High accuracy error estimates of a Galerkin finite element method for nonlinear time fractional diffusion equation
High accuracy error estimates of a Galerkin finite element method for nonlinear time fractional diffusion equation
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DOI：
10.1002/num.22428
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发表时间：
2019-09
影响因子：
3.9
通讯作者：
Jincheng Ren;D. Shi;Seakweng Vong
中科院分区：
文献类型：
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作者：
Jincheng Ren;D. Shi;Seakweng Vong
在这项工作中，一个有效的和快速的有限元数值方法与高阶精度进行了讨论，解决非线性时间分数阶扩散方程。构造了一个两层线性化有限元格式，并建立了时空误差分裂参数，将误差分裂为时间误差和空间误差两部分。基于时间离散系统的正则性，导出了时间误差估计。利用里兹投影算子的性质，推导出了空间误差的表达式.在不对问题的精确解作任何额外的正则性假设的情况下，得到了H1范数下的无条件超逼近结果.然后通过插值后处理技术得到全局超收敛误差估计。为了减少存储和计算时间，提出了一种求解非线性时间分数阶扩散方程的快速有限元计算格式。为了证实理论误差分析，提供了一些数值结果。
In this work, an effective and fast finite element numerical method with high‐order accuracy is discussed for solving a nonlinear time fractional diffusion equation. A two‐level linearized finite element scheme is constructed and a temporal–spatial error splitting argument is established to split the error into two parts, that is, the temporal error and the spatial error. Based on the regularity of the time discrete system, the temporal error estimate is derived. Using the property of the Ritz projection operator, the spatial error is deduced. Unconditional superclose result in H1‐norm is obtained, with no additional regularity assumption about the exact solution of the problem considered. Then the global superconvergence error estimate is obtained through the interpolated postprocessing technique. In order to reduce storage and computation time, a fast finite element method evaluation scheme for solving the nonlinear time fractional diffusion equation is developed. To confirm the theoretical error analysis, some numerical results are provided.