An Accurate and Efficient Scheme for Function Extension on Smooth Domains

Abstract

SIAM Journal on Numerical Analysis, Volume 63, Issue 4, Page 1427-1453, August 2025.
Abstract. A new scheme is proposed to construct an [math]-times differentiable function extension of an [math]-times differentiable function defined on a smooth domain, [math] in [math]-dimensions. The extension scheme relies on an explicit formula consisting of a linear combination of [math] function values in [math] 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 part of a numerical solver for the Poisson equation on domains with complex geometry in both two and three dimensions. Although the cost of extending the function increases mildly with the extension order, it remains a small fraction of the overall algorithm. Moreover, the modest additional work required for high order function extensions leads to considerably more accurate solutions of the partial differential equation.