Application of the gradient projection method for numerical solving the three-dimensional Stokes problem

An efficient numerical approach is developed to solve the three-dimensional Stokes problem in natural variables by the finite element method combined with the gradient projection method. It has the advantages of both the vector potential-vorticity method and the pressure-velocity method. The incompressibility condition is fulfilled with a high degree of accuracy, and there is no difficulty in solving free boundary problems. The approach allows us to reduce sufficiently the dimension of the finite element problem and preserve the sparse structure of the stiffness matrix. Numerical results obtained for free boundary problems confirm the advantages of the proposed procedure.