非线性拟双曲方程的有限元配置法数值分析

Numerical Analysis of the Finite Element Collocation Method for the Nonlinear Pseudo-Hyperbolic Equation

基于有限元配置法,采用分片双三次Hermite插值多项式空间作为逼近函数空间,本文对粘性振动及神经传播过程中涉及的一类非线性拟双曲方程的初边值问题建立了二维半离散和全离散格式.并对两种格式证明了数值解的存在唯一性,应用微分方程先验估计的理论和技巧得到了L2模最佳阶误差估计.数值实验结果表明:所提方法在保证整体误差估计要求且不增加计算量的前提下,比传统有限元方法有更高的逼近精度,并扩展了配置法的应用范围.

Based on the finite element collocation method,this paper discusses the initial boundaryvalue problems of some nonlinear pseudo-hyperbolic equations arising from viscous material wave of the mechanics and neuro transport of biological mathematics.The semi-discrete and the fully discrete forms of the problem are proposed by using piecewise bicubic Hermite interpolation polynomial space,and the unique existence of numerical solution for two discrete forms is proved.Optimal L2-error estimate is obtained based on the theory and technique of differential equations’ priori estimates.Numerical experiment results further verify that the proposed method has a better approximation accuracy than the general finite element methods without increasing computational cost,and thus extends the applications of the collocation method.