Two-grid algorithms for some linear and nonlinear elliptic systems

Authors:
Shishun Li;Zhengda Huang
Affiliations:
Department of Mathematics, Zhejiang University, Hangzhou, People's Republic of China;Department of Mathematics, Zhejiang University, Hangzhou, People's Republic of China
Venue:
Computing
Year:
2010

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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 (L2 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.