Multigrid Methods For Combined Hybrid Elements Of Elasticity Problems

Combined hybrid element method with high performance for elasticity problems can achieve high accuracy on coarse grid,avoid the locking phenomenon and be insensitive to mesh distor-tion.A solving operator is introduced on the element to express the stress by the displacement,then the stress parameters can be eliminated.So the discrete system from the method involves merely the displacement parameters.With the decrease of the mesh size,the condition num-ber of this system dramatically increases,just the same as the classical finite element method.Hence the research of the efficient solver to the system has great significance in both theory and application.In this thesis multigrid methods are developed for combined hybrid elements ap-proximation of elasticity problems and the convergence theories for the methods are obtained.The crucial part in this work is to overcome the difficulties caused by the stress solving opera-tor and the unnested interpolation spaces of the displacement.The main contents and research results are given as follows:1.A W-cycle multigrid method is developed for the discrete system arising from the com-bined hybrid rectangle element.It is first proved that the intergrid transfer operator defined on rectangular elements is stable in energy norm.Then the convergence of W-cycle multigrid method is obtained in this norm.The results of a numerical example conform the theoretical conclusion.2.A multigrid method is proposed for solving the combined hybrid quadrilateral elements discretization system.A bounded intergrid transfer operator in L~2norm is defined on quadri-lateral elements.Then the convergence of W-cycle multigrid method in L~2norm is proved,and the approximate solution of the full multigrid method satisfies the same type of error esti-mates as the discretization error.Moreover,the results of numerical tests are consistent with the theoretical analysis.3.Multigrid preconditioners are constructed for the linear system resulting from the com-bined hybrid quadrilateral elements discretization.Firstly,two stable intergrid transfer opera-tors in the mesh-dependent semi-norm are established on quadrilateral meshes,and the uniform equivalence of the semi-norm and the energy norm with respect to the mesh size is proved.As a result of these,the bounded properties of the intergrid transfer operators in energy norm are concluded.Secondly,the condition number of the variable V-cycle multigrid preconditioned system is shown to be uniformly bounded by a constant independent of mesh size and mesh level.Finally,numerical experiments are reported to justify the theoretical results and illustrate the efficiency of W-cycle and V-cycle multigrid preconditioned conjugate gradient method for the combined hybrid element with high performance in the case of the mesh distortion.4.A new V-cycle multigrid method is presented for the approximation system of the com-bined hybrid quadrilateral elements.The discrete system on the finest grid is arisen from the corresponding combined hybrid quadrilateral elements,but on the coarse grid the displacement and stress discrete space are replaced by the bilinear conforming finite element space and the piecewise linear space satisfying the energy compatibility condition respectively.The con-vergence of this V-cycle multigrid method is theoretically obtained only for combined hybrid scheme of high performance.Numerical experiments are supplied to demonstrate the validity of the method for the other combined hybrid elements.5.Multigrid methods are established for the combined hybrid hexahedron elements ap-proximation systems.The trilinear interpolation operator is adopted as the intergrid transfer operator on hexahedron elements.It is proved that the operator is bounded in L~2norm.Then the error estimate for the full multigrid method is obtained in this norm.The numerical results agree with the theoretical analysis.Furthermore,based on the intergrid transfer operator,W-cycle,V-cycle and variable V-cycle multigrid preconditioned conjugate gradient methods are developed.A numerical example is given to show the effectiveness of the presented methods.