Least-Squares Finite Element Method for the Stokes Problem with Zero Residual of Mass Conservation

In this paper the simulation of incompressible flow in two dimensions by the least-squares finite element method (LSFEM) in the vorticity-velocity-pressure version is studied. In the LSFEM, the equations for continuity of mass and momentum and a vorticity equation are minimized on a discretization of the domain of interest. A problem is these equations are minimized in a global sense. Thus this method may not enforce that$div\underline{u} = 0$at every point of the discretization. In this paper a modified LSFEM is developed which enforces near zero residual of mass conservation, i.e.,$\operatorname{div}\overline{u}$is nearly zero at every point of the discretization. This is accomplished by adding an extra restriction in the divergence-free equation through the Lagrange multiplier strategy. In this numerical method the inf-sup or say LBB condition is not necessary, and the matrix resulting from applying the method on a discretization is symmetric; the uniqueness of the solution and the application of the conjugate gradient method are also valid. Numerical experience is given in simulating the flow of a cylinder with diameter 1 moving in a narrow channel of width 1.5. Results obtained by the LSFEM show that mass is created or destroyed at different points in the interior of discretization. The results obtained by the modified LSFEM show the mass is nearly conserved everywhere.