A predictor-corrector scheme for approximating signed distances using finite element methods

In this article, we introduce a finite element method designed for the robust computation of approximate signed distance functions to arbitrary boundaries in two and three dimensions. Our method employs a novel prediction-correction approach, involving first the solution of a linear diffusion-based prediction problem, followed by a nonlinear minimization-based correction problem associated with the eikonal equation. The prediction step efficiently generates a suitable initial guess, significantly facilitating convergence of the nonlinear correction step. We validate the robustness of our approach using complex initial level set functions featuring arbitrary steep or flat regions in both two and three dimensions, as well as transient test cases like the vortex test. We compare our results with a standard finite difference Hamilton-Jacobi scheme, demonstrating improved interface preservation. Furthermore, we illustrate the method’s scalability for large-scale 3D problems using distributed memory parallelism (MPI), establishing its accuracy, efficiency, and broad applicability for reinitializing diverse level set functions.