Acceleration of nonlinear solvers for natural convection problems
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Sara Pollock
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Mengying Xiao
Abstract
This paper develops an efficient and robust solution technique for the steady Boussinesq model of non-isothermal flow using Anderson acceleration applied to a Picard iteration. After analyzing the fixed point operator associated with the nonlinear iteration to prove that certain stability and regularity properties hold, we apply the authors’ recently constructed theory for Anderson acceleration, which yields a convergence result for the Anderson accelerated Picard iteration for the Boussinesq system. The result shows that the leading term in the residual is improved by the gain in the optimization problem, but at the cost of additional higher order terms that can be significant when the residual is large. We perform numerical tests that illustrate the theory, and show that a 2-stage choice of Anderson depth can be advantageous. We also consider Anderson acceleration applied to the Newton iteration for the Boussinesq equations, and observe that the acceleration allows the Newton iteration to converge for significantly higher Rayleigh numbers that it could without acceleration, even with a standard line search.
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Funding: The author SP acknowledges support from National Science Foundation through the grant DMS 2011519. The author LR acknowledges support from National Science Foundation through the grant DMS 2011490.
References
[1] H. An, X. Jia, and H. F. Walker, Anderson acceleration and application to the three-temperature energy equations. J. Comput. Phys. 347 (2017), 1–19.10.1016/j.jcp.2017.06.031Suche in Google Scholar
[2] M. Akbas, S. Kaya, and L. Rebholz, On the stability at all times of linearly extrapolated BDF2 timestepping for multiphysics incompressible flow problems. Num. Meth. P.D.E.s 33 (2017), No. 4, 995–1017.10.1002/num.22061Suche in Google Scholar
[3] D. G. Anderson. Iterative procedures for nonlinear integral equations. J. Assoc. Comput. Mach. 12 (1965), No. 4, 547–560.10.1145/321296.321305Suche in Google Scholar
[4] D. Arnold, F. Brezzi, and M. Fortin, A stable finite element for the Stokes equations. Calcolo 21 (1984), No. 4, 337–344.10.1007/BF02576171Suche in Google Scholar
[5] D. Arnold and J. Qin, Quadratic velocity/linear pressure Stokes elements. In: Advances in Computer Methods for Partial Differential Equations VII (Eds. R. Vichnevetsky, D. Knight, and G. Richter), IMACS, 1992, pp. 28–34.Suche in Google Scholar
[6] M. Benzi, G. Golub, and J. Liesen, Numerical solution of saddle point problems. Acta Numerica 14 (2005), 1–137.10.1017/S0962492904000212Suche in Google Scholar
[7] M. Benzi and M. Olshanskii, An augmented Lagrangian-based approach to the Oseen problem. SIAM J. Sci. Comput. 28 (2006), 2095–2113.10.1137/050646421Suche in Google Scholar
[8] S. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 3rd edition. Springer-Verlag, 2008.10.1007/978-0-387-75934-0Suche in Google Scholar
[9] A. Cibik and S. Kaya, A projection-based stabilized finite element method for steady state natural convection problem. J. Math. Anal. Appl. 381 (2011), 469–484.10.1016/j.jmaa.2011.02.020Suche in Google Scholar
[10] H. Elman, D. Silvester, and A. Wathen, Finite Elements and Fast Iterative Solvers with applications in incompressible fluid dynamics. Numerical Mathematics and Scientific Computation, Oxford Univ. Press, Oxford, 2014.10.1093/acprof:oso/9780199678792.001.0001Suche in Google Scholar
