Universal Meshes for Problems With Moving Boundaries

Evan Gawlik, Stanford University

Science and engineering are replete with instances of moving-boundary problems: partial differential equations posed on domains that change with time. Problems of this type, which arise in areas as diverse as fluid-structure interaction, multiphase flow physics, and fracture mechanics, are inherently challenging to solve numerically.

In this talk, I will present a new framework for the design of finite element methods for moving-boundary problems (with prescribed boundary evolution) that have an 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 suitable finite-element space on the universal mesh for the problem at hand, and a suitable time integrator for ordinary differential equations. We derive error estimates for the method in the context of a model parabolic problem, and we showcase the method on numerical examples in one and two dimensions.

Abstract Author(s): Evan Gawlik, Adrian Lew