非线性双曲型积分微分方程的各向异性非协调有限元逼近

The Nonconforming Finite Element Approximation to the Nonlinear Hyperbolic Integro-differential Equation on Anisotropic Meshes

在各向异性网格下,讨论了一类非线性双曲型积分微分方程的一个矩形非协调有限元方法逼近,给出了半离散格式下的有限元解的收敛性分析和误差估计。在精确解适当光滑的前提下,利用新的技巧和精细估计得到了其超逼近性质。同时利用插值后处理技术导出了整体超收敛结果。本文的结论表明传统有限元分析中对网格的正则性要求和对Ritz-Volterra投影的依赖不是必要的,从而进一步扩展了非协调有限元方法的应用范围。

The approximation of a rectangular nonconforming finite element method for a kind of nonlinear hyperbolic integro-differential equations is discussed under the anisotropic meshes, the con- vergence analysis and the error estimate of finite element solution are presented for the semi-discrete scheme. The superclose property is derived through the new technique and sharp estimates when the exact solution is appropriately smooth. At the same time, based on the interpolated postprocessing trick, the global superconvergence is obtained. The results of this paper show that the regular condition on the meshes and the dependence on the Ritz-Volterra projection in traditional finite element analysis axe not necessary, and thus extend the applications of nonconforming finite element methods.