An SPD stabilized finite element method for the Stokes equations

A new residual-based stabilized finite element method is analysed for solving the Stokes equations in terms of velocity and pressure, where the $H^{-1}$ norm is introduced in the measurement of the residuals to obtain a symmetric positive-definite method. The $H^{-1}$ norm is computable and can always be easily realized offline by the continuous linear finite element solution or the preconditioner counterpart of the Poisson Dirichlet problem. Although the $H^{-1}$ norm is computed in the linear element space, no matter what the finite element spaces for the velocity and the pressure are, optimal error bounds can be established when using continuous finite element pairs $R_l-R_m$ for velocity and pressure for any $l,m\ge 1$. Numerical experiments are performed to confirm the theoretical results obtained.