Isogeometric Compatible Discretizations for Viscous Incompressible Flow

In this chapter, isogeometric discretizations for viscous incompressible flow are presented that satisfy the incompressibility constraint in a pointwise manner. As incompressibility is satisfied pointwise, these discretizations replicate the geometric structure of the Navier-Stokes equations and properly balance energy, enstrophy, and helicity. The result is a method with enhanced accuracy and robustness as compared with classical finite element methods for incompressible flow. Within the chapter, we review the geometric structure of the Navier-Stokes equations, outline the construction of compatible B-spline spaces which allow for pointwise mass conservation, and present a suite of illustrative numerical results demonstrating the potential of compatible B-splines in computational fluid dynamics.

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