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Erschienen in:
01.10.2013
Two-Level Newton’s Method for Nonlinear Elliptic PDEs
Erschienen in: Journal of Scientific Computing | Ausgabe 1/2013
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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.