Numerical solution of a time-fractional nonlinear Rayleigh-Stokes problem

TL;DR

This paper investigates numerical methods for solving a time-fractional nonlinear Rayleigh-Stokes problem, providing stability, regularity, and error estimates for finite element and convolution quadrature schemes, supported by numerical experiments.

Contribution

It introduces and analyzes a fully discrete numerical scheme combining finite element methods with convolution quadrature for the fractional Rayleigh-Stokes problem, including optimal error estimates.

Findings

Stability and regularity results for solutions.

Optimal error estimates for discretization schemes.

Numerical experiments confirming theoretical predictions.

Abstract

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.