A Functionally Connected Element Method for Solving Boundary Value Problems

Abstract

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.

Figures (10)

• Figure 1: 1D Helmholtz equation: $l^{2}$ errors of $C^1$ FCEs as a function of (a) the element size, and (b) the polynomial order. In (a), the polynomial order ($p$) is fixed while the element size is varied (h-refinement), showing a convergence rate of ($p+1$). In (b), the number of elements ($N$) is fixed while the polynomial order $p$ is varied (p-refinement), showing an exponential convergence rate. $(p+2)$ collocation points per element (Gauss-Lobatto-Legendre points).
• Figure 2: IVP: $l^{\infty}$ errors of FCE-$C^0$ as a function of (a) the element size, and (b) the polynomial order. $q=p+2$ Gauss-Lobatto-Legendre collocation points per element.
• Figure 3: 1D nonlinear Helmholtz equation: $l^{\infty}$ errors of FCE-$C^1$ as a function of (a) the element size, and (b) the polynomial order. $N$ uniform elements, and $q=p+2$ uniform collocation points per element.
• Figure 4: 2D Helmholtz equation: $l^{\infty}$ errors of FCE-$C^0$ versus (a) the element size in each direction, and (b) the polynomial order. $N$ uniform elements per direction.
• Figure 5: Advection equation: $l^{\infty}$ errors of FCE-$C^0$ versus (a) the element size in each direction, and (b) the polynomial order. $q=p+2$ Gauss-Lobatto-Legendre collocation points; $N$ denotes the number of elements per direction.

Theorems & Definitions (13)

• proof
• Remark 1
• Remark 2
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• Remark 3
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• Remark 4
• Remark 5