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Many geometry processing techniques require the solution of partialdifferential equations (PDEs) on surfaces. Such surface PDEs often involveboundary conditions prescribed on the surface, at points or curves on itsinterior or along the geometric (exterior) boundary of an open surface.However, input surfaces can take many forms (e.g., meshes, parametric surfaces,point clouds, level sets, neural implicits). One must therefore generate a meshto apply finite element-type techniques or derive specialized discretizationprocedures for each surface representation.  We propose instead to address such problems through a novel extension of theclosest point method (CPM) to handle interior boundary conditions specified atsurface points or curves. CPM solves the surface PDE by solving a volumetricPDE defined over the Cartesian embedding space containing the surface; only aclosest point function is required to represent the surface. As such, CPMsupports surfaces that are open or closed, orientable or not, and of anycodimension or even mixed-codimension. To enable support for interior boundaryconditions, we develop a method to implicitly partition the embedding spaceacross interior boundaries. CPM's finite difference and interpolation stencilsare adapted to respect this partition while preserving second-order accuracy.Furthermore, an efficient sparse-grid implementation and numerical solver isdeveloped that can scale to tens of millions of degrees of freedom, allowingPDEs to be solved on more complex surfaces. We demonstrate our method'sconvergence behaviour on selected model PDEs. Several geometry processingproblems are explored: diffusion curves on surfaces, geodesic distance, tangentvector field design, and harmonic map construction. Our proposed approach thusoffers a powerful and flexible new tool for a range of geometry processingtasks on general surface representations.

A Closest Point Method for Surface PDEs with Interior Boundary Conditions for Geometry Processing