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

翻译：本文提出并分析了一类无网格、超代数收敛的曲面偏微分方程（PDE）数值方法，该方法通过最小化非光滑性度量（如Sobolev范数）的Fourier延拓实现。现有曲面PDE谱方法主要局限于球面子域等少数曲面类型，而其他高阶方法通常采用径向基函数（RBF）。这些方法在曲面PDE的解析分析方面存在不足，且高度病态。我们的方法通过将曲面PDE延拓至矩形域，使得延拓函数的微分算子与曲面微分算子一致（类似最近点方法）。对于适定的线性PDE，可证明该方法超代数收敛，且大量数值实验表明谱精度可达机器误差水平。本方法适用于由点云定义的任意光滑曲面（闭合或非闭合），仅需极低的条件要求。