Partition of Unity Extension of Functions on Complex Domains

TL;DR

The paper presents PUX, an efficient partition of unity extension algorithm for functions on complex domains, enabling high-accuracy boundary integral solutions for PDEs.

Contribution

It introduces a novel, parallelizable extension method using radial basis functions and partitions of unity for complex multiply connected domains.

Findings

Achieves error convergence up to tenth order to 10^{-14}

Effective for solving Poisson equations with boundary integral methods

Parallelizable local extension process enhances computational efficiency

Abstract

We introduce an efficient algorithm, called partition of unity extension or PUX, to construct an extension of desired regularity of a function given on a complex multiply connected domain in $2D$. Function extension plays a fundamental role in extending the applicability of boundary integral methods to inhomogeneous partial differential equations with embedded domain techniques. Overlapping partitions are placed along the boundaries, and a local extension of the function is computed on each patch using smooth radial basis functions; a trivially parallel process. A partition of unity method blends the local extrapolations into a global one, where weight functions impose compact support. The regularity of the extended function can be controlled by the construction of the partition of unity function. We evaluate the performance of the PUX method in the context of solving the Poisson equation on multiply connected domains using a boundary integral method and a spectral solver. With a suitable choice of parameters the error converges as a tenth order method down to $10^{-14}$.