Volumetric variational principles for a class of partial differential equations defined on surfaces and curves

Jay Chu, Richard Tsai

Published 2017-06-09Version 1

In this paper, we propose simple numerical algorithms for partial differential equations (PDEs) defined on closed, smooth surfaces (or curves) via discretization of the suitably extended versions of these equations in a thin tubular neighborhood around the surfaces (or curves), with some corresponding boundary conditions. In particular, we consider PDEs that originate from variational principles defined on the surfaces; these include Laplace-Beltrami equations and surface wave equations. The objective is to systematically formulate an extension, including the boundary conditions, that can be easily implemented on uniform Cartesian grids or adaptive meshes, on which the surfaces are defined implicitly by the distance functions or by the closest point mapping. As such extensions are not unique, we investigate how a class of simple extensions can influence the resulting PDEs. In particular, we reduce the surface PDEs to model problems defined on a periodic strip and the corresponding boundary conditions, and use classical Fourier and Laplace transform methods to study the well-posedness of the resulting problem. For elliptic and parabolic problems, our boundary closure in most yields stable algorithms to solve nonlinear surface PDEs. For hyperbolic problems, the proposed boundary closure is unstable in general, but the instability can be easily controlled by either adding a higher order regularization term or by periodically but infrequently "reinitializing" the computed solutions. Some numerical examples for each representative surface PDEs are presented.