论文标题
具有Dirichlet边界的精确非局部歧管泊松模型的截断误差分析
Truncation Error Analysis for an Accurate Nonlocal Manifold Poisson Model with Dirichlet Boundary
论文作者
论文摘要
在这项工作中，我们引入了一类非局部模型，以准确地将泊松模型近似于嵌入具有Dirichlet边界的高维欧几里得空间的歧管上。与现有的非局部泊松模型相比，我们依靠沿边界沿边界的泊松方程本身将截短误差降低到本地对应物中，以将poisson方程式自身依赖于$ \ \ \ \ \ \ \ trunce prom} $ {$ frours}（$ \δ）（$ \Δ $ \ MATHCAL {O}（δ^2）$在内部，$δ$是非局部相互作用范围。我们的浓度是这种非局部模型的构建和截断误差分析。目前，对所有非局部模型的截断误差的控制是最佳的，并且足以达到将在我们后续工作中得出的二阶定位率。
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 $\mathcal{O}(δ)$ truncation error along the $2δ-$boundary layer and $\mathcal{O}(δ^2)$ at interior, with $δ$ 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.
