一类非线性四阶双曲方程扩展的混合元方法的超收敛分析

对一类非线性四阶双曲方程利用双线性元Q_(1)及Nedelec's元建立一个扩展的协调混合元逼近格式.首先证明了逼近解的存在唯一性.其次,基于上述两个单元的高精度结果,给出了插值和投影之间的误差估计,再利用对时间t的导数转移技巧和插值后处理技术,在半离散和全离散格式下分别导出了原始变量u和中间变量v=-△u在H~1模及中间变量q=▽u,σ=-▽(△u)在(L~2)~2模意义下单独利用插值和投影所无法得到的具有O(h~2)和O(h~2+τ~2)阶的超收敛结果.最后通过数值算例,表明逼近格式是行之有效的.这里,h和τ分别表示空间剖分参数及时间步长.

SUPERCONVERGENCE ANALYSIS OF AN EXPANDED MIXED FINITE ELEMENT METHOD FOR NONLINEAR FOURTH-ORDER HYPERBOLIC EQUATION

With the help of bilinear elementQ₁₁and Nédélec's element, an expanded conforming mixed finite element approximation scheme is proposed for nonlinear fourth-order hyperbolic equation. Firstly, the existence and uniqueness of approximation solution are proved. Secondly, based on integral indentity results of above two elements, an error estimate is established between interpolation and Ritz projection. Moreover, by use of derivative delivery techniques and postprocessing approach, the superconvergence results with order O(h²) and O(h²+τ²) of original variable u and intermediate variable v=-Δu in H¹-norm and intermediate variable q=-?u,σ=-?(Δu) in (L²)²-norm are derived for semi-discrete and fully-discrete schemes, which cannot be derived by interpolation and Ritz projection alone. Finally,it is shown that the proposed approximate schemes are effective by numerical example. Here, h and τ are parameter of subdivision in space and time step, respectively.