Numerical investigation of convergence in the $ L^{\infty} $ norm for modified SGFEM applied to elliptic interface problems

Pengfei Zhu; Kai Liu; Pengfei Zhu; Kai Liu

doi:10.3934/math.20241507

AIMS Mathematics

2024, Volume 9, Issue 11: 31252-31273. doi: 10.3934/math.20241507

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Numerical investigation of convergence in the $ L^{\infty} $ norm for modified SGFEM applied to elliptic interface problems

Pengfei Zhu ^{1
,
,},
Kai Liu ²

1.
School of Computer and Information Engineering, Guizhou University of Commerce, Guiyang 550014, China
2.
School of Science, East China University of Technology, Nanchang 330013, China

Received: 23 August 2024 Revised: 11 October 2024 Accepted: 21 October 2024 Published: 04 November 2024
MSC : 65N12, 65N30

Convergence in the $ L^{\infty} $ norm is a very important consideration in numerical simulations of interface problems. In this paper, a modified stable generalized finite element method (SGFEM) was proposed for solving the second-order elliptic interface problem in the two-dimensional bounded and convex domain. The proposed SGFEM uses a one-side enrichment function. There is no stability term in the weak form of the model problem, and it is a conforming finite element method. Moreover, it is applicable to any smooth interface, regardless of its concavity or shape. Several nontrivial examples illustrate the excellent properties of the proposed SGFEM, including its convergence in both the $ L^2 $ and $ L^{\infty} $ norms, as well as its stability and robustness.
- elliptic interface problem,
- SGFEM,
- convergence,
- $ L^{\infty} $ norm,
- scaled condition number
Citation: Pengfei Zhu, Kai Liu. Numerical investigation of convergence in the $ L^{\infty} $ norm for modified SGFEM applied to elliptic interface problems[J]. AIMS Mathematics, 2024, 9(11): 31252-31273. doi: 10.3934/math.20241507

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Abstract

Convergence in the $ L^{\infty} $ norm is a very important consideration in numerical simulations of interface problems. In this paper, a modified stable generalized finite element method (SGFEM) was proposed for solving the second-order elliptic interface problem in the two-dimensional bounded and convex domain. The proposed SGFEM uses a one-side enrichment function. There is no stability term in the weak form of the model problem, and it is a conforming finite element method. Moreover, it is applicable to any smooth interface, regardless of its concavity or shape. Several nontrivial examples illustrate the excellent properties of the proposed SGFEM, including its convergence in both the $ L^2 $ and $ L^{\infty} $ norms, as well as its stability and robustness.

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