Superconvergence analysis of linear FEM based on polynomial preserving recovery for Helmholtz equation with high wave number

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:372

Article

Du, Yu^1,3  Wu, Haijun²  Zhang, Zhimin^1,4 

[1] Beijing Computat Sci Res Ctr, Beijing 100193, Peoples R China

[2] Nanjing Univ, Dept Math, Nanjing 210093, Jiangsu, Peoples R China

[3] Xiangtan Univ, Dept Math, Xiangtan 411105, Hunan, Peoples R China

[4] Wayne State Univ, Dept Math, Detroit, MI 48202 USA

关键词: Helmholtz equation;    Large wave number;    Pollution errors;    Superconvergence;    Polynomial preserving recovery;    Finite element methods;   

DOI  :  10.1016/j.cam.2020.112731

来源: Elsevier

【 摘 要 】

Although superconvergence properties of the polynomial preserving recovery (PPR) for finite element methods (FEM) for coercive elliptic problems are well-understood, the superconvergence behavior of PPR for the Helmholtz equation with high wave number still remains unclear. We study the superconvergence property of the linear FEM with PPR for the two dimensional Helmholtz equation on triangulations satisfying the 0(h(1+alpha)) approximate parallelogram property for some constant alpha > 0. We prove the supercloseness on the H-1-seminorm between the finite element solution and the linear interpolant of the exact solution under k(kh)(2) <= C-0, where k is the wave number and h is the mesh size. Then, we obtain the superconvergence result based on the PPR technique which says that the PPR improves only the interpolation error but keeps the pollution error unchanged. Finally, we provide numerical tests to verify the superconvergence property and to demonstrate that the PPR combining with the continuous interior penalty technique is much effective for improving both the interpolation error and the pollution error. (C) 2020 Elsevier B.V. All rights reserved.

【 授权许可】

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