Partition of unity extension of functions on complex domains

• Published in: Journal of computational physics 2018-12, Vol.375, p.57-79
• Main Authors: Fryklund, Fredrik, Lehto, Erik, Tornberg, Anna-Karin
• Format: Article
• Language: English
• Subjects: Algorithms; Basis functions; Boundary conditions; Boundary integral method; Computational physics; Domains; Embedded domain; Embedded systems; Function extension; Integrals; Linear elliptic partial differential equation; Mathematical functions; Partial differential equations; Partition of unity; Partitions; Poisson equation; Radial basis function; Regularity; Unity; Weight
• ISSN: 0021-9991; 1090-2716; 1090-2716
• DOI: 10.1016/j.jcp.2018.08.012

•Extends the applicability of boundary integral methods to inhomogeneous PDEs.•A simple to implement and efficient algorithm for numerical function extension.•The most computationally costly part can be precomputed and reused.•The extended function has compact support and user specified regularity.•Solution to the Poisson equation converges as a tenth order method to round off. 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.