAM J. Numer. Anal. 52(6), 2931–2956 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Grand, S.P., van der Hilst, R.D., Widiyantoro, S.: Global seismic tomography: a snapshot of convection in the earth. GSA Today 7, 1–7 (1997)Google Scholar
  21. 21.
    Gross, S., Reusken, A.: Numerical Methods for Two-phase Incompressible Flows, vol. 40. Springer, Berlin (2011)zbMATHGoogle Scholar
  22. 22.
    Hager, B., Clayton, R.: Mantle convection. In: Peltier, W.R. (ed.) Constraints on the Structure of Mantle Convection Using Seismic Observations, Flow Models, and the Geoid, pp. 657–764. Gordon and Breach, New York (1989)Google Scholar
  23. 23.
    Hansbo, P., Larson, M.G., Zahedi, S.: A cut finite element method for a Stokes interface problem. Appl. Numer. Math. 85, 90–114 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Hartley, R., Roberts, G., White, N., Richardson, C.: Transient convective uplift of an ancient buried landscape. Nat. Geosci. 4, 562–565 (2011)CrossRefGoogle Scholar
  25. 25.
    Haskell, N.A.: The motion of a fluid under a surface load. Physics 6, 265–269 (1935)CrossRefzbMATHGoogle Scholar
  26. 26.
    Heywood, J.G., Rannacher, R., Turek, S.: Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 22(5), 325–352 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Hughes, T.J.R., Franca, L.P.: A new finite element formulation for computational fluid dynamics: VII. the Stokes problem with various well-posed boundary conditions: symmetric formulations that converge for all velocity/pressure spaces. Comput. Methods Appl. Mech. Eng. 65(1), 85–96 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ito, K., Li, Z.: Interface conditions for Stokes equations with a discontinuous viscosity and surface sources. Appl. Math. Lett. 19(3), 229–234 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Limache, A., Idelsohn, S., Rossi, R., Oñate, E.: The violation of objectivity in Laplace formulations of the Navier-Stokes equations. Int. J. Numer. Methods Fluids 54(6–8), 639–664 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Logg, A., Mardal, K.A., Wells, G.N.: DOLFIN: a C++/Python finite element library. In: Lecture Notes in Computational Science and Engineering, vol. 84, chap. 10. Springer (2012)Google Scholar
  31. 31.
    May, D., Brown, J., Pourhiet, L.L.: A scalable, matrix-free multigrid preconditioner for finite element discretizations of heterogeneous Stokes flow. Comput. Methods Appl. Mech. Eng. 290, 496–523 (2015). doi: 10.1016/j.cma.2015.03.014 MathSciNetCrossRefGoogle Scholar
  32. 32.
    Mitrovica, J.X.: Haskell [1935] revisited. J. Geophys. Res. 101, 555–569 (1996)CrossRefGoogle Scholar
  33. 33.
    Müller, E.H., Scheichl, R.: Massively parallel solvers for elliptic partial differential equations in numerical weather and climate prediction. Q. J. R. Meteorol. Soc. 140(685), 2608–2624 (2014)CrossRefGoogle Scholar
  34. 34.
    Müller, R.D., Sdrolias, M., Gaina, C., Roest, W.R.: Age, spreading rates, and spreading asymmetry of the world’s ocean crust. Geochem. Geophys. Geosyst. 9, 1525–2027 (2008)CrossRefGoogle Scholar
  35. 35.
    Neff, P., Pauly, D., Witsch, K.J.: Poincaré meets Korn via Maxwell: extending Korn’s first inequality to incompatible tensor fields. J. Differ. Equ. 258(4), 1267–1302 (2015)CrossRefzbMATHGoogle Scholar
  36. 36.
    Neilan, M.: Discrete and conforming smooth de Rham complexes in three dimensions. Math. Comput. 84, 2059–2081 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Notay, Y., Napov, A.: A massively parallel solver for discrete Poisson-like problems. J. Comput. Phys 281(C), 237–250 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Olshanskii, M.A., Reusken, A.: Analysis of a Stokes interface problem. Numer. Math. 103(1), 129–149 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Parnell-Turner, R., White, N., Henstock, T., Murton, B., Maclennan, J., Jones, S.M.: A continuous 55 million year record of transient mantle plume activity beneath Iceland. Nat. Geosci. 7, 914–919 (2014)CrossRefGoogle Scholar
  40. 40.
    Pironneau, O.: Finite Element Methods for Fluids. Wiley, Chichester (1989)zbMATHGoogle Scholar
  41. 41.
    Pompe, W.: Counterexamples to Korn’s inequality with non-constant rotation coefficients. Math. Mech. Solids 16, 172–176 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Quarteroni, A., Valli, A.: Domain Decomposition Methods for Partial Differential Equations. Numerical Mathematics and Scientific Computation. Clarendon Press, Wotton-under-Edge (1999)zbMATHGoogle Scholar
  43. 43.
    Rannacher, R.: On the numerical solution of the incompressible Navier-Stokes equations. ZAMM - J. Appl. Math. Mech. / Z. Angew. Math. Mech. 73(9), 203–216 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Röhrig-Zöllner, M., Thies, J., Kreutzer, M., Alvermann, A., Pieper, A., Basermann, A., Hager, G., Wellein, G., Fehske, H.: Increasing the performance of the Jacobi-Davidson method by blocking. SIAM J. Sci. Comput. 37(6), C697–C722 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Rudi, J., Malossi, A., Isaac, T., Stadler, G., Gurnis, M., Staar, P., Ineichen, Y., Bekas, C., Curioni, A., Ghattas, O.: An extreme-scale implicit solver for complex PDEs: Highly heterogeneous flow in Earth’s mantle. In: Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, pp. 5:1–5:12 (2015)Google Scholar
  46. 46.
    Schöberl, J., Zulehner, W.: On Schwarz-type smoothers for saddle point problems. Numer. Math. 95(2), 377–399 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Scott, L., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Scott, L.R., Vogelius, M.: Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. Modélisation mathématique et analyse numérique 19(1), 111–143 (1985)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Stixrude, L., Lithgow-Bertelloni, C.: Thermodynamics of mantle minerals - I. Physical properties. Geophys. J. Int. 162, 610–632 (2005)CrossRefGoogle Scholar
  50. 50.
    Weismüller, J., Gmeiner, B., Ghelichkhan, S., Huber, M., John, L., Wohlmuth, B., Rüde, U., Bunge, H.P.: Fast asthenosphere motion in high-resolution global mantle flow models. Geophys. Res. Lett. 42(18), 7429–7435 (2015)CrossRefGoogle Scholar
  51. 51.
    Zhang, S.: A new family of stable mixed finite elements for the 3d Stokes equations. Math. Comput. 74(250), 543–554 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Zulehner, W.: Analysis of iterative methods for saddle point problems: a unified approach. Math. Comput. 71(238), 479–505 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Markus Huber
    • 1
  • Ulrich Rüde
    • 2
  • Christian Waluga
    • 3
    Email author
  • Barbara Wohlmuth
    • 1
  1. 1.Institute for Numerical Mathematics (M2), Technische Universität MünchenGarching bei MünchenGermany
  2. 2.Department of Computer Science 10FAU Erlangen-NürnbergErlangenGermany
  3. 3.liNear GmbHAachenGermany

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