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.
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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.
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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
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DOI: https://doi.org/10.1007/s10915-013-9699-7