A new scheme is proposed to construct a $\mathcal{C}^n$ function extension for smooth functions defined on a smooth domain $D\in \mathbb{R}^d$. Unlike the PUX scheme, which requires the extrapolation of the volume grid via an expensive ill-conditioned least squares fitting, the scheme relies on an explicit formula consisting of a linear combination of function values in $D,$ which only extends the function along the normal direction. To be more precise, the $\mathcal{C}^n$ extension requires only $n+1$ function values along the normal directions in the original domain and ensures $\mathcal{C}^n$ smoothness by construction. When combined with a shrinking function and a smooth window function, the scheme can be made stable and robust for a broad class of domains with complex smooth boundary.
翻译:提议一个新的方案, 为平滑域名 $D\ in\ mathbb{R ⁇ d$ 定义的平滑函数构建一个 $mathcal{C ⁇ n$ 函数扩展。 与PUX 方案不同的是, PUX 方案要求通过条件最差、条件最差、条件最便宜的方形组合对体积网格进行外推, 方案依赖于一个明确的公式, 由以$D 表示的函数线性组合构成, 仅将函数沿着正常方向延伸。 更精确地说, $\ mathcal{C ⁇ n$ $ la$ 扩展只需要在原始域的正常方向上加1$, 并通过构造确保$\mathcal{C ⁇ n$的顺畅性。 当与一个条件较窄的功能和光滑的窗口功能相结合时, 方案可以稳定而稳健地适用于具有复杂光边界的广大域。