A fourth-order unfitted characteristic finite element method for solving the advection-diffusion equation on time-varying domains

TL;DR

This paper introduces a high-order unfitted characteristic finite element method for accurately solving advection-diffusion equations on complex, moving domains, combining advanced interface tracking, temporal, and spatial discretization techniques.

Contribution

It presents a novel fourth-order method integrating interface tracking, temporal, and spatial discretization for advection-diffusion problems on dynamic domains.

Findings

Achieves optimal convergence rates in numerical experiments.

Effectively handles complex, deforming domains.

Provides comprehensive error analysis including boundary representation.

Abstract

We propose a fourth-order unfitted characteristic finite element method to solve the advection-diffusion equation on time-varying domains. Based on a characteristic-Galerkin formulation, our method combines the cubic MARS method for interface tracking, the fourth-order backward differentiation formula for temporal integration, and an unfitted finite element method for spatial discretization. Our convergence analysis includes errors of discretely representing the moving boundary, tracing boundary markers, and the spatial discretization and the temporal integration of the governing equation. Numerical experiments are performed on a rotating domain and a severely deformed domain to verify our theoretical results and to demonstrate the optimal convergence of the proposed method.