具有变系数的时间分布阶扩散方程的高效新混合有限元逼近

Efficient New Mixed Finite Element Approximation for Distributed-order Time Fractional Diffusion Equations with Variable Coefficients

运用新混合有限元方法分析含有变系数的二维时间分布阶扩散方程.首先,利用Gauss积分对分布阶算子进行逼近,将问题转化为一个多项时间分数阶偏微分方程.其次,利用修正的L1格式和协调的新混合元(双线性元Q_(11)(K)与Q_(01)(K)×Q_(10)(K))以及中间变量p,构造了其全离散逼近格式,又由混合元的相关性质,给出了全离散格式的稳定性分析.最后,借助于双线性元和Q_(01)(K)×Q_(10)(K)元的高精度结果及插值后处理技术得出了超逼近和超收敛结果.

This paper mainly analyzes the two-dimensional distributed-order time fractional diffusion equations with variable coefficients by applying the new mixed finite element methods(MFEMs).Firstly, approximating the distributed-order operator D^(w)_(t)u by Gauss integral, then translating the initial problem into a multi-term time fractional diffusion equation.Secondly, the fully discrete approximation scheme of the variational form is constructed by the correctional L1 scheme, the new MFEMs(the bilinear element Q_(11)(K) and Q_(01)(K)×Q_(10)(K)) and the intermediate variable p.Moreover, the stability analysis of the fully discrete scheme is derived by applying some related properties of the MFEMs.Finally, the superclose and superconvergence results are obtained by using the high precision results of the conforming MFEMs and the interpolation postprocessing technique.