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We propose and analyze a class of meshfree, super-algebraically convergentmethods for partial differential equations (PDEs) on surfaces using Fourierextensions minimizing a measure of non-smoothness (such as a Sobolev norm).Current spectral methods for surface PDEs are primarily limited to a smallclass of surfaces, such as subdomains of spheres. Other high order methods forsurface PDEs typically use radial basis functions (RBFs). Many of these methodsare not well-understood analytically for surface PDEs and are highlyill-conditioned. Our methods work by extending a surface PDE into a box-shapeddomain so that differential operators of the extended function agree with thesurface differential operators, as in the Closest Point Method. The methods canbe proven to converge super-algebraically for certain well-posed linear PDEs,and spectral convergence to machine error has been observed numerically for avariety of problems. Our approach works on arbitrary smooth surfaces (closed ornon-closed) defined by point clouds with minimal conditions.

Underdetermined Fourier Extensions for Partial Differential Equations on Surfaces