Kernel-based methods for solving surface partial differential equations

A convergence analysis technique in our previous work is extended to various theoretically proven convergent kernel-based least-squares collocation methods for surface elliptic equation, projection methods for surface elliptic equation, and recently for surface parabolic equations. These partial differential equations (PDEs) on surfaces closely resemble their Euclidean counterparts, except that the problem domains change from bulk regions with a flat geometry to some manifolds, on which curvatures plays an important role in the physical processes. We do not focus on proofs in this paper, but on implementation details instead. First, we present an embedding formulation to solve a surface PDE in a narrow-band domain containing the surface. Next, we present another extrinsic projection formulation that works solely on data points on the surface. Lastly, we solve surface diffusion problem using kernel and the method of lines.