High-order Finite element Methods for Moving Boundary Problems

Evan Gawlik, Stanford University

We develop a framework for the design of finite element methods for moving boundary problems with arbitrarily high order of accuracy, both in space and in time. At the core of our approach is the use of a universal background mesh containing the domain of interest, which is adjusted in such a way that the moving boundary coincides with a union of curved element interfaces at any instant in time. The motion of these element interfaces over a given time step introduces convective terms to the governing equations that can be understood from an Arbitrary Lagrangian Eulerian (ALE) standpoint. These governing equations can be spatially discretized with a finite element method of one's choosing, and the resulting semidiscrete equations can then be integrated with any desired numerical integration scheme. The framework so described stands independently of the numerical representation of the boundary, be it with splines, level sets or the like. To evolve the numerical representation of the boundary, we propose a framework for approximating dynamics on the space of planar curves, where the relations governing the boundary velocity are enforced weakly within a subspace of the full tangent space to this manifold. We illustrate our approach through an application to the Stefan problem, a moving-boundary problem for which the aim is to predict the evolution of a solid-liquid interface during a melting process. We present numerical evidence for the order of accuracy of our schemes in one and two dimensions, both with prescribed boundary evolution and with the boundary treated as an unknown.

Abstract Author(s): Evan Gawlik, Ramsharan Rangarajan, Adrian Lew