A Parallel Grad-div Stabilized Finite Element Algorithms For Incompressible Flows With Damping

The Stokes or Navier-Stokes equations with damping,one of the main systems investigated in geophysics and ocean acoustics,are broadly employed in the resistance to the motion of the flow,and play a very important role in fluid mechanics.However,there are typically two problems encountered in modern scientific and engineering computations.Firstly,due to the complexity of the fluid flow region,the use of traditional numerical methods has the problems of large solution size and limited computer storage.Secondly,when simulating velocity-pressure coupled incompressible flow problems,classical finite element methods often lead to poor velocity simulation results when the viscosity coefficient is small.Therefore,the design of computationally efficient numerical methods with high velocity accuracy is essential to simulate incompressible flow problems.Based on the grad-div stabilization method and the fully overlapping domain decomposition technique,this work studies parallel grad-div stabilized finite element algorithms for the damped Stokes and Navier-Stokes equations.In the algorithms,we solve a global grad-div stabilized problem to compute a local solution in an intersecting subdomain on a global composite mesh,which is fine in the subdomain and rough elsewhere.As there is no information exchange between processors during the computation process and existing solvers can be easily implemented,the communication requirements are low.We use the local a priori error estimate theory tool of the grad-div stabilized finite element solution to derive the optimal error bounds of the velocity and pressure approximate solutions obtained by this algorithm,as well as the selection of coarse and fine grid sizes to achieve optimal convergence rates.We develop finite element parallel programs and use the proposed method to calculate known analytical solutions,backward-facing step flows,forward-facing step flows,and lid-driven cavity flows,and compare the numerical results with standard finite element methods and parallel algorithms without the grad-div stabilization term.The numerical results verify the correctness of the theoretical analysis,the necessity of adding the grad-div stabilization term,and the efficiency of the proposed algorithm.On the one hand,compared with the counterpart one excluding grad-div stabilization,the algorithms can reduce significantly the effect of pressure on the approximate velocities,and hence,yields much better approximate velocities in the case of small viscosities.On the other hand,the algorithms take much less computational time in getting approximate solutions with a comparable accuracy than the standard grad-div stabilization method.