A Conservative Eulerian Finite Element Method for Scalar Transport and Diffusion in Moving Domains

Core Concepts

The paper introduces a conservative Eulerian finite element method for the transport and diffusion of a scalar quantity in a time-dependent domain. The method is based on a reformulation of the partial differential equation to derive a scheme that conserves the quantity under consideration exactly on the discrete level.

Abstract

The paper presents a finite element method for an Eulerian formulation of partial differential equations governing the transport and diffusion of a scalar quantity in a time-dependent domain. The key aspects are:

The method follows the idea of a solution extension to realize the Eulerian time-stepping scheme, but introduces a reformulation of the PDE to derive a scheme that conserves the scalar quantity exactly on the discrete level.

For the spatial discretization, an unfitted finite element method (CutFEM) is considered, using ghost-penalty stabilization to handle arbitrary intersections between the mesh and geometry interface.

The stability of both first-order (BDF1) and second-order (BDF2) backward differentiation formula versions of the scheme is analyzed.

Numerical examples in 2D and 3D are provided to illustrate the potential of the method, including convergence studies demonstrating optimal convergence rates.

The key innovation is the reformulation of the PDE problem using an identity derived from the Reynolds transport theorem, which allows the method to conserve the total mass of the scalar variable exactly on the discrete level, unlike previous Eulerian approaches.

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Key Insights Distilled From

A conservative Eulerian finite element method for transport and diffusion in moving domains

by Maxim Olshan... at arxiv.org 04-11-2024

https://arxiv.org/pdf/2404.07130.pdf

Deeper Inquiries

How would the method perform for more complex moving geometries, such as those involving topology changes

The method may face challenges when dealing with more complex moving geometries that involve topology changes. In such cases, the extension of the solution into the active domain may become more intricate. The accuracy of the geometry approximation, especially in regions where the topology changes occur, could significantly impact the performance of the method. Ensuring the stability and conservation properties of the scheme in the presence of evolving topologies would be crucial. Additionally, handling the interactions between the moving geometry and the mesh interface during topology changes could pose significant computational challenges.

What are the potential challenges in extending this conservative Eulerian approach to other types of PDEs, such as the Navier-Stokes equations

Extending the conservative Eulerian approach to other types of PDEs, such as the Navier-Stokes equations, would introduce additional complexities and challenges. The Navier-Stokes equations involve coupled nonlinear terms and require more sophisticated numerical methods for accurate solutions. Adapting the conservative Eulerian method to handle the nonlinear convective terms, pressure terms, and the coupling between velocity and pressure fields in the Navier-Stokes equations would require careful consideration. Ensuring stability, conservation, and accuracy while dealing with the complex dynamics of fluid flow would be a significant challenge. Additionally, the computational cost of solving the Navier-Stokes equations on moving domains could be substantial, requiring efficient algorithms and high-performance computing resources.

The paper mentions that deriving error estimates for the proposed scheme is not obvious, even for the BDF1 case. What are the key difficulties in establishing rigorous error bounds for this type of Eulerian finite element method on moving domains

Establishing rigorous error bounds for Eulerian finite element methods on moving domains, such as the proposed scheme, presents several key difficulties. One major challenge is the evaluation of the error consistency in the presence of moving geometries and evolving domains. The interaction between the moving domain, the mesh interface, and the numerical solution introduces additional sources of error that need to be carefully analyzed. The complexity of the geometry approximation, the stability of the numerical scheme, and the accuracy of the solution extension all contribute to the difficulty in deriving error estimates. Furthermore, the nonlinearity of the PDEs, such as the convective terms in transport equations, can introduce additional challenges in error analysis. Overall, addressing these difficulties requires a comprehensive understanding of the numerical method, the underlying PDEs, and the dynamics of the moving domains.