非线性四阶双曲方程一个低阶混合元方法的超收敛和外推

对一类非线性四阶双曲方程利用双线性元Q₀₁及Q₀₁×Q₁₀ 元给出了一个低阶协调混合元逼近格式。证明了逼近解的存在唯一性。基于上述两个单元的高精度结果,利用对时间t的导数转移技巧, 导出了原始变量u和扩散项p=-Δu 在H¹模及流量=-∇u在L²模意义下具有Q(h²)阶的超逼近结果。进一步地, 借助插值后处理技术,得到了整体超收敛性。通过建立Q₀₁×Q₁₀元的一个新的渐近展开式,并构造一个合适的外推格式,得到O(h³)阶的外推解。这里,h表示空间剖分参数。

Superconvergence and extrapolation of a lower order mixed finite method for nonlinear fourth-order hyperbolic equation

With the help of 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. Firstly, the existence and uniqueness of approximation solution are proved. Secondly, Based on the known high accuracy results of the about two elements, by use of derivative delivery techniques, the superclose with order O(h²) for both scalar unknown u and the diffusion term v=-Δu in H¹-norm and the flux p=-∇u in L²-norm are derived, respectively. Moreover, the global superconvergence is obtained through interpolation post-processing technique. Finally, throught constructing a new asymptotic expansion formula of Q₀₁×Q₁₀ element and a suitable extrapolation scheme, the extrapolation solutions with order O(h³) are derived. Here, h is the subdivision parameter for the space.