Alternative to evolving surface finite element method
This provides an alternative numerical method for solving PDEs on evolving surfaces, potentially simplifying implementation by avoiding moving grids, though it is incremental as it transforms an existing method.
The authors propose an alternative to the evolving surface finite element method (ESFEM) by transforming the advection-diffusion equation on an evolving surface to an equivalent equation on a fixed initial surface, allowing solution on a fixed grid. Numerical examples show comparable accuracy between the two approaches.
ESFEM is a method introduced in order to solve a linear advection-diffusion equation on an evolving two-dimensional surface with finite elements by using a moving grid with nodes sitting on and evolving with the surface. The evolution of the surface is assumed to be given as a smooth one-parameter family of embeddings of a fixed initial surface into $\mathbb{R}^3$ satisfying uniform $C^4$ bounds. We calculate an equivalent transformed equation which is defined on the fixed initial surface and can hence be solved numerically on a fixed grid. We present numerical examples which indicate that both approaches are essentially of the same accuracy.