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Two-grid algorithms for some linear and nonlinear elliptic systems

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Abstract

In this paper, several two-grid algorithms are presented. For nonsymmetric linear systems, we propose a two-grid algorithm by using the information of the adjoint operator. The solution of the original systems is mainly reduced to a solution of symmetric positive definite (SPD) systems. For nonlinear systems, we present a two-grid algorithm based on the modified Newton method. The solution of the original systems on the fine space is reduced to the solution of two small systems on the coarse space and two similar linear systems (with same stiffness matrix) on the fine space. It is shown that the accuracy (\({\mathcal{L}^2}\) norm) obtained by this algorithm is as same as the optimal accuracy derived by using two full Newton steps. Additionally, for more practically applications, the ideas of these algorithms can be also extended to the multilevel case. Numerical experiments are given for these new algorithms.

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou, 310027, People’s Republic of China

    Shishun Li & Zhengda Huang

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  1. Shishun Li
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  2. Zhengda Huang
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Correspondence to Zhengda Huang.

Additional information

Communicated by C.C. Douglas.

This research was supported by the National Natural Science Foundation of China (Grant No. 10731060).

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Li, S., Huang, Z. Two-grid algorithms for some linear and nonlinear elliptic systems. Computing 89, 69–86 (2010). https://doi.org/10.1007/s00607-010-0095-y

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  • DOI: https://doi.org/10.1007/s00607-010-0095-y

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