An adaptive finite element method based on Superconvergent Cluster Recovery for the Cahn-Hilliard equation

Wenyan Tian; Yaoyao Chen; Zhaoxia Meng; Hongen Jia; Wenyan Tian; Yaoyao Chen; Zhaoxia Meng; Hongen Jia

doi:10.3934/era.2023068

Electronic Research Archive

2023, Volume 31, Issue 3: 1323-1343. doi: 10.3934/era.2023068

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Research article

An adaptive finite element method based on Superconvergent Cluster Recovery for the Cahn-Hilliard equation

1.
College of Mathematics, Taiyuan University of Technology, Tai'yuan 030024, China
2.
School of Mathematics and Statistics, Anhui Normal University, Wu'hu 241000, China
3.
Department of energy and power engineering, Shanxi Institute of Energy, Tai'yuan 030024, China

Received: 22 October 2022 Revised: 12 December 2022 Accepted: 26 December 2022 Published: 09 January 2023

In this study, we construct an error estimate for a fully discrete finite element scheme that satisfies the criteria of unconditional energy stability, as suggested in ^[1]. Our theoretical findings, in more detail, demonstrate that this system has second-order accuracy in both space and time. Additionally, we offer a powerful space and time adaptable approach for solving the Cahn-Hilliard problem numerically based on the posterior error estimation. The major goal of this technique is to successfully lower the calculated cost by controlling the mesh size using a Superconvergent Cluster Recovery (SCR) approach in accordance with the error estimation. To demonstrate the effectiveness and stability of the suggested SCR-based algorithm, numerical results are provided.
- error estimate,
- the Cahn-Hilliard equation,
- adaptive,
- SCR
Citation: Wenyan Tian, Yaoyao Chen, Zhaoxia Meng, Hongen Jia. An adaptive finite element method based on Superconvergent Cluster Recovery for the Cahn-Hilliard equation[J]. Electronic Research Archive, 2023, 31(3): 1323-1343. doi: 10.3934/era.2023068

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Abstract

In this study, we construct an error estimate for a fully discrete finite element scheme that satisfies the criteria of unconditional energy stability, as suggested in ^[1]. Our theoretical findings, in more detail, demonstrate that this system has second-order accuracy in both space and time. Additionally, we offer a powerful space and time adaptable approach for solving the Cahn-Hilliard problem numerically based on the posterior error estimation. The major goal of this technique is to successfully lower the calculated cost by controlling the mesh size using a Superconvergent Cluster Recovery (SCR) approach in accordance with the error estimation. To demonstrate the effectiveness and stability of the suggested SCR-based algorithm, numerical results are provided.

References

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