二阶双曲型问题C～0有限元的构造及其超收敛

【作者】 肖春霞；

【导师】 陈传淼；

【作者基本信息】 湖南师范大学 ， 数学学科计算数学， 2002， 硕士

【摘要】 本文为二阶常微分方程及二阶双曲型问题的时间方向构造了C⁰有限元，在节点及单元内部的一些特征点上获得了超收敛结果。 全文分为三部分： 第一部分：我们考虑以下二阶常微分方程其中a，b，g足够光滑。 我们构造了一个超逼近函数u_n，证明了在节点处C⁰有限元解u_h有如下超收敛估计并且已证明了在单元内部的一些特征点上u_n，u_h有超收敛结果。 第二部分：我们考虑以下二阶双曲型问题算子A一致椭圆算子且与t无关。 我们采用了张量积并在时间方向应用C⁰有限元，令0=t₀＜t₁＜t₂＜（?）t_N=T是[0，T]的一个剖分，令S₀^h∈H₀^l（Ω）是Ω上的二次有限元空间。h为网格参数，令Z_h为空间上的剖分节点，则C⁰全离散有限元解U在点积Z_h（?）{t_j}上有如下超收敛估计更多还原

【Abstract】 In this paper, we construct C0 finite elements for second-order ordinary differential equations and second-order hyperbolic equations in time, and at the nodes and some characteristic points several new superconvergence results are derived.This paper is divided in three parts.Part 1 We consider the following second-order ordinary differential equationwith a, b, f sufficiently smooth.We construct a superapproximation function un and have proved that at the nodes the C?finite element solution uh for the equation has following optimal order superconvergence resultsWe also have proved that at some characteristic points in the elements, both uh and uh have superconvergence results.Part 2 We consider the following second-order hyperbolic equationwith A a uniformly elliptical operator independent of t.In this part we use the tensor product and construct C0 finite elements for the equation in time. let 0=t0<t1<t2< A tN=T be a partition of [0,T],let S0h ∈ H01(Ω) be a finite element space of continuous piecewise 2th degree polynomials on Ω with mesh parameter h, and let Zh be the partition nodes of the space, then the C0 complete discrimination solution U has the following superconvergence results on the dot product Zh {tj}we also proved that at some characteristic points in the elements Ij=(tj,tj+1),DtU have superconvergence results.Part 3 In this part we present the results of some numerical experiments to test the theorems in section 1, section2. Our numerical results show that the superconvergence results of previous two sections hold.更多还原