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Based upon full domain decomposition, local and parallel stabilized finite element methods for the stationary Stokes equations are proposed and analysed, where the quadratic equal-order finite elements are employed for the velocity and pressure approximations, and a stabilized term based on two local Gauss integrations is used to offset the discrete pressure space to circumvent the discrete  inf-sup  condition. In the proposed parallel method, all of the computations are performed on the locally refined global grids that are fine around the interested subdomain and coarse elsewhere, making the method easy to implement based on a sequential solver with low communication cost. Stability and optimal error estimates of the present methods are deduced. Numerical results on examples including a problem with known analytic solution, lid-driven cavity flow, backward-facing step flow and flow around a cylinder are given to verify the theoretical predictions and demonstrate the high efficiency of the method. Results show that our parallel method can provide an approximate solution with the convergence rate of the same order as the solution computed by the standard stabilized finite element method, with a substantial reduction in computational time.

Local and parallel stabilized finite element methods based on full domain decomposition for the stationary Stokes equations