This report analyzes a multirate, decoupling algorithm, which allows different time steps in the fluid region and the porous region for the nonstationary Stokes–Darcy problem. The method presented requires only one, uncoupled Stokes and Darcy subphysics and subdomain solve per time step. Under a time step restriction of the form △t ≤ C (physical parameters) we prove stability and convergence of the method. Numerical tests are given to show the convergence result and demonstrate the computational efficiency of the partitioned method. They also show that in the expected case of greater fluid velocities in the free‐flow region than in the porous media region, allowing smaller time steps in the subregion with the faster velocities increases both accuracy and efficiency. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013

No abstract

By combining the techniques of two-grid method and the partition of unity, two local and parallel finite element algorithms are presented for the Stokes problem. The most interesting features of these algorithms are: (1) the partition of unity technique introduces a framework for domain decomposition, (2) only a series of local residual problems need to be solved on these subdomains in parallel, meanwhile require very little communication, (3) a globally continuous finite element solution is constructed by combining all the local solutions via the partition of unity functions. The optimal error estimates in L 2 and energy norms are proved under some assumptions. Also, several numerical simulations are presented to demonstrate the effectiveness and flexibility of the new algorithms.

By combining two-grid method with domain decomposition method, a new local and parallel finite element algorithm based on the partition of unity is proposed for the incompressible flows. The interesting points in this algorithm lie in (1) a class of partition of unity is derived by a given triangulation, which guides the domain decomposition (2) the globally fine grid correction step is decomposed into a series of local linearized residual problems on some subdomains and (3) the global continuous finite element solution is obtained by assembling all local solutions together using the partition of unity functions. Some numerical simulations are presented to demonstrate the high efficiency and flexibility of the new algorithm.