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Two-grid methods for semi-linear elliptic interface problems by immersed finite element methods

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  • 1. School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan Province, China;
    2. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China;
    3. College of Science, Hunan Agricultural University, Changsha 410128, China

Received date: 2019-02-18

  Revised date: 2019-05-19

  Online published: 2019-10-28

Supported by

Project supported by the National Natural Science Foundation of China (Nos. 11671157 and 11826212)

Abstract

In this paper, two-grid immersed finite element (IFE) algorithms are proposed and analyzed for semi-linear interface problems with discontinuous diffusion coefficients in two dimension. Because of the advantages of finite element (FE) formulation and the simple structure of Cartesian grids, the IFE discretization is used in this paper. Two-grid schemes are formulated to linearize the FE equations. It is theoretically and numerically illustrated that the coarse space can be selected as coarse as H=O(h1/4) (or H=O(h1/8)), and the asymptotically optimal approximation can be achieved as the nonlinear schemes. As a result, we can settle a great majority of nonlinear equations as easy as linearized problems. In order to estimate the present two-grid algorithms, we derive the optimal error estimates of the IFE solution in the Lp norm. Numerical experiments are given to verify the theorems and indicate that the present two-grid algorithms can greatly improve the computing efficiency.

Cite this article

Yang WANG, Yanping CHEN, Yunqing HUANG, Ying LIU . Two-grid methods for semi-linear elliptic interface problems by immersed finite element methods[J]. Applied Mathematics and Mechanics, 2019 , 40(11) : 1657 -1676 . DOI: 10.1007/s10483-019-2538-7

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