时间分数阶非线性Rayleigh-Stokes问题的数值解
Numerical solution of a time-fractional nonlinear Rayleigh-Stokes problem
摘要：
本文研究了具有Lipschitz连续非线性源项和初始数据$u_0\in\dot{H}^ν(Ω)$，$ν\in[0,2]$的广义二阶流体的半线性分数时间Rayleigh-Stokes问题.我们讨论的解决方案的稳定性，并提供正则性结果。分别基于标准Galerkin和集中质量有限元法分析了两种空间半离散格式。此外，一个完全离散的计划，通过应用卷积正交的时间产生的向后欧拉法，并获得最佳的误差估计光滑和非光滑的初始数据。最后，数值例子来说明理论结果。
英文摘要：
We study a semilinear fractional-in-time Rayleigh-Stokes problem for a generalized second-grade fluid with a Lipschitz continuous nonlinear source term and initial data $u_0\in\dot{H}^ν(Ω)$, $ν\in[0,2]$. We discuss stability of solutions and provide regularity results. Two spatially semidiscrete schemes are analyzed based on standard Galerkin and lumped mass finite element methods, respectively. Further, a fully discrete scheme is obtained by applying a convolution quadrature in time generated by the backward Euler method, and optimal error estimates are derived for smooth and nonsmooth initial data. Finally, numerical examples are provided to illustrate the theoretical results.
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