Abstract
We study the Rayleigh–Stokes problem for a generalized second-grade fluid which involves a Riemann–Liouville fractional derivative in time, and present an analysis of the problem in the continuous, space semidiscrete and fully discrete formulations. We establish the Sobolev regularity of the homogeneous problem for both smooth and nonsmooth initial data v, including vin L^2(Omega ). A space semidiscrete Galerkin scheme using continuous piecewise linear finite elements is developed, and optimal with respect to initial data regularity error estimates for the finite element approximations are derived. Further, two fully discrete schemes based on the backward Euler method and second-order backward difference method and the related convolution quadrature are developed, and optimal error estimates are derived for the fully discrete approximations for both smooth and nonsmooth initial data. Numerical results for one- and two-dimensional examples with smooth and nonsmooth initial data are presented to illustrate the efficiency of the method, and to verify the convergence theory.
Highlights
In this paper, we study the homogeneous Rayleigh–Stokes problem for a generalized second-grade fluid with a fractional derivative model
In this paper we develop a Galerkin finite element method (FEM) for problem (1.1) and derive optimal with respect to data regularity error estimates for both smooth and nonsmooth initial data
We develop two fully discrete schemes based on the backward Euler method and the second-order backward difference method and the related convolution quadrature for the fractional derivative term, which achieves respectively first and second-order accuracy in time
Summary
We study the homogeneous Rayleigh–Stokes problem for a generalized second-grade fluid with a fractional derivative model. Lubich et al [15] developed two fully discrete schemes for the problem ∂t u + ∂t−α Au = f with u(0) = v and 0 < α < 1 based on the convolution quadrature of the fractional derivative term, and derived optimal error estimates for nonsmooth initial data and right hand side. In Theorem 2.1, using an operator approach from [25], we develop the theoretical foundations for our study by establishing the smoothing property and decay behavior of the solution to problem (1.1) For both smooth initial data v ∈ H 2( ) and nonsmooth initial data v ∈ L2( ), we derive error estimates for the space semidiscrete scheme, cf Theorems 3.1 and 3.2: u(t) − uh(t) L2( ) + h ∇(u(t) − uh(t)) L2( ) ≤ ch2t (q/2−1)(1−α) v Hq ( ), q = 0, 2. Throughout, the notation c denotes a constant which may differ at different occurrences, but it is always independent of the solution u, mesh size h and time step-size τ
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