Advancements in Divergence-Free Approximations for Incompressible Flow

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Accurate computations of fluid flow models have a direct impact on simulations and predictions in various applications, e.g., weather and climate, aircraft design, etc. Typically, such computations are based on discretizations of partial differential equations that model the physics of the underlying system. These approximations introduce errors into the simulations that need to be rigorously quantified in order to make reliable simulations and predictions. This project will construct accurate, structure-preserving computational schemes that exactly enforce the underlying physical laws at the discrete level in fluid flow models. This attribute leads to high fidelity schemes that are robust with respect to several model parameters. A particular focus of the project is to develop and analyze computational algorithms that reduce geometric error and thus yield provably accurate answers. In addition, the project aims to provide user-friendly methods; in particular, the project will incorporate modules into current computational and open source software to improve usability, portability, and outreach.This project will construct finite element methods for the Stokes and Navier-Stokes equations that exactly enforce the divergence-free constraint at the discrete level. Such discretizations have several desirable properties, for example, improved stability and error estimates with respect to model parameters, exact conservation properties for any discretization parameter, and characterizations of discrete divergence-free subspaces. However, current divergence-free finite element methods are impractically high-order and arduous to implement. This project will identify and construct simple divergence-free finite element methods with an emphasis on the three-dimensional setting and on methods that can be incorporated into current finite element software. Divergence-free finite element methods on domains with curved boundaries that are robust with respect to both model parameters and the geometry will also be developed and analyzed. This will consist of studying both isoparametric divergence-free finite element methods and fictitious domain approaches.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.