A non-conforming least-squares finite element method for incompressible fluid flow problems

In this paper, we develop least-squares finite element methods (LSFEMs) for incompressible fluid flows with improved mass conservation. Specifically, we formulate a new locally conservative LSFEM for the velocity-vorticity-pressure Stokes system, which uses a piecewise divergence-free basis for the velocity and standard C⁰ elements for the vorticity and the pressure. The new method, which we term dV-VP improves upon our previous discontinuous stream-function formulation in several ways. The use of a velocity basis, instead of a stream function, simplifies the imposition and implementation of the velocity boundary condition, and eliminates second-order terms from the least-squares functional. Moreover, the size of the resulting discrete problem is reduced because the piecewise solenoidal velocity element is approximately one-half of the dimension of a stream-function element of equal accuracy. In two dimensions, the discontinuous stream-function LSFEM [1] motivates modification of our functional, which further improves the conservation of mass. We briefly discuss the extension of this modification to three dimensions. Computational studies demonstrate that the new formulation achieves optimal convergence rates and yields high conservation of mass. We also propose a simple diagonal preconditioner for the dV-VP formulation, which significantly reduces the condition number of the LSFEM problem.