Stability, Convergence, and Pressure-Robustness Analysis of Hybrid Numerical Schemes for Incompressible Flows

Core Concepts

This work provides a rigorous framework for analyzing the stability, convergence, and pressure-robustness of hybrid discretization methods for incompressible Stokes flows, where both the velocity and pressure are approximated using hybrid spaces.

Abstract

The paper focuses on the analysis of a hybrid discretization scheme for the Stokes problem, where the velocity and pressure are approximated using hybrid spaces. The key aspects of the analysis are: Identifying a set of assumptions on the discrete spaces and operators that guarantee (generalized) inf-sup stability of the pressure-velocity coupling. Deriving error estimates that distinguish the velocity- and pressure-related contributions to the error. Identifying the conditions under which the pressure-related contributions vanish in the estimate of the velocity, leading to pressure-robustness. Several examples of existing and new hybrid schemes are provided, and their theoretical properties are validated through extensive numerical experiments. The analysis starts by introducing the hybrid velocity and pressure spaces, along with the necessary interpolators and reconstructions of key quantities. An abstract hybrid discretization scheme is then formulated, and its stability and convergence are analyzed under the identified assumptions. The stability analysis shows that the scheme satisfies a uniform inf-sup condition, leading to a priori error estimates. The convergence analysis further distinguishes the velocity and pressure contributions to the error, and identifies the conditions for pressure-robustness, where the velocity error does not depend on the pressure. The framework is then applied to analyze the classical Botti-Massa scheme and the Rhebergen-Wells method, as well as to derive new pressure-robust schemes on standard and polyhedral meshes.

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Key Insights Distilled From

Stability, convergence, and pressure-robustness of numerical schemes for incompressible flows with hybrid velocity and pressure

by Lorenzo Bott... at arxiv.org 04-22-2024

https://arxiv.org/pdf/2404.12732.pdf

Deeper Inquiries

How can the proposed framework be extended to analyze hybrid discretization methods for more complex fluid flow problems, such as the Navier-Stokes equations or non-Newtonian fluids

The proposed framework can be extended to analyze hybrid discretization methods for more complex fluid flow problems by adapting the existing methodology to accommodate the additional complexities of the Navier-Stokes equations or non-Newtonian fluids. For the Navier-Stokes equations, the key modifications would involve incorporating the convective terms and the nonlinear nature of the equations into the discretization scheme. This may require redefining the velocity-pressure coupling, revisiting the stability and error analysis to account for the new terms, and potentially introducing additional stabilization techniques to handle the increased complexity of the equations. When dealing with non-Newtonian fluids, the framework would need to be adjusted to capture the specific rheological behavior of the fluid. This could involve modifying the viscosity term to account for non-Newtonian effects, adapting the pressure-velocity coupling to handle the non-linear constitutive equations governing the fluid behavior, and ensuring that the discretization scheme remains stable and convergent in the presence of these additional complexities. Overall, extending the framework to analyze more complex fluid flow problems would require a careful consideration of the specific characteristics of the equations governing the flow, and a systematic approach to incorporating these features into the existing hybrid discretization methodology.

What are the potential trade-offs between the pressure-robustness property and other desirable features, such as optimal convergence rates or computational efficiency, when designing hybrid discretization schemes

The pressure-robustness property in hybrid discretization schemes is a desirable feature as it ensures that the error estimates for the velocity component remain accurate and independent of the pressure approximation. However, achieving pressure-robustness may come with certain trade-offs when compared to other desirable features such as optimal convergence rates or computational efficiency. One potential trade-off is the complexity of the discretization scheme. Introducing mechanisms to maintain pressure-robustness, such as specialized stabilization techniques or additional constraints on the pressure approximation, can increase the overall complexity of the numerical method. This complexity may impact the implementation, computational cost, and scalability of the scheme. Another trade-off could be in terms of computational efficiency. Ensuring pressure-robustness may require additional computational resources, such as increased memory usage or higher computational time, especially if the pressure-robustness mechanisms involve solving additional equations or introducing more sophisticated numerical techniques. Furthermore, there may be trade-offs in terms of convergence rates. While pressure-robust schemes aim to provide accurate results for the velocity component, the trade-off could be in the overall convergence behavior of the method. Balancing pressure-robustness with achieving optimal convergence rates for the entire system of equations can be a delicate balance that requires careful consideration and possibly compromises in one aspect to enhance the other. In summary, while pressure-robustness is a valuable property in hybrid discretization schemes, achieving it may involve trade-offs in terms of complexity, computational efficiency, and convergence rates that need to be carefully evaluated and balanced based on the specific requirements of the problem at hand.

Can the insights from this work on hybrid velocity-pressure approximations be leveraged to develop novel numerical methods for other types of partial differential equations, beyond fluid mechanics

The insights gained from the analysis of hybrid velocity-pressure approximations in the context of incompressible flows can indeed be leveraged to develop novel numerical methods for other types of partial differential equations beyond fluid mechanics. The key principles and techniques, such as the pressure-robustness property, stability analysis, and error estimates, can be applied to a wide range of PDEs in various fields of science and engineering. For example, in structural mechanics, the concept of hybrid approximations could be utilized to develop efficient and accurate numerical methods for problems involving solid mechanics, elasticity, or structural dynamics. By adapting the framework to account for the specific characteristics of these problems, such as material properties, boundary conditions, and loading conditions, novel hybrid discretization schemes could be designed to provide reliable solutions with improved computational efficiency and accuracy. Similarly, in electromagnetics or heat transfer, the principles of hybrid discretization could be applied to develop numerical methods for solving Maxwell's equations or the heat conduction equation. By incorporating the appropriate field equations, boundary conditions, and material properties into the framework, researchers can create innovative numerical techniques that offer robustness, accuracy, and efficiency in simulating electromagnetic or thermal phenomena. Overall, the versatility and applicability of the insights from hybrid velocity-pressure approximations make them valuable tools for developing novel numerical methods for a wide range of partial differential equations in diverse scientific and engineering disciplines.