Caputo型时间分数阶变系数扩散方程的局部间断Galerkin方法

Local discontinuous Galerkin finite element method for the Caputo-type diffusion equation with variable coefficient

提出一种带有Caputo导数的时间分数阶变系数扩散方程的数值解法.方程的解在初始时刻附近通常具有弱正则性,采用非一致网格上的L1公式离散时间分数阶导数,并使用局部间断Galerkin(local discontinuous Galerkin,LDG)方法离散空间导数,给出方程的全离散格式.基于离散的分数阶Gronwall不等式,证明了格式的数值稳定性和收敛性,且所得结果关于α是鲁棒的,即当α→1^(-)时不会发生爆破.最后,通过数值算例验证理论分析的结果.

We present an efficient method for seeking the numerical solution of a Caputotype diffusion equation with a variable coefficient.Since the solution of such an equation is likely to have a weak singularity near the initial time,the time-fractional derivative is discretized using the L1 formula on nonuniform meshes.For spatial derivative,we employ the local discontinuous Galerkin method to derive a fully discrete scheme.Based on a discrete fractional Gronwall inequality,the numerical stability and convergence of the derived scheme are proven which are both α-robust,that is,the bounds obtained do not blow up as α→1^(-).Finally,numerical experiments are displayed to confirm the theoretical results.