Finite Element Methods for Moving-Boundary Problems: High-Order Methods and Analysis
Evan Gawlik, Stanford University
We develop a framework for the design of finite element methods for 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, yet is immune to large distortions of the mesh under large deformations of the domain. The framework is general, allowing one to achieve any desired order of accuracy in space and time by selecting a suitable finite-element space on the universal mesh and a suitable time integrator for ordinary differential equations.
In the process of deriving our method, we present a unified, geometric framework that puts our method and conventional deforming-mesh methods on a common footing suitable for analysis. The main idea is to recast the governing equations on a sequence of cylindrical spacetime slabs that span short intervals of time. The clarity brought about by this geometric viewpoint renders the analysis of numerical methods for moving-boundary problems more tractable, as it reduces the task to a standard analysis of fixed-domain problems with time-dependent PDE coefficients. Using the analytical framework so described, we prove a general error estimate for a class of finite-element methods for moving boundary problems that includes our universal-meshing method and existing deforming-mesh methods as special cases. We verify the aforementioned error estimates with several numerical examples in one and two dimensions.
Abstract Author(s): Evan Gawlik, Adrian Lew