非线性四阶双曲方程低阶混合元方法的超收敛分析

摘要/Abstract

摘要：

对一类非线性四阶双曲方程利用双线性元Q₁₁及Q₀₁×Q₁₀元给出了一个低阶混合元格式. 基于上述两个单元的高精度结果，采用插值和投影相结合的方法，利用对时间t的导数转移技巧，借助插值后处理技术，在半离散格式下导出了原始变量u和中间变量v=-Δu在H¹模意义下及流量=-▽u在（L²）²模意义下具有O（h²）阶的超逼近和超收敛结果. 与此同时，在全离散格式下，证明了u和v在H¹模意义下及???20160405-1???在（L²）²模意义下单独利用插值或投影所无法得到的具有O（h²+（Δt）²）阶的超逼近和超收敛结果.

关键词: 非线性四阶双曲方程, 混合元方法, 半离散和全离散格式, 超逼近和超收敛

Abstract:

Based on the bilinear element Q₁₁ and the Q₀₁×Q₁₀ element, a lower order conforming mixed finite element approximation scheme is proposed for nonlinear fourth-order hyperbolic equation. With the help of the known high accuracy results of the about two elements, by use of derivative delivery techniques and the post-processing technique, the superclose and superconvergence with order O(h²) for both scalar original variable u and the intermediate variable v=-Δu in H¹ norm and the flux =-▽u in (L²)² norm are derived in semi-discrete schemes through interpolation and Ritz projection approach, respectively. Meanwhile, the superclose and superconvergence with order O(h²+(Δt)²) for both u and v in HM¹ norm and ???20160405-1??? in (L²)² norm are proved in fully-discrete schemes, which cannot be deduced by the interpolation and Ritz projection alone.

Key words: Nonlinear fourth-order hyperbolic equation, Mixed finite element method, Semi-discrete and fully-discrete schemes, Superclose and superconvergence

中图分类号:

O242.21

张厚超, 石东洋. 非线性四阶双曲方程低阶混合元方法的超收敛分析[J]. 数学物理学报, 2016, 36(4): 656-671.

Zhang Houchao, Shi Dongyang. Superconvergence Analysis of a Lower Order Mixed Finite Element Method for Nonlinear Fourth-Order Hyperbolic Equation[J]. Acta mathematica scientia,Series A, 2016, 36(4): 656-671.

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