Numerical Methods for Incompressible Viscous Flows: A Comparative Review of Boundary Element, Domain Discretization, and Meshfree Approaches

The numerical simulation of incompressible viscous flows remains a central pillar of modern computational fluid dynamics (CFD). Over the past decades, a wide spectrum of numerical methodologies has been developed, reflecting fundamentally different mathematical formulations and discretization philosophies. Among these, domain-based approaches—such as finite difference, finite element, finite volume, and meshfree methods—have emerged as versatile and general-purpose frameworks, while boundary element methods provide efficient alternatives for problems governed by linear physics, particularly in unbounded domains. This review presents a comprehensive examination of the historical development, mathematical foundations, and computational characteristics of these approaches for Newtonian incompressible flows. Emphasis is placed on the conceptual distinctions between boundary-integral and domain-based formulations, their applicability to internal and external flow regimes, and their compatibility with turbulence modeling strategies, including Reynolds-averaged Navier–Stokes (RANS), large-eddy simulation (LES), and direct numerical simulation (DNS). The intention is to provide a unified perspective that clarifies the strengths and limitations of the principal CFD methodologies and offers guidance on their suitability for different classes of flow problems.