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 Plus annual

Presents the access of premium content as premium feature

Premium Content

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 Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Boundary integral equations are an efficient and accurate tool for thenumerical solution of elliptic boundary value problems. The solution isexpressed as a layer potential; however, the error in its evaluation growslarge near the boundary if a fixed quadrature rule is used. Firstly, we analyzethis error for Laplace's equation with analytic density and the global periodictrapezoid rule, and find an intimate connection to the complexification of theboundary parametrization. Our main result is then a simple and efficient schemefor accurate evaluation up to the boundary for single- and double-layerpotentials for the Laplace and Helmholtz equations, using surrogate localexpansions about centers placed near the boundary. The scheme---which alsounderlies the recent QBX Nystr\"om quadrature---is asymptotically exponentiallyconvergent (we prove this in the analytic Laplace case), requires noadaptivity, generalizes simply to three dimensions, and has O(N) complexitywhen executed via a locally-corrected fast multipole sum. We give an example ofhigh-frequency scattering from an obstacle with perimeter 700 wavelengths long,evaluating the solution at $2\times 10^5$ points near the boundary with11-digit accuracy in 30 seconds in MATLAB on a single CPU core.

Evaluation of layer potentials close to the boundary for Laplace and Helmholtz problems on analytic planar domains