Researches On Some Problems Of Finite Element Methods

It is well known that the classical finite element approximation theory relies on the regular or quasi-uniform assumption, i.e., there exists a constant c＞0, such that for all element K,(h_k/ρ_k)≤c, or (?)≤C, where h=(?)h_k, (?)=(?)h_k, h_K andρ_K are the diameter and the supremum of the largest inscribed circle in K recpectively. However, the domain considered may be narrow or irregular. For example, in modelling a gap between rotor and stator in an electrical machine, or in modelling a cartilage between a joint and hip, if we seek their approximate solution with numerical calculus methods on the domain by employing the regular partition, the computing cost will be very high or can not be dealt with, it is better to employ the anisotropic triangulation in the practical application. On the other hand, solutions of some elliptic boundary problems may generate sharp boundary or interior layers, that means that the solution varies significantly in certain direction. Examples include diffusion problems in domains with edges and singularly perturbed convention-diffusion-reaction problems. In such cases it had better reflect this anisotropy in the discretization by using anisotropic meshes with a smell mesh size in the direction of the rapid variation of the solution and a large mesh size in the perpendicular direction. When dealing with the above two cases, we subdivide domainΩin anisotropic meshes, and then either h_K/ρ_K or h_k/ρ_k might be very large or even more infinite and the above regular assumption or qusi-uniform assumption is no longer valid, therefore some basic theories and techniques of the classical finite element methods are not fit. For example, when the consistency error of nonconforming element is estimated with traditional technique, (meas(F))/(measK) is presented and might be infinite if F is long edge, new ways or techniques must be employed in order to obtain convergence in such case. On the other hand, the Sobolev interpolation theory can not be directly used, hence the researches on the posed-well and the stability and LBB condition(one of the key points with mixed finite element methods) of interpolation operator are very difficult. T.Apel et al. presented an anisotropic interpolation theorem that can be used to check the anisotropy of an element. But it is not convenient in practical computing. S.C. Chen et al. supplied an improved one which is much easier to be used than that of [2,3], and then it was used to check the anisotropy of Lagrange type, Hermite type, Crouzeix-Raviart type, quasiWilson element, ACM's element, Carey elenent, and so on(refer to [4,～,11,39,41]). The above models and results suggest that it is not necessary to require meshes to satisfy regular condition in classical finite element methods. Therefore, the researcg on anisotropic finite elements is becoming a highlight issue in theoretical analysis and engineering practices and there have been a lot of important results [2,～, 11, 39, 40, 41, 65,～, 70] on such aspect in recent years.The researches on eigenvalue problems with the finite element methods have been received considerable attention. J.Fix [12], K.Ishihara [13], I.Babuska and J.Osborn[14], B.Mercier[15], Y.D.Yang [16], D.S.Wu [17] and H.P.Liu [18] have been studying on such aspect and derived optimal estimates or presented high accuracy in their works respectively. As we all know, most of the researches were done on the meshes satisfying the regular condition, but few are treated on the anisotropic meshes. One of the key points is the error estimate of L~2-norm between the exact solution and the finite element solution of eigenvalue problems. The work is not easy for some finite elememts, for example, Crouzeix-Raviart type nonconforming element, and it was not reported in previous literature on such aspect, but the problem is solved in this paper. The main aim is focused on the approximate of some eigenvalue problems for element finite methods on anisotropic meshes from 2nd chapter to 6th chapter of this paper, elements include Lagrange element, Hermite type, Crouzeix-Raviart type, Wilson or quasi-Wilson element, Carey element, and so on. By employing a set of novel techniques, we have obtained the optimal error estimate for eigenpair, so the application scope is enriched. Especially, Q.Lin et al. presented the eigenvalue problem of the primitive variables provided the so-called BB compatibility condition as an open problem for readers[71], and they solved such problem associated with the Stokes equation by the stream function and the vorticity and reported a superconvergence for eigenvalue with mixed method, however, one of the main aims of the 6th chapter of this paper presents a nonconforming finite element method for eigenvalue problems of stationary Stokes equations on anisotropic meshes.The study of superclose property and superconvergence with finite element methods is another important branch in numerical mathematics and practical application, and have been received considerable attention. The problems are dealt with not only for elliptic boundary problem, Stokes equation, evolution equation and diffusion-convection-reaction model, but also eigenvalue problem, there are many papers on them, such as 5,7,9,16,17, 19,20,42,49,50,51,58,63,64, and so on. A lot of scientists, including Q.Lin, C.M.Chen, Q.D.Zhu, N.N.Yan, S.H.Zhang, have made the advantage of the world in this aspect. But presented as above, regular condition or quasi-uniform assumption plays a very important role in the error estimates, and the similar results are hardly dealt with on anisotropic meshes. Do these results still hold for anisotropic meshes? By employing Bramble-Hilbert Lemma and Taloy expansion, we also obtain the superconvergence result to eigenvalue for a class of Crouzeix-Raviart type element on anisotropic meshes in 2.2.3 subsection. We obtain the superclose property and superconvergence results for source problems corresponding to eigenvalue one in 2.3.3 and 6.3. These superconvergence properties are also presented completely both for a class of viscoelasticity type equations on anisotropic meshes in the 7th chapter and for diffusion-convection-reaction on quasi-uniform meshes in the 8th one respectively. Because it is very complex or diffullt to construct postprocessing impolation operators(especially used on anisotropic meshes), and most of postprocessing impolation operators have not been seen in previous literatures, the results of this paper are creative on some branches.At last, the numerical results are given to investigate an elliptic boundary problem whose solution generates sharp boundary or interior layers. The Carey element is employed, the result demenstrates that the theory analysis coincides with the calculated results and the Carey element is fit to be used on such problems.In a word, the regularity or quasi-uniform assumption in classical finite element methods is not necessary for meshes. Therefore, the anisotropic finite element methods are more useful in practical computings, the researches of this paper enlarge and develop the theory of the finite element methods.