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.

翻译：我们给出了在分区域上满足子域边界处固有$C^0$或$C^1$连续性的分段函数的一般形式。这些一般形式基于函数连接理论中的策略构建，我们将赋予这些一般形式的分区域称为函数连接单元（FCE）。我们进一步提出了一种方法，结合函数连接单元和最小二乘配点法，用于求解边值和初值问题。该方法具有谱精度，其中FCE形式中的自由函数由多项式基或由准随机正弦函数的非多项式基表示。对于涉及相对边界条件的边值问题，FCE方法相较于传统基于单元的方法具有独特优势。我们通过一维和二维的多个线性和非线性数值示例，展示了本文所发展的FCE方法的性能。