Eulerian finite element method for parabolic PDEs on implicit surfaces

Abstract

We define an Eulerian level set method for parabolic partial differential equations on a stationary hypersurface $\Gamma$ contained in a domain $\Omega \subset {\mathbb{R}}^{n+1}$. The method is based on formulating the partial differential equations on all level surfaces of a prescribed function $\Phi$ whose zero level set is $\Gamma$. Eulerian surface gradients are formulated by using a projection of the gradient in ${\mathbb{R}}^{n+1}$ onto the level surfaces of $\Phi$. These Eulerian surface gradients are used to define weak forms of surface elliptic operators and so generate weak formulations of surface elliptic and parabolic equations. The resulting equation is then solved in one dimension higher but can be solved on a mesh which is unaligned to the level sets of $\Phi$. We consider both second order and fourth order elliptic operators with natural second order splittings. The finite element method is applied to the weak form of the split system of second order equations using piece-wise linear elements on a fixed grid. The computation of the mass and element stiffness matrices are simple and straightforward. Numerical experiments are described which indicate the power of the method. We describe how this framework may be employed in applications.