一类非线性对流扩散方程两重网格特征有限元方法及误差估计

Analysis of a Two-grid Method for Characteristic Finite Element Solutions of Nonlinear Convection-diffusion Equations

主要研究了一类非线性对流扩散方程的全离散特征有限元方法的两重网格算法及其误差估计.首先在网格步长为H的粗网格上计算一个较小的非线性问题,然后利用一阶牛顿迭代和粗网格解将网格步长为h的细网格上的非线性问题转化为线性问题求解.由于非线性问题的求解仅在粗网格上进行,该两重网格算法可以节省大量的计算工作量,同时具有较高的精度,证明了该两重网格算法L^2模先验误差估计结果为O(△t+h^2+H^(4-d/2)),其中d为空间维数.

A two-grid characteristic finite element approximation is presented and discussed for nonlinear convection-diffusion equations.Piecewise linear trial functions are used to develop the finite element scheme.In this two-grid scheme,the full nonlinear problem is solved only on a coarse grid with grid size H.The nonlinearities are expanded by one Newton-like iteration about the coarse grid solution on a fine gird of size h.The resulting linear system is solved only on the fine grid.A prior error estimates are derived with the L2-norm O（△t ＋ h2 ＋ H4-d/2）,where d 〉 1 is the spacial dimension.