Analysis of the L² least-squares finite element method for the velocity-vorticity-pressure Stokes equations with velocity boundary conditions

A theoretical analysis of the L² least-squares finite element method (LSFEM) for solving the Stokes equations in the velocity-vorticity-pressure (VVP) first-order system formulation with the Dirichlet velocity boundary condition is given. The least-squares energy functional is defined to be the sum of the squared L²-norms of the residuals in the partial differential equations, weighted appropriately by the viscosity constant ν. It is shown that, with many advantages, the method is stable and convergent without requiring extra smoothness of the exact solution, and the piecewise linear finite elements can be used to approximate all the unknowns. Furthermore, with respect to the order of approximation for smooth exact solutions, the method exhibits an optimal rate of convergence in the H¹-norm for velocity and a suboptimal rate of convergence in the L²-norm for vorticity and pressure. Some numerical experiments in two and three dimensions are given, which confirm the a priori error estimates. Since the boundary of the bounded domain under consideration is polygonal in ℛ² or polyhedral in ℛ³ instead of C¹-smooth, the authors adopt the more direct technique of Bramble-Pasciak and Cai-Manteuffel-McCormick, which departs from the Agmon-Douglis-Nirenberg theory, in showing the coercivity bound of the least-squares functional.