[11] C. Evans, S. Pollock, L. Rebholz, and M. Xiao, A proof that Anderson acceleration increases the convergence rate in linearly converging fixed point methods (but not in quadratically converging ones). SIAM J. Numer. Anal. 58 (2020), 788–810.10.1137/19M1245384Suche in Google Scholar
[12] V. Eyert, A comparative study on methods for convergence acceleration of iterative vector sequences. J. Comput. Phys. 124 (1996), No. 2, 271–285.10.1006/jcph.1996.0059Suche in Google Scholar
[13] H. Fang and Y. Saad, Two classes of multisecant methods for nonlinear acceleration. Numer. Linear Algebra Appl. 16 (2009), No. 3, 197–221.10.1002/nla.617Suche in Google Scholar
[14] G. Fu, J. Guzman, and M. Neilan, Exact smooth piecewise polynomial sequences on Alfeld splits. Math. Comput. 89 (2020), No. 323, 1059–1091.10.1090/mcom/3520Suche in Google Scholar
[15] G. H. Golub, C. F. Van Loan, Matrix Computations, 3rd ed. Johns Hopkins University Press, Baltimore, 1996.Suche in Google Scholar
[16] R. Haelterman, A. E. J. Bogaers, K. Scheufele, B. Uekermann, and M. Mehl, Improving the performance of the partitioned QN-ILS procedure for fluid–structure interaction problems: filtering. Comput. Struct. 171 (2016), 9–17.10.1016/j.compstruc.2016.04.001Suche in Google Scholar
[17] C. T. Kelley, Numerical methods for nonlinear equations. Acta Numerica 27 (2018), 207–287.10.1017/S0962492917000113Suche in Google Scholar
[18] W. Layton, An Introduction to the Numerical Analysis of Viscous Incompressible Flows. SIAM, Philadelphia, 2008.10.1137/1.9780898718904Suche in Google Scholar
[19] S. Pollock, L. Rebholz, and M. Xiao, Anderson-accelerated convergence of Picard iterations for incompressible Navier–Stokes equations. SIAM J. Numer. Anal. 57 (2019), No. 2, 615–637.10.1137/18M1206151Suche in Google Scholar
[20] S. Pollock and L. G. Rebholz, Anderson acceleration for contractive and noncontractive iterations. IMA J. Numer. Anal. 41 (2021), No.4, 2841–287210.1093/imanum/draa095Suche in Google Scholar
[21] S. Pollock and H. Schwartz, Benchmarking results for the Newton–Anderson method. Results Appl. Math. 8 (2020), 100095, 1–11.10.1016/j.rinam.2020.100095Suche in Google Scholar
[22] V. Pták, The rate of convergence of Newton’s process. Numer. Math. 25 (1976), 279–285.10.1007/BF01399416Suche in Google Scholar
[23] A. Toth and C. T. Kelleyr, Convergence analysis for Anderson acceleration. SIAM J. Numer. Anal. 53 (2015), No. 2, 805–819.10.1137/130919398Suche in Google Scholar
[24] H. F. Walker and P. Ni, Anderson acceleration for fixed-point iterations. SIAM J. Numer. Anal. 49 (2011), No. 4, 1715–1735.10.2172/1240289Suche in Google Scholar
[25] S. Zhang, A new family of stable mixed finite elements for the 3D Stokes equations. Math. Comp. 74 (2005), No. 250, 543–554.10.1090/S0025-5718-04-01711-9Suche in Google Scholar
[26] S. Zhang, A family of Qk+1,k × Qk,k+1 divergence-free finite elements on rectangular grids. SIAM J. Numer. Anal. 47 (2009), No. 3, 2090–2107.10.1137/080728949Suche in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- A reduced basis method for fractional diffusion operators II
- Schur complement spectral bounds for large hybrid FETI-DP clusters and huge three-dimensional scalar problems
- Entropy stabilization and property-preserving limiters for ℙ1 discontinuous Galerkin discretizations of scalar hyperbolic problems
- Acceleration of nonlinear solvers for natural convection problems
Artikel in diesem Heft
- Frontmatter
- A reduced basis method for fractional diffusion operators II
- Schur complement spectral bounds for large hybrid FETI-DP clusters and huge three-dimensional scalar problems
- Entropy stabilization and property-preserving limiters for ℙ1 discontinuous Galerkin discretizations of scalar hyperbolic problems
- Acceleration of nonlinear solvers for natural convection problems
