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An Analysis of the Rayleigh-Stokes problem for a Generalized Second-Grade Fluid
Emilia Bazhlekova, Bangti Jin, Raytcho Lazarov, Zhi Zhou
TL;DR
The paper addresses numerical analysis of the fractional Rayleigh–Stokes problem for a generalized second-grade fluid, including regularity for smooth and nonsmooth initial data. It develops Galerkin finite element and convolution-quadrature schemes, obtaining data-regularity-optimal error estimates and numerical confirmation of the convergence analysis.
Problem
The study concerns efficient numerical approximation of a fractional Rayleigh–Stokes model whose closed-form solutions are inconvenient for numerical evaluation and available only for restricted settings.
Method
The paper establishes Sobolev regularity and develops a continuous piecewise linear Galerkin finite element method with backward Euler and second-order backward difference convolution quadrature.
Results
The semidiscrete and fully discrete schemes achieve error estimates optimal with respect to initial-data regularity for smooth and nonsmooth data, with numerical experiments confirming the convergence analysis.
Takeaways & Limitations
The analysis provides a unified treatment of regularity and discretization errors for smooth and nonsmooth initial data in this fractional fluid model.
Abstract
from arXiv · showhide
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 $v\in L^2(Ω)$. 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.
1. Introduction
The paper develops and analyzes finite element and convolution-quadrature methods for a fractional Rayleigh–Stokes problem in generalized second-grade fluids. It establishes regularity and data-regularity-optimal error estimates for smooth and nonsmooth initial data, supported by numerical experiments.
- The model uses a Riemann–Liouville fractional derivative to describe viscoelastic behavior in generalized second-grade, non-Newtonian fluids.
- Closed-form solutions involve infinite series and special functions, making numerical evaluation inconvenient and motivating efficient numerical algorithms.
- The paper develops a Galerkin finite element method using continuous piecewise linear functions over shape-regular quasi-uniform meshes.
- The semidiscrete analysis covers smooth v ∈ ˙H2(Ω) and nonsmooth v ∈ L2(Ω), using Ritz and L2 projections for the initial data.
- Backward Euler and second-order backward difference convolution-quadrature schemes provide respectively first- and second-order time accuracy, with data-regularity-optimal error estimates.
- The paper establishes solution regularity, analyzes semidiscrete and fully discrete schemes, and uses one- and two-dimensional numerical examples to illustrate convergence theory.
2. Regularity of the solution
The paper establishes Sobolev regularity, existence, stability, and qualitative time behavior for the homogeneous Rayleigh–Stokes problem, including nonsmooth initial data in L2(Ω). The solution lacks a smoothing property and exhibits fractional short-time behavior with algebraic long-time decay.
- Existence and regularity: The operator-theoretic analysis establishes a unique homogeneous solution for every v ∈L2(Ω) together with regularity estimates.The analysis uses solution-operator representations, Laplace transforms, sectorial operator bounds, and interpolation.
- Qualitative behavior: The time-dependent eigencomponents are completely monotone, positive, and monotonically decreasing for α ∈(0,1) and positive λj, γ.Complete monotonicity follows from positivity of Kj(r) and Bernstein’s theorem.
- Qualitative behavior: The solution does not have a smoothing property, despite its positive-time Sobolev regularity.The paper states this explicitly after deriving the eigenfunction representation.
- Qualitative behavior: For v ∈L2(Ω), ∥u(t)∥˙H2(Ω) behaves like t^(α−1) as t →0, while u(t) decays like t^−1 as t →∞.The short-time behavior matches subdiffusion, whereas the long-time decay is algebraic and slower than diffusion’s exponential decay.
- Existence and regularity: For any t > 0, the solution belongs to ˙H2+q(Ω) when the initial data belongs to ˙Hq(Ω), while converging to v in ˙Hq(Ω) as t →0.This gives spatial regularity at positive times and recovery of the initial data in the corresponding Sobolev scale.
3. Semidiscrete Galerkin Finite element method
The paper develops a continuous piecewise linear finite element Galerkin method for the homogeneous problem and derives optimal semidiscrete error estimates across initial-data regularity classes. The analysis treats nonsmooth, smooth, and very weak initial data using projection operators, resolvent estimates, and interpolation.
- Method: The space semidiscrete method uses a finite element space Xh with L2 and Ritz projections and a discrete Laplacian Ah.The semidiscrete solution operator Sh(t) is represented through Ah and the same contour framework used for the continuous problem.
- Error estimates: The operator-trick analysis avoids the additional logarithmic factor | ln h| that can arise from an alternative technique for nonsmooth data.The paper explicitly contrasts the two proof approaches in the semidiscrete analysis.
