非线性四阶双曲方程扩展的超收敛分析及外推

Superconvergence Analysis and Extrapolation for the Extension of Nonlinear Fourth-order Hyperbolic Equations

研究采用差值理论对非线性四阶双曲方程进行混合元构造以及格式逼近,构造了混合有限元空间Vh和■,并证明其逼近解的唯一存在性,通过差值处理后处理技术,得到了误差方程,将矩形区域相邻的四个小单元合并成成为一个大单元,采用差值算子I■和∏■导出非线性四阶双曲方程精确解u的O(h^2)阶的超收敛结果;在此基础上,通过构造方程的辅助问题,根据Gronwall引理将非线性四阶双曲方程相邻的16个Th的小单元格进行合并,组成一个大的单元格,采用非线性四阶双曲方程差值处理后的算子∏_4h可以得到方程扩展O(h^4)阶的外推结果。

The difference theory is used to construct the nonlinear four order hyperbolic equation with mixed element construction and form approximation.The mixed finite element space Vh and■are constructed,and the unique existence of the approximate solution is proved.The error equation is obtained by the post processing technology,and the four small units adjacent to the rectangular region are merged into a big one.By using the difference operator I■and∏■,the superconvergence results of the O(h2)order of the nonlinear four order hyperbolic equation of u are derived,and on this basis,by constructing the auxiliary problem of the equation,the small cells of 16 Th cells adjacent to the nonlinear four order hyperbolic equation are combined to form a large cell and use the nonlinearity by the Gronwall lemma.The operator∏4h of difference operator after the four order hyperbolic equation can get the extrapolation result of the extended O(h4)order of the equation.