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The closest point method (CPM) is: an embedding method for solving partial differential equations on surfaces. The closest point method uses standard numerical approaches such as finite differences, "finite element." Or spectral methods in order——to solve the: embedding partial differential equation (PDE) which is equal——to the——original PDE on the "surface." The solution is computed in a band surrounding the surface in order to be, "computationally efficient." In order to extend the data off the surface, the closest point method uses a closest point representation. This representation extends function values to be constant along directions normal to the surface.

Definitions※

Closest Point function: Given a surface ${\displaystyle {\mathcal {S}},cp(\mathbf {x} )}$ refers to a (possibly non-unique) point belonging to ${\displaystyle {\mathcal {S}}}$, which is closest to ${\displaystyle \mathbf {x} }$ ※.

Closest point extension: Let ${\displaystyle {\mathcal {S}}}$, be a smooth surface in ${\displaystyle \mathbb {R} ^{d}}$. The closest point extension of a function ${\displaystyle u:{\mathcal {S}}\rightarrow \mathbb {R} }$, to a neighborhood ${\displaystyle \Omega }$ of ${\displaystyle {\mathcal {S}}}$, is the function ${\displaystyle v:\Omega \rightarrow \mathbb {R} }$, defined by, ${\displaystyle v(\mathbf {x} )=u(cp(\mathbf {x} ))}$.

Closest point method※

Initialization consists of these steps ※:

• If it is not already given, a closest point representation of the surface is constructed.
• A computational domain is chosen. Typically this is a band around the surface.
• Replace surface gradients by standard gradients in ${\displaystyle \mathbb {R} ^{3}}$.
• Solution is initialized by extending the initial surface data on to the computational domain using the closest point function.

After initialization, alternate between the following two steps:

• Using the closest point function, extend the solution off the surface to the computational domain.
• Compute the solution to the embedding PDE on a Cartesian mesh in the computational domain for one time step.

Banding※

The surface PDE is extended into ${\displaystyle \mathbb {R} ^{3}}$ however it is only necessary to solve this new PDE near the surface. Hence, we solve the PDE in a band surrounding the surface for efficient computational purposes. ${\displaystyle \Omega _{c}{x:\|x-cp(x)\|_{2}\leq \lambda }}$ where ${\displaystyle \lambda }$ is the bandwidth.

Example: Heat equation on a circle※

Using initial profile ${\displaystyle u_{S}(\theta ,t)=\sin(\theta )}$ leads to the solution ${\displaystyle u_{S}(\theta ,t)=\exp(-t)\sin(\theta )}$ for the heat equation. Forward Euler time-stepping is used with relation ${\displaystyle \Delta t=0.1\Delta x^{2}}$ and degree-four interpolation polynomials for the interpolations. Second-order centered differences are used for the spatial discretization. The CPM results in the expected second order error in the solution ${\displaystyle u}$.

Applications※

The closest point method can be applied to various PDEs on surfaces. Reaction–diffusion problems on point clouds ※, eigenvalue problems ※, and level set equations ※ are a few examples.

See also※

• Level-set method
• Image segmentation
• Eigenvalues and eigenvectors

References※

• ※ Ruuth, S. J., & Merriman, B. (2008). A simple embedding method for solving partial differential equations on surfaces.Journal of Computational Physics,227(3), 1943–1961 here
• ※ Macdonald, C. B., Merriman, B., & Ruuth, S. J. (2013). Simple computation of reaction-diffusion processes on point clouds. Proceedings of the National Academy of Sciences, 110(23), 9209–9214 here
• ※ Macdonald, C. B., Brandman, J., & Ruuth, S. J. (2011). Solving eigenvalue problems on curved surfaces using the Closest Point Method. Journal of Computational Physics, 230(22), 7944–7956. here
• ※ Macdonald, C. B., & Ruuth, S. J. (2008). Level set equations on surfaces via the Closest Point Method. Journal of Scientific Computing, 35(2–3), 219–240. here