Mapped Coercivity for the Stationary Navier–Stokes Equations and Their Finite Element Approximations

• Published in: Journal of computational methods in applied mathematics Vol. 25; no. 3; pp. 547 - 560
• Main Authors: Becker, Roland, Braack, Malte
• Format: Journal Article
• Language: English
• Published: Minsk De Gruyter 01.07.2025; Walter de Gruyter GmbH
• Subjects: 35J50; 35J66; 35Q35; 65N30; 76D05; Banach spaces; Coercivity; Computational fluid dynamics; Discretization; Finite element method; Finite Elements; Fixed points (mathematics); Fluid flow; Navier-Stokes equations; Nonlinear differential equations; Nonlinear equations; Nonlinear Operators; Nonlinear systems; Operators (mathematics); Parabolic differential equations; Partial differential equations; Saddle points; Saddle-Point Problems; Stabilization
• ISSN: 1609-4840; 1609-9389
• DOI: 10.1515/cmam-2024-0187

This paper addresses the challenge of proving the existence of solutions for nonlinear equations in Banach spaces, focusing on the Navier–Stokes equations and their discretizations. 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–Pitkäranta, 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.