Chen, L., Huang, X., & Zhang, C. (2024). DISTRIBUTIONAL FINITE ELEMENT CURL DIV COMPLEXES AND APPLICATION TO QUAD CURL PROBLEMS. arXiv preprint arXiv:2311.09051v4.
This paper aims to develop a new finite element method for solving the quad-curl problem in three dimensions, which arises in various applications like magnetohydrodynamics. The challenge lies in constructing conforming finite element spaces for the curl-div operator due to its high smoothness requirements.
The authors introduce the concept of tangential-normal continuity for finite element spaces to address the smoothness challenges associated with the curl-div operator. They construct a distributional finite element curl-div complex using piecewise polynomials with relaxed smoothness requirements. This complex incorporates tangential-normally continuous finite elements for the curl-div operator and N´ed´elec elements for tangential continuity. The authors then apply this complex to discretize the quad-curl problem, formulating a mixed finite element method.
The paper presents a novel and efficient approach to solving the quad-curl problem in three dimensions using distributional finite element methods. The proposed tangential-normal continuous finite elements and the corresponding curl-div complex offer a significant advancement in computational efficiency and accuracy compared to traditional methods.
This research contributes significantly to the field of scientific computing, particularly in the area of finite element methods for electromagnetic problems. The proposed method provides a practical and efficient solution for simulating complex physical phenomena governed by the quad-curl operator.
The paper primarily focuses on theoretical analysis and numerical experiments on simplified domains. Further research could explore the application of this method to more complex geometries and real-world engineering problems. Additionally, investigating the extension of this approach to higher-order curl problems could be a promising direction for future work.
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by Long Chen, X... at arxiv.org 11-04-2024
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