标题： 静止纳维叶-斯托克斯方程及其有限元逼近的映射各向异性 显示英文标题

标题： Mapped coercivity for the stationary Navier-Stokes equations and their finite element approximations

摘要： 本文解决了在巴拿赫空间中证明非线性方程解的存在性的挑战，重点研究了纳维-斯托克斯方程和thom的离散化方法。 传统方法，如基于单调性的方法和不动点定理，在处理一般的非线性算子或有限元离散化时常常面临限制。 一种新的概念，映射强制性，提供了一个统一的框架，通过连续映射来分析非线性算子。 我们将这些思想应用于巴拿赫空间中的鞍点问题，强调无限维公式和有限元离散化。 我们的分析包括稳定化技术，以在有限维设置中恢复强制性，确保稳定性与解的存在性。 对于线性问题，我们探讨了inf-sup条件与映射强制性之间的关系，以斯托克斯方程为例进行研究。 对于非线性鞍点系统，我们通过满射映射将框架扩展到映射强制性，从而为各种稳定化纳维-斯托克斯有限元方法提供了简洁的解存在性证明。 这些方法包括Brezzi-Pitkäranta方法、一个简单变体以及局部投影稳定化（LPS）技术，并扩展到对流主导流动。 所提出的方法为分析非线性偏微分方程及其离散化提供了一个强大的工具，绕过了传统的分解方法，并为计算流体力学的未来发展奠定了基础。 显示英文摘要

摘要： This paper addresses the challenge of proving the existence of solutions for nonlinear equations in Banach spaces, focusing on the Navier-Stokes equations and discretizations of thom. Traditional methods, such as monotonicity-based approaches and fixed-point theorems, often face limitations in handling general nonlinear operators or finite element discretizations. A novel concept, mapped coercivity, provides a unifying framework to analyze nonlinear operators through a continuous mapping. We apply these ideas to saddle-point problems in Banach spaces, emphasizing both infinite-dimensional formulations and finite element discretizations. Our analysis includes stabilization techniques to restore coercivity in finite-dimensional settings, ensuring stability and existence of solutions. For linear problems, we explore the relationship between the inf-sup condition and mapped coercivity, using the Stokes equation as a case study. For nonlinear saddle-point systems, we extend the framework to mapped coercivity via surjective mappings, enabling concise proofs of existence of solutions for various stabilized Navier-Stokes finite element methods. These include Brezzi-Pitk\"aranta, a simple variant, and local projection stabilization (LPS) techniques, with extensions to convection-dominant flows. The proposed methodology offers a robust tool for analyzing nonlinear PDEs and their discretizations, bypassing traditional decompositions and providing a foundation for future developments in computational fluid dynamics.