Adaptive and parallel variational multiscale method for the Navier-Stokes equations

Abstract

Navier-Stokes equations are basic equations in fluid dynamic. The problem is important both in practice and theory. As it is difficult to find their accuracy solutions, numerical simulations and experimentations have become important approaches to solve the problem.Variational multiscale finite element method is one of most useful methods. In order to guarantee the effectiveness, adaptive algorithm has been developed,which makes use of the solutions in the progress to automatically control the computing progress. In this thesis we first present an adaptive variational multiscale method for the Stokes equations. Then we develop two kinds of variational multiscale method based on the partition of unity for the Navier-Stokes equations. First, we propose some a posterior error indicators for the variational multiscale method for the Stokes equations and prove the equivalence between the indicators and the error of the finite element discretization. Some numerical experiments are presented to show their efficiency on constructing adaptive meshes and controlling the error. Secondly, a parallel variational multiscale method based on the partition of unity is proposed for incompressible flows. Based on two-grid method, this algorithm localizes the global residual problem of variational multiscale method into a series of local linearized residual problems. To decrease the undesirable effect of the artificial homogeneous Dirichlet boundary condition of local sub-problems, an oversampling technique is also introduced. The globally continuous finite element solutions are constructed by assembling all local solutions together using the partition of unity functions. Especially, we add an artificial stabilization term in the local and parallel procedure by considering the residual as a subgrid value, which keeps the sub-problems stable. We present the theoretical analysis of the method and numerical simulations demonstrate the high efficiency and flexibility of the new algorithm. Another a partition of unity parallel variational multiscale method is proposed. The main difference lies in that in this algorithm we propose two kinds of refinement method.It is diffcult to obtain the theoretical result as the above method. However, the numerical simulations show that the error of this algorithm decays exponentially with respect to the oversampling parameter.