An accurate and efficient scheme for function extensions on smooth domains

TL;DR

This paper introduces a new explicit scheme for extending functions smoothly beyond their original domain, improving the accuracy of numerical solutions to PDEs on complex geometries.

Contribution

The paper presents a novel explicit formula for n-times differentiable function extension relying on boundary-normal directions, enhancing PDE solver accuracy.

Findings

Extension scheme is efficient and easy to implement.

Significant accuracy improvements in Poisson equation solutions.

Automatic smoothness tangent to the boundary achieved.

Abstract

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.