STABILIZED MIXED FINITE ELEMENT METHODS WITH SYMMETRIC TENSORS FOR ELASTIC STRUCTURE AND THE FAST SOLVERS
主 题: STABILIZED MIXED FINITE ELEMENT METHODS WITH SYMMETRIC TENSORS FOR ELASTIC STRUCTURE AND THE FAST SOLVERS
报告人: Xuehai Huang (温州大学)
时 间: 2016-09-01 14：00-15：00
地 点: 理科一号楼 1418
Firstly, we design two classes of stabilized mixed finite element methods for linear elasticity on simplicial grids to attack the lower order case of Hu-Zhang element. In the first class of elements, we use H(div;S)-Pk and L2(Rn)-Pk-1 to approximate the stress and displacement spaces, respectively, for 1≤k≤n, and employ a stabilization technique in terms of the jump of the discrete displacement over the faces of the triangulation under consideration; in the second class of elements, we use H1(Rn)-Pk to approximate the displacement space for 1≤k≤n and adopt the stabilization technique suggested by Brezzi, Fortin, and Marini. The feature of these methods is the low number of global degrees of freedom in the lowest order case. Secondly, a block diagonal preconditioner with the MINRES method and an approximate block factorization preconditioner with the GMRES method are developed for Hu-Zhang mixed finite element methods of linear elasticity. They are based on a new stability result of the saddle point system in mesh-dependent norms. The mesh-dependent norm for the stress corresponds to the mass matrix which is easy to invert while the displacement it is spectral equivalent to Schur complement. A fast auxiliary space preconditioner based on the H1 conforming linear element of the linear elasticity problem is then designed for solving the Schur complement. For both diagonal and triangular preconditioners, it is proved that the conditioning numbers of the preconditioned systems are bounded above by a constant independent of both the crucial Lamé constant and the mesh-size. Thirdly, a V-cycle multigrid method for the Hellan-Herrmann-Johnson (HHJ) discretization of the Kirchhoff plate bending problems is developed. It is shown that the contraction number of the V-cycle multigrid HHJ method is bounded away from one uniformly with respect to the mesh-size. The key is a stable decomposition of the kernel space which is derived from an exact sequence of the HHJ method. The uniform convergence is achieved for V-cycle multigrid method with only one smoothing step and without full elliptic regularity. The exact sequences of the HHJ method and the corresponding commutative diagram is of some interest independent of the current context.
