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

By combining the techniques of the two-grid method and the partition of unity, we derive two local and parallel finite element algorithms 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 requiring 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.

[1]  Yanren Hou,et al.  Adaptive Local Postprocessing Finite Element Method for the Navier-Stokes Equations , 2013, J. Sci. Comput..

[2]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[3]  Chunxiong Zheng,et al.  Partition of unity refinement for local approximation , 2011 .

[4]  Yinnian He,et al.  Newton Iterative Parallel Finite Element Algorithm for the Steady Navier-Stokes Equations , 2010, J. Sci. Comput..

[5]  M. Holst.  Applications of Domain Decomposition and Partition of Unity Methods in Physics and Geometry , 2010, 1001.1364.

[6]  Yanren Hou,et al.  Local and parallel finite element algorithms for time-dependent convection-diffusion equations , 2009 .

[7]  Yanren Hou,et al.  Postprocessing Fourier Galerkin Method for the Navier-Stokes Equations , 2009, SIAM J. Numer. Anal..

[8]  Jinchao Xu,et al.  Local and parallel finite element algorithms for the stokes problem , 2008, Numerische Mathematik.

[9]  M. Larson,et al.  Adaptive variational multiscale methods based on a posteriori error estimation: Energy norm estimates for elliptic problems , 2007 .

[10]  Michael J. Holst,et al.  Adaptive Numerical Treatment of Elliptic Systems on Manifolds , 2001, Adv. Comput. Math..

[11]  Yanren Hou,et al.  An AIM and one-step Newton method for the Navier–Stokes equations☆ , 2001 .

[12]  Jinchao Xu,et al.  Local and parallel finite element algorithms based on two-grid discretizations , 2000, Math. Comput..

[13]  Michael J. Holst,et al.  A New Paradigm for Parallel Adaptive Meshing Algorithms , 2000, SIAM J. Sci. Comput..

[14]  H. K. Lee,et al.  Numerical Solution of the Stationary Navier-Stokes Equations Using a Multilevel Finite Element Method , 1998, SIAM J. Sci. Comput..

[15]  Lutz Tobiska,et al.  A Two-Level Method with Backtracking for the Navier--Stokes Equations , 1998 .

[16]  I. Babuska,et al.  The Partition of Unity Method , 1997 .

[17]  I. Babuska,et al.  The partition of unity finite element method: Basic theory and applications , 1996 .

[18]  Jinchao Xu.  Two-grid Discretization Techniques for Linear and Nonlinear PDEs , 1996 .

[19]  M. Marion,et al.  Nonlinear Galerkin methods and mixed finite elements: two-grid algorithms for the Navier-Stokes equations , 1994 .

[20]  Jinchao Xu,et al.  Iterative Methods by Space Decomposition and Subspace Correction , 1992, SIAM Rev..

[21]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[22]  Yanren Hou.  Local and Parallel Finite Element Algorithms for the Time-Dependent Convection-Diffusion Equations , 2009 .

[23]  Douglas N. Arnold,et al.  Local error estimates for finite element discretization of the Stokes equations , 1995 .

[24]  Jean E. Roberts,et al.  Mixed and hybrid finite element methods , 1987 .