- Error estimates: For nonsmooth initial data v ∈L2(Ω), Theorem 3.1 gives semidiscrete error estimates with vh = Phv.Interpolation extends the estimate to all q ∈[0,2], while very weak data with −1 < q < 0 is also treated.
- Error estimates: For smooth initial data v ∈˙H2(Ω), Theorem 3.2 gives the corresponding semidiscrete estimate with vh = Rhv.The proof uses the resolvent estimate together with the identity AhRh = PhA.
4. Fully discrete schemes
The paper develops backward Euler and second-order backward difference convolution-quadrature schemes for the fully discrete problem, with error analyses covering smooth and nonsmooth initial data. The SBD method requires correction to attain second-order accuracy, while the estimates reveal initial-time singular behavior.
- Fully discrete schemes: The fully discrete schemes combine convolution quadrature with the spatial semidiscrete Galerkin approximation and analyze errors for smooth and nonsmooth initial data.The analysis includes v ∈ L2(Ω), v ∈ ˙H2(Ω), and intermediate Sobolev regularity through interpolation.
- Convolution quadrature: Convolution quadrature discretizes the fractional convolution in time using multistep methods, here backward Euler and second-order backward difference.The time interval is divided into equal steps, and the resulting discrete convolution is rewritten in time-stepping form.
- Second-order backward difference method: The SBD method is only first-order accurate when g(0) ≠ 0, so a correction is introduced to recover second-order accuracy.The correction follows an established approach and preserves the initial-value contribution needed for second-order convergence.
- Nonsmooth initial data: For v ∈ L2(Ω), the fully discrete analyses provide error estimates for both backward Euler and SBD schemes, with the latter governed by second-order convolution-quadrature bounds.The theorem and lemma sequence treats nonsmooth initial data separately for the two time-stepping methods.
- Backward Euler error analysis: The error estimates exhibit a t^-α singularity as t →0+, even for smooth initial data v ∈ ˙H2(Ω).As α →0+, the problem approaches the standard parabolic equation and this singular behavior disappears for smooth data.
5. Numerical results
Numerical experiments in one and two dimensions test temporal and spatial convergence for smooth, nonsmooth, and very weak initial data. The observed rates agree with the theoretical predictions, while errors near t=0 deteriorate for nonsmooth data.
- Experimental setup: The experiments use one- and two-dimensional domains with smooth, nonsmooth, very weak, and two-dimensional nonsmooth initial data.Errors are normalized in L2(Ω) and H1(Ω) norms, using continuous piecewise linear finite elements and uniform temporal meshes.
- Smooth initial data: For smooth data, backward Euler and second-order backward difference methods show temporal rates O(τ) and O(τ^2), respectively.The spatial mesh is fixed at h = 2−11 so temporal discretization dominates; the error decreases as α increases.
- Smooth initial data: Smooth-data spatial convergence is O(h^2) in L2(Ω) and O(h) in H1(Ω), with empirical rates almost independent of α.These results are reported for the backward Euler scheme at t = 0.1.
- Nonsmooth initial data: For nonsmooth data, temporal rates remain O(τ) for backward Euler and O(τ^2) for second-order backward difference, while spatial rates remain O(h^2) in L2(Ω) and O(h) in H1(Ω).The predicted rates are confirmed at t = 0.1, 0.01, and 0.001.
- Behavior near t = 0: As t → 0, spatial error stays essentially unchanged for smooth data but grows like O(t−3α/4) for nonsmooth data.For α = 0.5, the predicted empirical rate is −3α/4 = −0.375.
- Very weak and two-dimensional data: Very weak one-dimensional data exhibit superconvergence at rates O(h^2) in L2(Ω) and O(h) in H1(Ω), while two-dimensional tests confirm first- and second-order temporal and O(h^2)/O(h) spatial rates.The superconvergence is attributed to solution smoothness on both sides of the Dirac support and good finite-element approximation properties.
6. Concluding remarks
The study analyzes the homogeneous Rayleigh–Stokes problem using operator-theoretic regularity results and Galerkin-based discretizations. Numerical error plots support the computational study across fractional orders and times.
- The solution Sobolev regularity was established using an operator-theoretic approach.
- A space semidiscrete Galerkin finite element scheme was developed for the problem.
- Backward Euler and second-order backward difference schemes with convolution quadrature were developed for fully discrete approximation.
- Figure 5 presents error plots for α = 0.1, 0.5, 0.9 at t = 0.1, 0.01, and 0.001 with N = 1000.