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Auxiliary Space Preconditioners for a \(C^{0}\) Finite Element Approximation of Hamilton–Jacobi–Bellman Equations with Cordes Coefficients

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Abstract

In the past decade, there are many works on the finite element methods for the fully nonlinear Hamilton–Jacobi–Bellman (HJB) equations with Cordes condition. The linearised systems have large condition numbers, which depend not only on the mesh size but also on the parameters in the Cordes condition. This paper is concerned with the design and analysis of auxiliary space preconditioners for the linearised systems of a \(C^0\) finite element discretization of HJB equations [Calcolo, 58, 2021]. Based on the stable decomposition on the auxiliary spaces, we propose both the additive and multiplicative preconditioners which converge uniformly in the sense that the resulting condition number is independent of both the number of degrees of freedom and the parameter \(\lambda \) in Cordes condition. Numerical experiments are carried out to illustrate the efficiency of the proposed preconditioners.

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References

  1. Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions, vol. 25. Springer, Switzerland (2006)

    MATH  Google Scholar 

  2. Maugeri, A., Palagachev, D.K., Softova, L.G.: Elliptic and Parabolic Equations with Discontinuous Coefficients, vol. 109. WILEY-VCH Verlag GmbH & Co., Berlin (2000)

    Book  Google Scholar 

  3. Smears, I., Süli, E.: Discontinuous Galerkin finite element approximation of Hamilton–Jacobi–Bellman equations with Cordes coefficients. SIAM J. Numer. Anal. 52(2), 993–1016 (2014)

    Article  MathSciNet  Google Scholar 

  4. Smears, I., Süli, E.: Discontinuous Galerkin finite element methods for time-dependent Hamilton–Jacobi–Bellman equations with Cordes coefficients. Numer. Math. 133(1), 141–176 (2016)

    Article  MathSciNet  Google Scholar 

  5. Neilan, M., Wu, M.: Discrete Miranda-Talenti estimates and applications to linear and nonlinear PDEs. J. Comput. Appl. Math. 356, 358–376 (2019)

    Article  MathSciNet  Google Scholar 

  6. Gallistl, D., Süli, E.: Mixed finite element approximation of the Hamilton–Jacobi–Bellman equation with Cordes coefficients. SIAM J. Numer. Anal. 57(2), 592–614 (2019)

    Article  MathSciNet  Google Scholar 

  7. Wu, S.: \(C^0\) finite element approximations of linear elliptic equations in non-divergence form and Hamilton-Jacobi-Bellman equations with Cordes coefficients. Calcolo 58(1), 1–26 (2021)

    Article  Google Scholar 

  8. Neilan, M.: Discrete and conforming smooth de Rham complexes in three dimensions. Math. Comput. 84(295), 2059–2081 (2015)

    Article  MathSciNet  Google Scholar 

  9. Christiansen, S.H., Hu, J., Hu, K.: Nodal finite element de Rham complexes. Numer. Math. 139(2), 411–446 (2018)

    Article  MathSciNet  Google Scholar 

  10. Smears, I., Süli, E.: Discontinuous Galerkin finite element approximation of nondivergence form elliptic equations with Cordes coefficients. SIAM J. Numer. Anal. 51(4), 2088–2106 (2013)

    Article  MathSciNet  Google Scholar 

  11. Kawecki, E.L.: A DGFEM for nondivergence form elliptic equations with Cordes coefficients on curved domains. Numerical Methods for Partial Differential Equations 35(5), 1717–1744 (2019)

    Article  MathSciNet  Google Scholar 

  12. Kawecki, E.L., Smears, I.: Unified analysis of discontinuous galerkin and c0-interior penalty finite element methods for Hamilton-Jacobi-Bellman and isaacs equations. ESAIM: Mathematical Modelling and Numerical Analysis 55(2), 449–478 (2021)

    Article  MathSciNet  Google Scholar 

  13. Brenner, S.C., Kawecki, E.L.: Adaptive \(C^0\) interior penalty methods for Hamilton-Jacobi-Bellman equations with Cordes coefficients. J. Comput. Appl. Math. 388, 113241 (2021)

    Article  Google Scholar 

  14. Kawecki, E.L., Smears, I.: Convergence of adaptive discontinuous Galerkin and \(C^0\)-interior penalty finite element methods for Hamilton-Jacobi-Bellman and Isaacs equations. Found. Comput. Math. 22(2), 315–364 (2022)

    Article  MathSciNet  Google Scholar 

  15. Peisker, P., Braess, D.: A conjugate gradient method and a multigrid algorithm for Morley’s finite element approximation of the biharmonic equation. Numer. Math. 50(5), 567–586 (1987)

    Article  MathSciNet  Google Scholar 

  16. Brenner, S.C.: An optimal-order nonconforming multigrid method for the biharmonic equation. SIAM J. Numer. Anal. 26(5), 1124–1138 (1989)

    Article  MathSciNet  Google Scholar 

  17. Brenner, S.: Convergence of nonconforming multigrid methods without full elliptic regularity. Math. Comput. 68(225), 25–53 (1999)

    Article  MathSciNet  Google Scholar 

  18. Stevenson, R.: An analysis of nonconforming multigrid methods, leading to an improved method for the Morley element. Math. Comput. 72(241), 55–81 (2003)

