Finite element method for Poisson equations with the Wang-Ball element

Lanyin Sun; Ziwei Dong; Lanyin Sun; Ziwei Dong

doi:10.3934/math.2025711

AIMS Mathematics

2025, Volume 10, Issue 7: 15867-15892. doi: 10.3934/math.2025711

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Research article

Finite element method for Poisson equations with the Wang-Ball element

Lanyin Sun ^,,
Ziwei Dong

School of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000, China

Received: 28 April 2025 Revised: 22 June 2025 Accepted: 01 July 2025 Published: 14 July 2025
MSC : 65D17, 65M60

In electrostatic field research, the Poisson equation is the core equation describing the relationship between electric potential and charge distribution. The finite element method (FEM) is an analytical engineering tool for accurately calculating various physical quantities in the electrostatic field. In this paper, we employ the Wang-Ball basis functions to construct the trial and test function spaces of FEM for solving the Poisson equation. In addition, we provide an error analysis based on the Wang-Ball operator. Several examples with different electrostatics backgrounds are also given to substantiate the effectiveness of this method. Furthermore, numerical results show that Wang-Ball elements work well for degree $ k > 3 $.
- finite element method,
- the Wang-Ball element,
- Poisson equation,
- unisolvence,
- error analysis
Citation: Lanyin Sun, Ziwei Dong. Finite element method for Poisson equations with the Wang-Ball element[J]. AIMS Mathematics, 2025, 10(7): 15867-15892. doi: 10.3934/math.2025711

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Abstract

In electrostatic field research, the Poisson equation is the core equation describing the relationship between electric potential and charge distribution. The finite element method (FEM) is an analytical engineering tool for accurately calculating various physical quantities in the electrostatic field. In this paper, we employ the Wang-Ball basis functions to construct the trial and test function spaces of FEM for solving the Poisson equation. In addition, we provide an error analysis based on the Wang-Ball operator. Several examples with different electrostatics backgrounds are also given to substantiate the effectiveness of this method. Furthermore, numerical results show that Wang-Ball elements work well for degree $ k > 3 $.

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