非线性抛物方程混合有限元方法的高精度分析

SUPERCONVERGENCE ANALYSIS OF A MIXED FINITE ELEMENT METHOD FOR NONLINEAR PARABOLIC EQUATION

采用双线性元及零阶Raviart-Thomas元(Q_(11)+Q_(10)×Q_(01))对非线性抛物方程讨论了一种H^1-Galerkin混合有限元方法.提出一个线性化的二阶格式,利用数学归纳法有技巧的导出了原始变量u在H^1(Ω)模意义下及流量■=▽u在L^2(Ω)模意义下的O(h^2+τ~2)阶超逼近性质.引入一个有关初始点的时间离散方程,并利用其得到了▽·■在L^2(Ω)模意义下的O(h^2+τ~2)阶的超逼近结果.同时利用插值后处理技巧得到整体超收敛.最后,数值算例结果验证了理论分析(其中,h是剖分参数,τ是时间步长).

An H1-Galerkin mixed finite element method is discussed for nonlinear parabolic equations with the bilinear element and the zero-order Raviart-Thomas element(Q11+Q10×Q01).A linearized second order fully-discrete scheme is proposed.The superclose results with O(h2+τ2)of original variant u in H1-norm and flux variant■in L2-norm are derived technically.A time semi-discrete equation at the starting point is introduced and the superclose property of▽·■in L2-norm is reduced.Furthermore,the corresponding global superconvergence results are obtained by the interpolated postprocessing technique.At last,numerical results are presented to illustrate the feasibility of the proposed method(Here,h is the subdivision parameter,andτ,the time step).