A Functionally Connected Element Method for Solving Boundary Value Problems
arxiv(2024)
摘要
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.
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