Finite Element Methods for Incompressible Flow Yielding Divergence-Free Approximations

University Of Pittsburgh, Pittsburgh PA

Investigators

Abstract

This research project aims to develop practical, structure-preserving computational methods to simulate incompressible flow. Systems with incompressible flow are ubiquitous in computational fluid dynamics (CFD) and arise in several engineering and biological models such as those used in aircraft design, weather prediction, and lipid membrane modeling. The main focus of the project is to identify compatible computational methods that inherit at the discrete level the intrinsic structure and invariants of the systems under study. Such discretizations produce numerical schemes with enhanced stability and conservation properties, resulting in physically relevant and accurate approximations. It is anticipated that the results of the project will directly benefit researchers and practitioners in CFD and be useful in applications that require faithful approximations. In addition to the construction of these numerical schemes and algorithms, the project will analyze the stability and convergence of the methods; the theoretical analysis will provide insight for future development of numerical schemes with improved efficiency, accuracy, and fidelity. The project is centered on developing finite element methods for the incompressible Stokes and Navier-Stokes equations that enforce the divergence-free constraint exactly. Such schemes have several desirable properties, including improved error estimates, enhanced long-time stability and accuracy of time-stepping schemes, explicit characterizations and local bases of divergence-free subspaces, and coupled-method accuracy when combined with projection methods. Specific objectives of this project include (i) developing stable finite element pairs for Navier-Stokes equations in three dimensions that strongly enforce exact conservation of mass; (ii) applying simplicial Bernstein-Bezier theory, a powerful analytical tool for polynomial splines, to the mixed finite element framework; (iii) constructing robust and computationally attractive methods for axisymmetric fluid models; and (iv) developing stable mixed finite element pairs on surfaces by incorporating fluid flow models in a finite element exterior calculus framework.