Abstract
A combination method of Newton’s method and two-level piecewise linear finite element algorithm is applied for solving second-order nonlinear elliptic partial differential equations numerically. Newton’s method is to find a finite element solution by solving \(m\) Newton equations on a fine mesh. The two-level Newton’s method solves \(m-1\) Newton equations on a coarse mesh and processes one Newton iteration on a fine mesh. Moreover, the optimal error estimates of Newton’s method and the two-level Newton’s method are provided to justify the efficiency of the two-level Newton’s method. If we choose \(H\) such that \(h=O(|\log h|^{1-2/{p}}H^2)\) for the \(W^{1,p}(\Omega )\)-error estimates, the two-level Newton’s method is asymptotically as accurate as Newton’s method on the fine mesh. Meanwhile, the numerical investigations provided a sufficient support for the theoretical analysis. Finally, these investigations also proved that the proposed method is efficient for solving the nonlinear elliptic problems.
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10915-013-9699-7/MediaObjects/10915_2013_9699_Fig1_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10915-013-9699-7/MediaObjects/10915_2013_9699_Fig2_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10915-013-9699-7/MediaObjects/10915_2013_9699_Fig3_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10915-013-9699-7/MediaObjects/10915_2013_9699_Fig4_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10915-013-9699-7/MediaObjects/10915_2013_9699_Fig5_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10915-013-9699-7/MediaObjects/10915_2013_9699_Fig6_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10915-013-9699-7/MediaObjects/10915_2013_9699_Fig7_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10915-013-9699-7/MediaObjects/10915_2013_9699_Fig8_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10915-013-9699-7/MediaObjects/10915_2013_9699_Fig9_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10915-013-9699-7/MediaObjects/10915_2013_9699_Fig10_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10915-013-9699-7/MediaObjects/10915_2013_9699_Fig11_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs10915-013-9699-7/MediaObjects/10915_2013_9699_Fig12_HTML.gif)
Similar content being viewed by others
References
Dennis Jr, J.E., Moré, J.J.: Quasi-Newton methods. Numer. Math. 11, 324–330 (1968)
Dembo, R.S., Eisenstat, S.C., Steihaug, T.: Inexact Newton methods. SIAM J. Numer. Anal. 19, 400–408 (1982)
Dennis Jr, J.E., Moré, J.J.: A characterization of superlinear convergence and its application to quasi-Newton methods. Math. Comput. 28, 549–560 (1974)
Douglas Jr, J., Dupont, T.: A Galerkin method for a nonlinear Dirichlet problem. Math. Comput. 29, 689–696 (1975)
Bank, R.E., Rose, D.J.: Analysis of a multilevel interative method for nonlinear finite element equations. Math. Comput. 39, 453–465 (1982)
Xu, J.C.: A novel two two-grid method for semilinear elliptic equations. SIAM J. Sci. Comput. 15, 231–237 (1994)
Xu, J.C.: Two-grid discretization techniques for linear and nonlinear PDEs. SIAM J. Numer. Anal. 33, 1759–1777 (1996)
He, Y., Li, J.: Convergence of three iterative methods based on the finite element discretization for the stationary Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 198, 1351–1359 (2009)
Layton, W.: A two-level discretization method for the Navier–Stokes equations. Comput. Math. Appl. 26, 33–38 (1993)
Layton, W., Tobiska, L.: A two-level method with backtraking for the Navier–Stokes equations. SIAM J. Numer. Anal. 35, 2035–2054 (1998)
Layton, W., Leferink, H.W.J.: Two-level Picard and modified Picard Methods for the Navier–Stokes equations. Appl. Math. Comput. 69, 263–274 (1995)
Girault, V., Lions, J.L.: Two-grid finite element scheme for the steady Navier–Stokes equations in polyhedra. Port. Math. 58, 25–57 (2001)
He, Y., Li, J., Yang, X.: Two-level penalized finite element methods for the stationary Navier–Stokes equations. Int. J. Inf. Syst. Sci. 2, 131–143 (2006)
He, Y., Li, K.: Two-level stabilized finite element methods for the steady Navier–Stokes problem. Computing 74, 337–351 (2005)
He, Y., Wang, A.: A simplified two-level for the steady Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 197, 1568–1576 (2008)
Layton, W., Leferink, H.W.J.: A multilevel mesh independence principle for the Navier–Stokes equations. SIAM J. Numer. Anal. 33, 17–30 (1996)
Layton, W., Lee, H.K., Peterson, J.: Numerical solution of the stationary Navier–Stokes equations using a multilevel finite element method. SIAM J. Sci. Comput. 20, 1–12 (1998)
Girault, V., Lions, J.L.: Two-grid finite element scheme for the transient Navier–Stokes problem. Math. Model. Numer. Anal. 35, 945–980 (2001)
Olshanskii, M.A.: Two-level method and some a priori estimates in unsteady Navier–Stokes calculations. J. Comput. Appl. Math. 104, 173–191 (1999)
He, Y.: Two-level method based on finite element and Crank–Nicolson extrapolation for the time-dependent Navier–Stokes equations. SIAM J. Numer. Anal. 41, 1263–1285 (2003)
He, Y.: A two-level finite element Galerkin method for the nonstationary Navier–Stokes equations, I: spatial discretization. J. Comput. Math. 22, 21–32 (2004)
He, Y., Miao, H., Ren, C.: A two-level finite element Galerkin method for the nonstationary Navier–Stokes equations, II: time discretization. J. Comput. Math. 22, 33–54 (2004)
He, Y., Liu, K.M.: Multi-level spectral Galerkin method for the Navier–Stokes equations II: time discretization. Adv. Comput. Math. 25, 403–433 (2006)
He, Y., Liu, K.M., Sun, W.W.: Multi-level spectral Galerkin method for the Navier–Stokes equations I: spatial discretization. Numer. Math. 101, 501–522 (2005)
He, Y., Liu, K.M.: A multi-level finite element method in space-time for the Navier–Stokes equations. Numer. Methods PDEs 21, 1052–1078 (2005)
Adams, R.A.: Sobolev Space. Academic Press, New York (1975)
Grisvard, P.: Elliptic Problem in Nonsmooth Domains. Pitman, Boston (1985)
Ciarlet, P.: The Finite Element Method for Elliptic Problems. North-Holland, New York (1978)
Schartz, A.: An observation concerning Ritz–Galerkin methods with indefinite bilinear forms. Math. Comput. 28, 959–962 (1974)
Rannacher, R., Scott, R.: Some optimal error estimates for piecewise linear finite element approximation. Math. Comput. 38, 437–445 (1982)
Thomée, V., Xu, J., Zhang, N.: Superconvergence of gradient in piecewise linear finite element approximation to a parabolic problem. SIAM J. Numer. Anal. 26, 553–573 (1989)
Xu, H., He, Y.: Some iterative finite element methods for steady Navier–Stokes equations with different viscosities. J. Comput. Phys. 232, 136–152 (2013)
Acknowledgments
This study subsidized by the NSF of China (No. 11271298). The authors would like to thank the editor and referees for their valuable comments and suggestions which helped to improve the quality of this paper.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
He, Y., Zhang, Y. & Xu, H. Two-Level Newton’s Method for Nonlinear Elliptic PDEs. J Sci Comput 57, 124–145 (2013). https://doi.org/10.1007/s10915-013-9699-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-013-9699-7