Calculus on surfaces with general closest point functions

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 this class. Finally, we construct some closest point functions and demonstrate their effectiveness numerically on surface PDEs

Integration over curves and surfaces defined by the closest point mapping

We propose a new formulation for integrating over smooth curves and surfaces that are described by their closest point mappings. Our method is designed for curves and surfaces that are not defined by any explicit parameterization and is intended to be used in combination with level set techniques. However, contrary to the common practice with level set methods, the volume integrals derived from our formulation coincide exactly with the surface or line integrals that one wish to compute. We study various aspects of this formulation and provide a geometric interpretation of this formulation in terms of the singular values of the Jacobian matrix of the closest point mapping. Additionally, we extend the formulation - initially derived to integrate over manifolds of codimension one - to include integration along curves in three dimensions. Some numerical examples using very simple discretizations are presented to demonstrate the efficacy of the formulation.Comment: Revised the pape

Solving eigenvalue problems on curved surfaces using the Closest Point Method

Eigenvalue problems are fundamental to mathematics and science. We present a simple algorithm for determining eigenvalues and eigenfunctions of the Laplace--Beltrami operator on rather general curved surfaces. Our algorithm, which is based on the Closest Point Method, relies on an embedding of the surface in a higher-dimensional space, where standard Cartesian finite difference and interpolation schemes can be easily applied. We show that there is a one-to-one correspondence between a problem defined in the embedding space and the original surface problem. For open surfaces, we present a simple way to impose Dirichlet and Neumann boundary conditions while maintaining second-order accuracy. Convergence studies and a series of examples demonstrate the effectiveness and generality of our approach

Finding a closest point in a lattice of Voronoi's first kind

We show that for those lattices of Voronoi's first kind with known obtuse superbasis, a closest lattice point can be computed in $O(n^4)$ operations where $n$ is the dimension of the lattice. To achieve this a series of relevant lattice vectors that converges to a closest lattice point is found. We show that the series converges after at most $n$ terms. Each vector in the series can be efficiently computed in $O(n^3)$ operations using an algorithm to compute a minimum cut in an undirected flow network

A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces

The closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961, [2008]) is an embedding method developed to solve a variety of partial differential equations (PDEs) on smooth surfaces, using a closest point representation of the surface and standard Cartesian grid methods in the embedding space. Recently, a closest point method with explicit time-stepping was proposed that uses finite differences derived from radial basis functions (RBF-FD). Here, we propose a least-squares implicit formulation of the closest point method to impose the constant-along-normal extension of the solution on the surface into the embedding space. Our proposed method is particularly flexible with respect to the choice of the computational grid in the embedding space. In particular, we may compute over a computational tube that contains problematic nodes. This fact enables us to combine the proposed method with the grid based particle method (Leung and Zhao, J. Comput. Phys. 228(8):2993-3024, [2009]) to obtain a numerical method for approximating PDEs on moving surfaces. We present a number of examples to illustrate the numerical convergence properties of our proposed method. Experiments for advection-diffusion equations and Cahn-Hilliard equations that are strongly coupled to the velocity of the surface are also presented

On closest-point maps in Banach spaces

Ph.D.W. J. Kammere