High-Order Finite Element Methods for Moving Boundary Problems with Prescribed Boundary Evolution

TL;DR

This paper presents a high-order finite element framework using a universal mesh for moving boundary problems, ensuring accurate boundary representation and robustness under large deformations.

Contribution

The authors develop a general high-order finite element method with a universal mesh approach for moving boundary problems, maintaining accuracy and mesh quality.

Findings

Achieves arbitrary high order accuracy in space and time.

Maintains exact or near-exact boundary representation.

Demonstrates effectiveness on Stefan problem in 1D and 2D.

Abstract

We introduce a framework for the design of finite element methods for two-dimensional moving boundary problems with prescribed boundary evolution that have arbitrarily high order of accuracy, both in space and in time. At the core of our approach is the use of a universal mesh: a stationary background mesh containing the domain of interest for all times that adapts to the geometry of the immersed domain by adjusting a small number of mesh elements in the neighborhood of the moving boundary. The resulting method maintains an exact representation of the (prescribed) moving boundary at the discrete level, or an approximation of the appropriate order, yet is immune to large distortions of the mesh under large deformations of the domain. The framework is general, making it possible to achieve any desired order of accuracy in space and time by selecting a preferred and suitable finite-element space on the universal mesh for the problem at hand, and a preferred and suitable time integrator for ordinary differential equations. We illustrate our approach by constructing a particular class of methods, and apply them to a prescribed-boundary variant of the Stefan problem. We present numerical evidence for the order of accuracy of our schemes in one and two dimensions.