MathSciDoc: An Archive for Mathematician ∫

A new residual-based stabilized finite element method is analyzed 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 (SPD) method. The $H^{−1}$ norm is computable and can be always easily realized offline by the continuous linear finite element solution or the preconditioned 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.

Stokes equations; stabilized finite element method; H^{−1} norm; symmetric positive definiteness; linear finite element solution of Poisson Dirichlet problem