Calculus on Surfaces with General Closest Point Functions

Abstract

The Closest Point Method for solving partial differential equations (PDEs) posed on surfaces was recently introduced by Ruuth and Merriman [J. Comput. Phys. 2008] and successfully applied to a variety of surface PDEs. In this paper we study the theoretical foundations of this method. The main idea is that surface differentials of a surface function can be replaced with Cartesian differentials of its closest point extension, i.e., its composition with a closest point function. We introduce a general class of these closest point functions (a subset of differentiable retractions), show that these are exactly the functions necessary to satisfy the above idea, and give a geometric characterization of this class. Finally, we construct some closest point functions and demonstrate their effectiveness numerically on surface PDEs.Comment: 22 pages, 3 figures, 4 table

Simple computation of reaction–diffusion processes on point clouds

“…In addition, this remains true in the general case where S is a smooth object of dimensions (3,4). The right-hand side of Eq.…”

An embedding technique for the solution of reaction–diffusion equations on algebraic surfaces with isolated singularities

“…Such a calculus has been introduced earlier (see for example [1,8,10,13,21,24]); here we will use much of the same notation and terminology as [21]. We point out here that throughout the paper surface functions, i.e., functions of the form f : S → R m , are not required to be polynomials.…”

“…We consider the following C 1 -smooth surface functions: a scalar function u : S → R, a vector-valued function f : S → R m , and a vector field g : S → R n with C 1 -smooth local extensions into R n called u E , f E , g E (see [21]). We define the surface gradient ∇ S , the surface Jacobian D S , and the surface divergence div S at the regular point x by:…”

“…Within the past decade several numerical embedding methods have been created for solving PDEs that are posed on smooth surfaces. These include level-set methods such as those presented in [15] and [14] as well as the Closest Point Method as presented in [23] and [26] and further extended in [20] and [21]. However, the existing embedding techniques are low-order accurate or inconsistent when applied to PDEs on surfaces with singularities.…”

“…It is therefore necessary to choose a numerical method that is effective for arbitrary co-dimensional embeddings. The Closest Point Method is one such embedding method, as shown in [21] and [26], and is our method of choice.…”