BDDC Preconditioners for Nonconforming FEMs for the Problems with Discontinuous Coefficients
|School||Nanjing Normal University|
|Keywords||discontinuous coefficients finite element method nonconforming ele-ment domain decomposition BDDC|
In this thesis, under the general framework of the additive Schwarz methods, we discuss the BDDC (the balancing domain decomposition by constraints method) preconditioners for the nonconforming finite element methods for several different partial differential equations with piecewise but discontinuous coefficients.Firstly, we consider the second order elliptic equations. We apply the rotated Q1finite element on both matching grids and nonmatching grids. Based on the two different coarse spaces, we propose two different BDDC preconditioners respectively. We show that our method has a quasi-optimal convergence behavior, i.e, the condi-tion number of the preconditioned problem is O((1+logH/h)n),(n=2,3), which is independent of the jumps of the coefficients.Secondly, the incompressible Stokes problem are considered, and the Qiot/Qo finite element method are used. On the benign subspace, we show our BDDC precon-ditioning method has a quasi-optimal convergence behavior, and the convergence rate is as strong as the second elliptic problem.Finally, we propose a BDDC preconditioner for the Morley finite element method for fourth order elliptic equations. Our BDDC preconditioning method is quasi-optimal convergence, i.e, the condition number of the preconditioned problem is in-dependent of the jumps of the coefficients, and depends only logarithmically on the ratio between the subdomain size and the mesh size.Numerical experiments are presented to confirm our theoretical analysis.