Geometrically Higher Order Unfitted Space-Time Methods for PDEs on Moving Domains (2202.02216v2)

Abstract: In this paper, we propose new geometrically unfitted space-time Finite Element methods for partial differential equations posed on moving domains of higher order accuracy in space and time. As a model problem, the convection-diffusion problem on a moving domain is studied. For geometrically higher order accuracy, we apply a parametric mapping on a background space-time tensor-product mesh. Concerning discretisation in time, we consider discontinuous Galerkin, as well as related continuous (Petrov-)Galerkin and Galerkin collocation methods. For stabilisation with respect to bad cut configurations and as an extension mechanism that is required for the latter two schemes, a ghost penalty stabilisation is employed. The article puts an emphasis on the techniques that allow to achieve a robust but higher order geometry handling for smooth domains. We investigate the computational properties of the respective methods in a series of numerical experiments. These include studies in different dimensions for different polynomial degrees in space and time, validating the higher order accuracy in both variables.

• The paper introduces higher order unfitted space-time methods that decouple the mesh from moving domains to enhance accuracy and robustness.
• It employs ghost penalty stabilization and Petrov-Galerkin enhancements to address cut configuration challenges and reduce computational complexity.
• Numerical experiments demonstrate expected high convergence rates and superconvergence effects, validating the method's practical efficacy for complex flow problems.

An Analysis of the Geometrically Higher Order Unfitted Space-Time Methods for PDEs on Moving Domains

The paper presents a comprehensive paper of geometrically unfitted space-time finite element methods (FEMs) for solving partial differential equations (PDEs) on moving domains. The work focuses specifically on a convection-diffusion problem set within such domains, where traditional mesh-fitted methods like Arbitrary Lagrangian-Eulerian (ALE) methods often face computational challenges due to mesh adaptation requirements. By contrast, unfitted FEMs, such as CutFEM, circumvent these issues by decoupling the computational mesh from the domain geometry.

Higher Order Unfitted Space-Time Methods

The authors explore higher order geometrical accuracy by employing parametric mappings over a background space-time tensor-product mesh. They emphasize discontinuous Galerkin and continuous Galerkin formulations in time, alongside a Galerkin-collocation approach to increase solution regularity and reduce the number of variables in each time step. The paper fundamentally aims to increase the practical applicability of unfitted methods by enhancing their robustness against varied cut configurations in space-time and addressing potential computational complexity challenges, especially when greater accuracy is desired.

Stabilization and Computational Complexity

A notable contribution of this paper is the employment of ghost penalty stabilization techniques to handle the ill-posedness of cut configurations, essential for maintaining method stability and bounded condition numbers. The methodology also extends existing DG-in-time methods to higher order in both space and time. One challenge with space-time methods lies in their increased computational burden due to the high number of unknowns involved with higher order expansions in time. Nevertheless, the proposed extension to Petrov-Galerkin methods, which feature higher order time continuity, mitigates this burden significantly. Through these advancements, the authors achieve arbitrary high order accuracy in both space and time.

Numerical Experiments and Results

The numerical experiments establish the proposed methods' accuracy and efficiency. Throughout various scenarios, including dynamically changing domains such as moving circles and complex domains involving topology changes like colliding circles, the methods demonstrate expected higher order convergence rates. Remarkably, there is a superconvergence effect observed, particularly with DG and CG methods with increased time discretization orders, indicating an improved convergence in time at nodes. These results substantiate the methods' capabilities for practical complex flow problems, showcasing their robustness in handling domain movements without necessitating mesh reconstruction.

Future Directions and Implications

The paper's findings promote the practicality of unfitted FEMs in applications where domain geometries are complex or constantly changing. Beyond the demonstrated higher order accuracy, one of the implications of this work is the potential impact on simulating physical problems, from multiphase flows to bio-mechanical applications, which involve intricate temporal domain interactions.

Future research directions may include a numerical analysis to substantiate the observed superconvergence in these methods. Additionally, while the topology-preserving numerical integration strategically aids in handling geometrical consistency, the paper hints at the adequacy of simpler alternatives (e.g., topology-insensitive integration), potentially simplifying stabilizations while maintaining accuracy. Lastly, there is an invitation to explore continuous mesh deformations that could simplify method implementation and analysis.

Conclusion

Overall, this paper represents a significant step in addressing higher order accuracy and computational challenges within unfitted space-time FEMs for PDEs on moving domains. Through innovative stabilization and temporal discretization strategies, the authors set the groundwork for further advancements within this domain, emphasizing robustness and computational feasibility for a range of complex applications.