Local and Parallel Finite Element Algorithms Based on the Partition of Unity for the Stokes Problem

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.

A stabilized coupled method and its optimal error estimates for elliptic interface problems

“…For many numerical methods, researchers often use stabilization to reduce the error [18], which inspires us to solve the interface problem in such way. By combining the two-level method and the partition of unity, the authors and their collaborators proposed two local and parallel algorithms for the Stokes problem [19], elliptic equations [20], the Stokes-Darcy model [21] and the fluid-fluid model [22,23]. In this paper, we consider the elliptic interface problem as a mixed elliptic-elliptic model.…”

Local and parallel finite element method for solving the biharmonic eigenvalue problem of plate vibration

“…In 2000, Xu and Zhou propose some local and parallel finite element algorithms by combining the two‐grid finite element discretization scheme with the local defect correction technique for elliptic boundary value problems. Later, this parallel‐computing technique has been developed (see, e.g., ). In 2013, Bi et al propose a local finite element discretization based on the shifted‐inverse power method for the second‐order elliptic eigenvalue problem.…”

Multiscale finite element discretizations based on local defect correction for the biharmonic eigenvalue problem of plate buckling

“…In 2000, Xu and Zhou propose some local and parallel finite element algorithms based two‐grid discretizations by using local defect correction technique. Since then, this technique has been developed by many scholars (see) and been applied to time‐dependent convection‐diffusion equations, quantum eigenvalue problems,() Navier‐Stokes equations, etc. On the basis of the work of Dai and Zhou (see Algorithm B0 and Algorithm B in), we establish a three‐scale and a multiscale discretization scheme in Section and the parallel version of multiscale discretization scheme in Section based on local defect correction technique.…”