Analysis of an implicitly extended Crank-Nicolson scheme for the heat equation on a time-dependent domain

TL;DR

This paper analyzes a Crank-Nicolson type scheme for the heat equation on moving domains, employing ghost-penalty extensions and cut finite elements, demonstrating second-order convergence and validating results with numerical experiments.

Contribution

It introduces an implicit ghost-penalty extension for the Crank-Nicolson scheme on moving domains combined with cut finite elements, with comprehensive error analysis.

Findings

Second-order convergence in time under CFL condition

Numerical validation in 2D and 3D confirms theoretical estimates

Effective handling of moving domains with ghost-penalty extensions

Abstract

We consider a time-stepping scheme of Crank-Nicolson type for the heat equation on a moving domain in Eulerian coordinates. As the spatial domain varies between subsequent time steps, an extension of the solution from the previous time step is required. Following Lehrenfeld \& Olskanskii [ESAIM: M2AN, 53(2):\,585-614, 2019], we apply an implicit extension based on so-called ghost-penalty terms. For spatial discretisation, a cut finite element method is used. We derive a complete a priori error analysis in space and time, which shows in particular second-order convergence in time under a parabolic CFL condition. Finally, we present numerical results in two and three space dimensions that confirm the analytical estimates, even for much larger time steps.