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 Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We design a monotone meshfree finite difference method for linear ellipticequations in the non-divergence form on point clouds via a nonlocal relaxationmethod. The key idea is a novel combination of a nonlocal integral relaxationof the PDE problem with a robust meshfree discretization on point clouds.Minimal positive stencils are obtained through a local $l_1$-type optimizationprocedure that automatically guarantees the stability and, therefore, theconvergence of the meshfree discretization for linear elliptic equations. Amajor theoretical contribution is the existence of consistent and positivestencils for a given point cloud geometry. We provide sufficient conditions forthe existence of positive stencils by finding neighbors within an ellipse (2d)or ellipsoid (3d) surrounding each interior point, generalizing the study forPoisson's equation by Seibold (Comput Methods Appl Mech Eng 198(3-4):592-601,2008). It is well-known that wide stencils are in general needed forconstructing consistent and monotone finite difference schemes for linearelliptic equations. Our result represents a significant improvement in thestencil width estimate for positive-type finite difference methods for linearelliptic equations in the near-degenerate regime (when the ellipticity constantbecomes small), compared to previously known works in this area. Numericalalgorithms and practical guidance are provided with an eye on the case of smallellipticity constant. At the end, we present numerical results for theperformance of our method in both 2d and 3d, examining a range of ellipticityconstants including the near-degenerate regime.

Monotone meshfree methods for linear elliptic equations in non-divergence form via nonlocal relaxation