High-order finite element methods for nonlinear convection-diffusion equation on time-varying domain

TL;DR

This paper introduces a high-order finite element method for solving nonlinear convection-diffusion equations on dynamically changing domains, achieving optimal convergence in complex, deforming geometries.

Contribution

It develops a semi-implicit high-order finite element approach combining surface tracking and backward differentiation for nonlinear, time-varying domain problems.

Findings

Optimal convergence orders achieved in energy norm for third- and fourth-order methods.

Method effectively handles severely deforming domains.

Numerical experiments validate high accuracy and stability.

Abstract

A high-order finite element method is proposed to solve the nonlinear convection-diffusion equation on a time-varying domain whose boundary is implicitly driven by the solution of the equation. The method is semi-implicit in the sense that the boundary is traced explicitly with a high-order surface-tracking algorithm, while the convection-diffusion equation is solved implicitly with high-order backward differentiation formulas and fictitious-domain finite element methods. By two numerical experiments for severely deforming domains, we show that optimal convergence orders are obtained in energy norm for third-order and fourth-order methods.