Edward J. Fuselier

A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces

In this paper we present a high-order kernel method for numerically solving diffusion and reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces embedded in

Stability and Error Estimates for Vector Field Interpolation and Decomposition on the Sphere with RBFs

\mathbb{R }^d

Improved stability estimates and a characterization of the native space for matrix-valued RBFs

. For two-dimensional surfaces embedded in

Error and Stability Estimates for Surface-Divergence Free RBF Interpolants on the Sphere

\mathbb{R }^3

Reduction in transport by the parallel velocity shear instability due to reversed magnetic shear

, these types of problems have received growing interest in biology, chemistry, and computer graphics to model such things as diffusion of chemicals on biological cells or membranes, pattern formations in biology, nonlinear chemical oscillators in excitable media, and texture mappings. Our kernel method is based on radial basis functions and uses a semi-discrete approach (or the method-of-lines) in which the surface derivative operators that appear in the PDEs are approximated using collocation. The method only requires nodes at “scattered” locations on the surface and the corresponding normal vectors to the surface. Additionally, it does not rely on any surface-based metrics and avoids any intrinsic coordinate systems, and thus does not suffer from any coordinate distortions or singularities. We provide error estimates for the kernel-based approximate surface derivative operators and numerically study the accuracy and stability of the method. Applications to different non-linear systems of PDEs that arise in biology and chemistry are also presented.

Sobolev-type approximation rates for divergence-free and curl-free RBF interpolants

In this paper we present error estimates for kernel interpolation at scattered sites on manifolds. The kernels we consider will be obtained by the restriction of positive definite kernels on