Direct and Inverse Results on Bounded Domains for Meshless Methods via Localized Bases on Manifolds
Provides theoretical foundations for meshless methods on manifolds, benefiting researchers in approximation theory and numerical analysis.
The authors develop direct and inverse approximation estimates for kernel-based meshless methods on bounded domains, using localized Lagrange functions that are computationally efficient and applicable to Sobolev-Matérn kernels on smooth manifolds.
This article develops direct and inverse estimates for certain finite dimensional spaces arising in kernel approximation. Both the direct and inverse estimates are based on approximation spaces spanned by local Lagrange functions which are spatially highly localized. The construction of such functions is computationally efficient and generalizes the construction given by the authors for restricted surface splines on $\mathbb{R}^d$. The kernels for which the theory applies includes the Sobolev-Matérn kernels for closed, compact, connected, $C^\infty$ Riemannian manifolds.