Truncation Error Analysis for an Accurate Nonlocal Manifold Poisson Model with Dirichlet Boundary

TL;DR

This paper develops a new nonlocal Poisson model on manifolds with Dirichlet boundary that achieves improved truncation error control by leveraging boundary geometry, promising higher accuracy in approximations.

Contribution

It introduces a novel nonlocal Poisson model that explicitly incorporates boundary geometry to reduce truncation errors, surpassing existing models in accuracy.

Findings

Achieves $ ext{O}( ext{delta})$ truncation error at the boundary

Attains $ ext{O}( ext{delta}^2)$ error in the interior

Provides optimal error control among nonlocal models

Abstract

In this work, we introduced a class of nonlocal models to accurately approximate the Poisson model on manifolds that are embedded in high dimensional Euclid spaces with Dirichlet boundary. In comparison to the existing nonlocal Poisson models, instead of utilizing volumetric boundary constraint to reduce the truncation error to its local counterpart, we rely on the Poisson equation itself along the boundary to explicitly express the second order normal derivative by some geometry-based terms, so that to create a new model with $\mathcal{O}(\delta)$ truncation error along the $2\delta-$boundary layer and $\mathcal{O}(\delta^2)$ at interior, with $\delta$ be the nonlocal interaction horizon. Our concentration is on the construction and the truncation error analysis of such nonlocal model. The control on the truncation error is currently optimal among all nonlocal models, and is sufficient to attain second order localization rate that will be derived in our subsequent work.