A stable, efficient scheme for 𝒞^n function extensions on smooth domains in ℝ^d

06/22/2022

∙

by   Charles Epstein, et al.

∙

0

∙

share

A new scheme is proposed to construct a 𝒞^n function extension for smooth functions defined on a smooth domain D∈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 𝒞^n extension requires only n+1 function values along the normal directions in the original domain and ensures 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.