非线性对流占优扩散方程经济型差分流线扩散法无网格比超收敛分析

UNCONDITIONAL SUPERCONVERGENCE ANALYSIS OF ECONOMICAL FINITE DIFFERENCE STREAMLINED DIFFUSION METHOD FOR NONLINEAR CONVECTION-DOMINANTED DIFFUSION EQUATION

本文主要用经济型差分流线扩散(EFDSD)法研究非线性对流占优扩散方程的向后Euler(BE)全离散有限元格式,并在时间步长τ和空间剖分参数h的比值无约束下,导出H1模意义下具有O(h2+τ)阶的超收敛性质.首先,引入时间离散系统,将误差分为时间误差和空间误差两部分,并利用数学归纳法,通过时间误差给出了时间离散方程解的正则性.其次利用空间误差导出有限元解的W0,∞模的有界性,再借助插值后处理技巧得到了H1模意义下的无网格比的超逼近和整体超收敛结果.最后,通过数值例子对理论分析的正确性和算法的高效性予以了验证.

In this article,the backward Euler(BE)fully discrete finite element method of the economical finite difference streamlined diffusion(EFDSD)method for nonlinear convectiondominated diffusion equation is mainly investigated and the superconvergence of order O(h~2+τ)in H~1 norm is derived without the restriction between the time stepτand the mesh size h.Firstly,a time discrete system is established to split the error into two parts,which are the temporal error and spatial error,and with the help of mathematical induction,the regularity of the time discrete system is reduced by the temporal error.Then the finite element solution in W~(0,∞)norm is bounded by the spatial error and the unconditional superclose and global superconvergence results are gained in H~1 norm through interpolation post-processing technique.Lastly,a numerical example is provided to verify the correctness of the theoretical analysis and the effectiveness of the method.