A Functionally Connected Element Method for Solving Boundary Value Problems
<|reference_start|>A Functionally Connected Element Method for Solving Boundary Value Problems: We present the general forms of piece-wise functions on partitioned domains satisfying an intrinsic $C^0$ or $C^1$ continuity across the sub-domain boundaries. These general forms are constructed based on a strategy stemming from the theory of functional connections, and we refer to partitioned domains endowed with these general forms as functionally connected elements (FCE). We further present a method, incorporating functionally connected elements and a least squares collocation approach, for solving boundary and initial value problems. This method exhibits a spectral-like accuracy, with the free functions involved in the FCE form represented by polynomial bases or by non-polynomial bases of quasi-random sinusoidal functions. The FCE method offers a unique advantage over traditional element-based methods for boundary value problems involving relative boundary conditions. A number of linear and nonlinear numerical examples in one and two dimensions are presented to demonstrate the performance of the FCE method developed herein.<|reference_end|>
@article{yang2024a,
    title={A Functionally Connected Element Method for Solving Boundary Value
  Problems},
    author={Jielin Yang, Suchuan Dong},
    journal={arXiv preprint arXiv:2403.06393},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.06393},
    primaryClass={math.NA cs.NA physics.flu-dyn}
}