    Article  Google Scholar 

  19. Carstensen, C., Hu, J.: Hierarchical argyris finite element method for adaptive and multigrid algorithms. Computational Methods in Applied Mathematics 21(3), 529–556 (2021)

    Article  MathSciNet  Google Scholar 

  20. Zhang, X.: Multilevel Schwarz methods for the biharmonic Dirichlet problem. SIAM J. Sci. Comput. 15(3), 621–644 (1994)

    Article  MathSciNet  Google Scholar 

  21. Brenner, S.C.: A two-level additive Schwarz preconditioner for nonconforming plate elements. Numer. Math. 72(4), 419–447 (1996)

    Article  MathSciNet  Google Scholar 

  22. Smears, I.: Nonoverlapping domain decomposition preconditioners for discontinuous Galerkin approximations of Hamilton-Jacobi-Bellman equations. J. Sci. Comput. 74(1), 145–174 (2018)

    Article  MathSciNet  Google Scholar 

  23. Peisker, P.: On the numerical solution of the first biharmonic equation. ESAIM: Mathematical Modelling and Numerical Analysis 22(4), 655–676 (1988)

    Article  MathSciNet  Google Scholar 

  24. Glowinski, R., Pironneau, O.: Numerical methods for the first biharmonic equation and for the two-dimensional Stokes problem. SIAM Rev. 21(2), 167–212 (1979)

    Article  MathSciNet  Google Scholar 

  25. Xu, J.: The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids. Computing 56(3), 215–235 (1996)

    Article  MathSciNet  Google Scholar 

  26. Zhang, S., Xu, J.: Optimal solvers for fourth-order PDEs discretized on unstructured grids. SIAM J. Numer. Anal. 52(1), 282–307 (2014)

    Article  MathSciNet  Google Scholar 

  27. Peisker, P., Rust, W., Stein, E.: Iterative solution methods for plate bending problems: Multigrid and preconditioned CG algorithm. SIAM J. Numer. Anal. 27(6), 1450–1465 (1990)

    Article  MathSciNet  Google Scholar 

  28. Elman, H.C.: Iterative methods for large, sparse, nonsymmetric systems of linear equations. PhD thesis, Yale University New Haven, Conn (1982)

  29. Grasedyck, L., Wang, L., Xu, J.: A nearly optimal multigrid method for general unstructured grids. Numer. Math. 134(3), 637–666 (2016)

    Article  MathSciNet  Google Scholar 

  30. Holst, M., Vandewalle, S.: Schwarz methods: to symmetrize or not to symmetrize. SIAM J. Numer. Anal. 34(2), 699–722 (1997)

    Article  MathSciNet  Google Scholar 

  31. Notay, Y., Napov, A.: Further comparison of additive and multiplicative coarse grid correction. Appl. Numer. Math. 65, 53–62 (2013)

    Article  MathSciNet  Google Scholar 

  32. Grisvard, P.: Elliptic Problems in Nonsmooth Domains. SIAM, Philadelphia (2011)

    Book  Google Scholar 

  33. Renardy, M., Rogers, R.C.: An Introduction to Partial Differential Equations, vol. 13. Springer, Switzerland (2006)

    MATH  Google Scholar 

  34. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Oxford (1978)

    MATH  Google Scholar 

  35. Brenner, S., Scott, R.: The Mathematical Theory of Finite Element Methods, vol. 15. Springer, Switzerland (2007)

    Google Scholar 

  36. Saad, Y.: Iterative Methods for Sparse Linear Systems. SIAM, Philadelphia (2003)

    Book  Google Scholar 

  37. Xu, J., Zikatanov, L.: Algebraic multigrid methods. Acta Numer. 26, 591–721 (2017)

    Article  MathSciNet  Google Scholar 

  38. Bramble, J.H., Pasciak, J.E., Xu, J.: Parallel multilevel preconditioners. Math. Comput. 55(191), 1–22 (1990)

    Article  MathSciNet  Google Scholar 

  39. Bramble, J., Pasciak, J., Vassilevski, P.: Computational scales of sobolev norms with application to preconditioning. Math. Comput. 69(230), 463–480 (2000)

    Article  MathSciNet  Google Scholar 

  40. Boffi, D., Brezzi, F., Fortin, M., et al.: Mixed Finite Element Methods and Applications, vol. 44. Springer, Berlin (2013)

    Book  Google Scholar 

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Acknowledgements

The authors would like to express their gratitude to Prof. Jun Hu in Peking University for his helpful discussions.

Funding

The work of Shuonan Wu is supported in part by the National Natural Science Foundation of China grant No. 11901016 and the startup grant from Peking University grant No. 7100601681.

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  1. School of Mathematical Sciences, Peking University, 100871, Beijing, China

    Guangwei Gao & Shuonan Wu

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  1. Guangwei Gao
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Correspondence to Shuonan Wu.

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Gao, G., Wu, S. Auxiliary Space Preconditioners for a \(C^{0}\) Finite Element Approximation of Hamilton–Jacobi–Bellman Equations with Cordes Coefficients. J Sci Comput 92, 105 (2022). https://doi.org/10.1007/s10915-022-01957-x

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