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The Closest Point method, initially developed by Ruuth and Merriman, allowsfor the numerical solution of surface partial differential equations withoutthe need for a parameterization of the surface itself. Surface quantities areembedded into the surrounding domain by assigning each value at a given spatiallocation to the corresponding value at the closest point on the surface. Thisembedding allows for surface derivatives to be replaced by their Cartesiancounterparts (e.g. $\nabla_s = \nabla$). This equivalence is only valid on thesurface, and thus, interpolation is used to enforce what is known as the sidecondition away from the surface. To improve upon the method, this work derivesan operator embedding that incorporates curvature information, making it validin a neighborhood of the surface. With this, direct enforcement of the sidecondition is no longer needed. Comparisons in $\mathbb{R}^2$ and $\mathbb{R}^3$show that the resulting Curvature-Augmented Closest Point method has betteraccuracy and requires less memory, through increased matrix sparsity, than theClosest Point method, while maintaining similar matrix condition numbers. Todemonstrate the utility of the method in a physical application, simulations ofinextensible, bi-lipid vesicles evolving toward equilibrium shapes are alsoincluded.

The Curvature-Augmented Closest Point Method with Vesicle Inextensibility Application