A Conservative High-Order Cut Finite Element Method for Convection-Diffusion Equations in Time-Dependent Domains

Core Concepts

The proposed space-time cut finite element method achieves mass conservation naturally by utilizing Reynolds' transport theorem, while attaining high-order convergence and controlling the condition number of the system matrix.

Abstract

The authors present a mass-conservative high-order unfitted finite element method for solving convection-diffusion equations in evolving domains. The key highlights are:

The method extends the space-time cut finite element method (CutFEM) proposed in previous work to naturally achieve mass conservation by using Reynolds' transport theorem.

A more efficient stabilization procedure for the cut finite element method in time-dependent domains is introduced by partitioning the time-dependent domain into macroelements. This macroelement stabilization controls the condition number of the system matrix and increases the sparsity compared to full stabilization.

Numerical experiments demonstrate that the method fulfills mass conservation, achieves high-order convergence, and the condition number of the resulting system matrix is controlled.

The method is applied to both bulk domain problems as well as coupled bulk-surface problems involving surfactants.

Stats

The change of mass in the domain is equal to the amount of mass produced inside the domain, as shown in equation (7).
The proposed conservative scheme satisfies a discrete mass conservation equation, as shown in equation (25).
The global conservation error 𝑒𝑐(𝑇) is of the order of machine epsilon for any polynomial order with the conservative scheme, while this is not the case for the non-conservative solution.

Quotes

"The strength of the cut finite element method in [11, 12] is its simple implementation, which is made possible by the ghost-penalty stabilization added in the method."
"Recently, a strategy on how to apply ghost-penalty only where necessary was developed for stationary domains in [23]. The idea is to construct a macroelement partition of the mesh, where macroelements have a large intersection with the domain of interest, and therefore stabilization is only needed on the internal faces."
"We are not aware of any other unfitted discretization that achieves both mass conservation and is higher order than second order."

Key Insights Distilled From

A High-Order Conservative Cut Finite Element Method for Problems in Time-Dependent Domains

by Seba... at arxiv.org 04-17-2024

https://arxiv.org/pdf/2404.10756.pdf

Deeper Inquiries

How can the proposed macroelement stabilization technique be extended to other types of time-dependent problems beyond convection-diffusion equations

The proposed macroelement stabilization technique can be extended to other types of time-dependent problems by adapting the concept of partitioning the domain into macroelements and applying stabilization only where necessary. This approach can be applied to problems involving advection-diffusion equations, reaction-diffusion systems, and even more complex systems like fluid-structure interaction or multiphysics problems. By constructing macroelements based on the evolving domain's characteristics and applying stabilization selectively within these macroelements, the method can ensure mass conservation and control the condition number of the system matrix. This extension allows for the efficient and accurate numerical solution of a wide range of time-dependent PDEs in evolving domains.

What are the potential challenges in applying the conservative cut finite element method to more complex real-world problems with irregular domain geometries and heterogeneous material properties

When applying the conservative cut finite element method to more complex real-world problems with irregular domain geometries and heterogeneous material properties, several challenges may arise. One challenge is the accurate representation of the evolving boundaries and interfaces in the numerical discretization. Irregular geometries may require adaptive mesh refinement strategies to capture fine details and ensure accurate solutions. Heterogeneous material properties can lead to discontinuities in the solution, requiring careful handling to maintain stability and accuracy. Additionally, the computational cost of implementing the method for complex problems may increase, requiring efficient algorithms and parallel computing techniques to handle the computational load effectively.

Can the ideas presented in this work be adapted to develop conservative high-order methods for other types of partial differential equations in evolving domains, such as Navier-Stokes equations or phase-field models

The ideas presented in this work can be adapted to develop conservative high-order methods for various types of partial differential equations in evolving domains, including Navier-Stokes equations and phase-field models. For Navier-Stokes equations, the conservative cut finite element method can be extended to handle the convection and diffusion terms in the momentum and continuity equations while ensuring mass conservation and stability. Similarly, for phase-field models describing phase transitions, the method can be modified to conserve the order parameter and energy while accurately capturing the evolving interfaces between different phases. By incorporating the principles of mass conservation and selective stabilization, high-order conservative methods can be developed for a wide range of time-dependent PDEs in evolving domains.