PPIFE Method with Non-Homogeneous Flux Jump Conditions and Its Efficient Numerical Solver for Elliptic Optimal Control Problems with Interfaces
Year: 2020
Author: Enlai Zheng, Quanxiang Wang, Zhiyue Zhang, Enlai Zheng
Numerical Mathematics: Theory, Methods and Applications, Vol. 13 (2020), Iss. 3 : pp. 719–744
Abstract
In this paper, we design a partially penalized immersed finite element method for solving elliptic interface problems with non-homogeneous flux jump conditions. The method presented here has the same global degrees of freedom as classic immersed finite element method. The non-homogeneous flux jump conditions can be handled accurately by additional immersed finite element functions. Four numerical examples are provided to demonstrate the optimal convergence rates of the method in $L^{\infty}$, $L^{2}$ and $H^{1}$ norms. Furthermore, the method is combined with post-processing technique to solve elliptic optimal control problems with interfaces. To solve the resulting large-scale system, block diagonal preconditioners are introduced. These preconditioners can lead to fast convergence of the Krylov subspace methods such as GMRES and are independent of the mesh size. Four numerical examples are presented to illustrate the efficiency of the numerical schemes and preconditioners.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.OA-2019-0088
Numerical Mathematics: Theory, Methods and Applications, Vol. 13 (2020), Iss. 3 : pp. 719–744
Published online: 2020-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 26
Keywords: Immersed finite element interface optimal control preconditioners fast solver.
Author Details
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Nonconforming immersed finite element method for solving elliptic optimal control problems with interfaces
Wang, Quanxiang
Xie, Jianqiang
Zhang, Zhiyue
Applicable Analysis, Vol. 101 (2022), Iss. 6 P.2197
https://doi.org/10.1080/00036811.2020.1802431 [Citations: 1]