【作者】 李畅；

【导师】 石东洋；

【作者基本信息】 郑州大学 ， 计算数学， 2013， 硕士

【摘要】 本论文由四部分组成.第一部分主要利用双线性元和零阶R—T元,对非线性sine-Gordon方程构造了一个新的协调矩形混合元变分形式,基于积分恒等式技巧,导数转移及插值算子的特性,给出了在半离散格式下相关变量的超逼近性质.同时使用插值后处理技术得到了相应的整体超收敛结果.第二部分主要对非线性sine-Gordon方程提出一个新的非协调混合有限元逼近格式,基于EQ₁^rot非协调元有两个特殊性质：（a）当精确解属于H3（Ω）时,其相容误差为O（h2）阶比它的插值误差高一阶；（b）插值算子与Ritz投影算子等价,再结合零阶R—T元的高精度结果,导出了半离散格式精确解u在H1模意义下和中间变量p在L2模意义下二阶超逼近及超收敛.最后,讨论了一个二阶全离散逼近格式,得到了精确解u在H1模意义下和流量p在L2模意义下的最优误差估计.第三部分对非线性sine-Gordon方程利用协调线性三角形元构造了一个新的H1—Galerkin混合有限元格式,利用上述的方法和技巧,给出了在半离散及全离散格式下解的超逼近性质及超收敛结果.第四部分主要利用双线性元和零阶R—T元及EQ₁^rot元对非线性伪双曲方程初边值问题分别构造了新的协调及非协调混合元格式,在抛弃传统有限元分析中不可缺少的工具—Ritz投影的前提下,借助平均值技巧,积分恒等式和插值后处理技术,给出了解的超逼近性质及整体超收敛结果.更多还原

【Abstract】 This dissertation includes four parts.In the first part, a new conforming rectangular mixed finite element viriational form is constructed for sine-Gordon equations with the bilinear element and zero order R-T element. By using integral identity techniques, derivative transfer and interpolation operator’s characteristics, we get the superclose properties of the in semi-discrete scheme. At the same time, by virtue of the technique of interpolation post-processing technique, the global superconvergence results are obtained.In the second part, a new nonconforming mixed finite element approximation scheme for nonlinear sine-Gordon equations is proposed. By use of two special properties of this element:(a) the consistency error is of order O(h2) one order higher than its interpolation error O(h), when the exact solution belongs to H3(Ω);(b) the interpolation operator is equivalent to its Ritz-pr ejection operator,the superclose properties and superconvergences of exact solution u in H1-norm and intermediate variable p in L2-norm are deduced for semi-discrete scheme, and the optimal order error estimates are obtained for a two order fully-discrete scheme.In the third part, by use of the given estimates of conforming linear trianglar element, a new H1-Galerkin mixed finite element viriational form is constructed for sine-Gordon equations. With the help of the aforementioned methods and skills, we get the superclose property and superconvergence in both semi-discrete form and fully discrete form.In the last part, a new conforming and nonconforming mixed finite element methods are proposed for Nonlinear pseudo-hyperbolic equations respectively, without using Ritz projection, which is an indispensable tool in the traditional finite element analysise, the superclose and the global superconvergence are derived directly through interpolation operater, mean-value tricle and post-processing technique.更多还原