求解奇异摄动问题混合有限元的最优一致收敛的统一方法_英文_

Received date: 2000- 11- 05
214
数学研究
2001 年
problems , higher- order equat ions, and the relations between the taditional SHishkin type meshes( denoted as S- mech , which use = O ( ln N ) as the transition point ) and a non - traditional Shishkin type mesh ( denoted as A- mesh , which use = O( | ln | ) as the transiition point). To the best knowledgeof the author , there exist very few results for solving SPP by using mixed FEM (MFEM )[ 14] . Also there are fewer results related to the A - mesh[ 2, 9, 29, 14] compared to the results based on the S - mesh[ 8, 12, 13, 16, 26] et c. More references for S - mesh can be found in[ 18, 23, 21, 13] . Though Roos claimed that the differences betwwen A- mesh and S - mesh seem in practice to be not important[ 21, P. 304] , there is no detailed comparison between these two type meshes up to now . Re cently the author found that both meshes perform very similarly in both the theoret ical error est imates and the numerical results for solving some two- dimensional singularly perturbed problems by using standard FEM and mixde methods [ 14] . Since A - mesh separates the boundary layers totally from other subre gions, the error estimates for the A - mesh only require the derivative estimates of the analytic solut ion , while the error analysis on the S- mesh have to resort to some nice but hardly proved Shishkin type de compositions [ 18, 23, 21, 16] or the equivalently asymptotic expansions[ 12, 13] In this paper , we extend the techniques of [ 14] to solve some one - dimensional problems by using MFEM. For simplicity we only present the analysis for a fourth- order problem [ 26, 22] on both the Smesh and the A- mesh , and steady [ 25] and non - steady semilinear problems [ 31, 5] on the A - mesh . By introducing a novel interpolation on the m - th ( m 1) order polynomial spaces, we obtain the opti mal uniform convergence rates in the L 2 - norm for the above mentioned problems . The results obtained show that it is much easier to obtain the error estimates on the A - mesh than on the S- mesh. However , the estimates obtained on both meshes are very similar , which justifies the comjecture of Roos[ 21] . Exten sions of the techniques to other models are straightforward, details will be discussed in our forthcoming pa per . The organization of the paper is as follows: in Section 2 we discuss a fourth - order problem on the A - mesh, ! 3 is devoted to a semilinear problem , a nonsteady semilinear problem is investigated in ! 4, in the last section we present the analysis of the fourth- order problem on the S - mesh. Throughout the paper , we denote = (0, 1) and = [ 0, 1] , C will denote a generic positive con stant , which is independent of the mesh and the perturbat ion parameter 0 < 1. We use the notation ∀ # ∀k , ! and | #| k , ! for the standard Sobolev space H ( !) ( or H 0(!)) norm and semi- norm on any
norm is presented for solving singularly perturbed problems by using an arbitrary order mixed finite element meth od on some Shishkin type meshes[ 21, 23, 19, 18] . The generality of our techniques is showed by investigations of a fourth - order problem, steady and non - steady semilinear problems . The results show that the error estimates on the non - traditional Shishkin type mesh are much easier to prove than on the tradition Shishkin type mesh. However, both meshes g ive comparable error estimates , which justifies the conjecture of Roos[ 21] . Key words finite element method; singular perturbed problem ; optimal uniform convergence. CLC number
O 241. 82
there was great interest shown in the investigation of numerical methods for singularly per turbed problems( SPP)[ 18, 23, 19] . Singular perturbation problems arise in many application areas, such as in chemical kinetics, fluid dynamics, system control[ 18, 23, 19] and thin plate and shell modeling [ 3, 4] etc . Such problems usually undergo rapid changes within very thin layers near the boundary or inside the problem domain. Such sharp transit ions require very fine meshes globally (which is very inefficient) or locally to resolve the boundary layers. T he challenging SPP[ 19, 20, 21] serve frequently as test models for new algorithms, e . g . . in multigrid methods[ 10, Ch. 10] , domain decompositi on methods([ 9, 11] and many papers in the proceedings of Domain Decomposition Methods) and adaptive methods[ 1, 20, 28, 24] . More details can be found in the above mentioned papers and references therein. Recent applicat ion of Shishkin type meshes to solve SPP by using both the finite different method and finite element method( FEM ) [ 18, 23, 21, 12, 13, 16, 2, 29] has shown surprising . results: the piece wise uniform Shishkin type meshes are very easy to implement and achieve global uniform convergence ( in dependent of the small perturbation parameter ), which is very important and interesting in many areas [ 28, 4, 27] Though significant progress has been made , many unsolved problems remain as described in the most recent survey by Roos[ 21] , such as the use of finite element techniques to derive opt imal esti mates for FEM on Shishkin meshes in both the L and L 2 norms, higher- order finite elements, nonlinear