Truncation Error Analysis for an Accurate Nonlocal Manifold Poisson Model with Dirichlet Boundary (2203.02120v2)

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.