A new scheme is proposed to construct an n-times differentiable function extension of an n-times differentiable function defined on a smooth domain D in d-dimensions. The extension scheme relies on an explicit formula consisting of a linear combination of n+1 function values in D, which extends the function along directions normal to the boundary. Smoothness tangent to the boundary is automatic. The performance of the scheme is illustrated by using function extension as a step in a numerical solver for the inhomogeneous Poisson equation on multiply connected domains with complex geometry in two and three dimensions. We show that the modest additional work needed to do function extension leads to considerably more accurate solutions of the partial differential equation.

翻译：本文提出了一种新方案，用于将定义在d维光滑域D上的n次可微函数，构造为n次可微的函数延拓。该延拓方案基于一个由D内n+1个函数值线性组合构成的显式公式，可沿边界法向方向实现函数延拓。切向方向的平滑性自然得以保持。通过将函数延拓作为数值求解器中的步骤，应用于二维和三维复杂几何多连通区域上的非齐次泊松方程，展示了该方案的性能。研究表明，引入函数延拓所需适度额外计算量可显著提升偏微分方程解的精度。