English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

The recently developed quadrature by expansion (QBX) technique accuratelyevaluates the layer potentials with singular, weakly or nearly singular, oreven hyper singular kernels in the integral equation reformulations of partialdifferential equations. The idea is to form a local complex polynomial orpartial wave expansion centered at a point away from the boundary to avoid thesingularity in the integrand, and then extrapolate the expansion at points nearor even exactly on the boundary. In this paper, in addition to the localcomplex Taylor polynomial expansion, we derive new representations of theLaplace layer potentials using both the local complex polynomial and plane waveexpansions. Unlike in the QBX, the local complex polynomial expansion in thenew quadrature by two expansions (QB2X) method only collects the far-fieldcontributions and its number of expansion terms can be analyzed using toolsfrom the classical fast multipole method. The plane wave type expansion in theQB2X method better captures the layer potential features near the boundary. Itis derived by applying the Fourier extension technique to the density andboundary geometry functions and then analytically utilizing the Residue Theoremfor complex contour integrals. The internal connections of the layer potentialwith its density function and curvature on the boundary are explicitly revealedin the plane wave expansion and its error is bounded by the Fourier extensionerrors. We present preliminary numerical results to demonstrate the accuracy ofthe QB2X representations and to validate our analysis.

Quadrature by Two Expansions: Evaluating Laplace Layer Potentials using Complex Polynomial and Plane Wave Expansions