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 $u_0\in\dot{H}^\nu(\Omega)$, $\nu\in[0,2]$. 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.
翻译:我们用利普西茨连续的非线性源术语和初始数据来研究通用二级流体的半线性分数-时间雷利-斯托克斯问题。我们讨论了解决方案的稳定性,并提供了定期性结果。我们分别根据标准Galerkin和整块质量限制元素方法分析了两种空间半分解方案。此外,通过对落后的Euler方法产生的时间在时间上进行演进二次计算,获得了完全独立的方案,并对光滑和非光滑的初步数据得出了最佳误差估计。最后,提供了数字例子来说明理论结果。