Meshless collocation methods for solving pdes on surfaces

We present three recently proposed kernel-based collocation methods in unified notations as an easy reference for practitioners who need to solve PDEs on surfaces S ⊂ ℝ^d. These PDEs closely resemble their Euclidean counterparts, except that the problem domains change from bulk regions with a flat geometry of some surfaces, on which curvatures play an important role in the physical processes. First, we present a formulation to solve surface PDEs in a narrow band domain containing the surface. This class of numerical methods is known as the embedding types. Next, we present another formulation that works solely on the surface, which is commonly referred to as the intrinsic approach. Convergent estimates and numerical examples for both formulations will be given. For the latter, we solve both the linear and nonlinear time-dependent parabolic equations on static and moving surfaces.