Method of moving frames to solve (an)isotropic diffusion equations on curved surfaces

As a continuing effort to develop the method of moving frames (MMF) ensuing (Chun in J Sci Comput 53(2):268-294, 2012), a novel MMF scheme is proposed to solve (an)isotropic diffusion equations on arbitrary curved surfaces. First we show that if the divergence of a vector is computed exactly on the surface, the mixed formulations expanded in the moving frames are equivalent to the Laplace-Beltrami operator. Otherwise, the divergence error dominates, but it can be made negligible by either way; the use of a higher order differentiation scheme more than the first order or the alignment of the moving frames. Moreover, the propagational property of the media along a specific direction, known as anisotropy, is represented by the rescaling of the moving frames, not by repetitive multiplications of the diffusivity tensor, without adding any schematic complexity nor deterioration of the accuracy and stability to the isotropic diffusion scheme. Convergence results for a spherical shell, an irregular surface, and a non-convex surface are displayed with several examples of modeling anisotropy on various curved surfaces. A computational simulation of atrial reentry is illustrated as an exemplary use of the MMF scheme for practical applications.