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Dynamic subscales and unconditional semi-discrete stability for non-residual VMS approximations of generalised Newtonian flows
G. R. Barrenechea, E. Castillo, R. Codina, J. Pastene, P. Vega
TL;DR
The paper studies the missing semi-discrete stability analysis for dynamic non-residual VMS approximations of incompressible generalised Newtonian flows. It uses orthogonal projections and dynamic subscales to separate unresolved mechanisms, proving unconditional linearised stability and nonlinear existence and error bounds. The pressure-related dynamic subscale supplies the time-derivative control needed for the projected pressure-gradient error.
Problem
Semi-discrete stability and error analysis is missing for dynamic non-residual VMS separation of pressure-gradient effects in nonlinear shear-dependent incompressible flows.
Method
The method uses orthogonal projections, term-by-term unresolved contributions, dynamic velocity subscales, and pressure stabilisation to separate pressure-divergence, convection, and pressure-gradient mechanisms.
Results
The linearised problem has well-posedness and unconditional stability in an anisotropic VMS norm, while the nonlinear formulation has fixed-point existence, optimal-order a priori bounds, and weak-in-time pressure-convection control.
Takeaways & Limitations
The pressure-related dynamic subscale controls the projected pressure-gradient contribution, avoiding the large-time restriction associated with quasi-static subscale formulations.
Takeaways & Limitations
The formulation uses equal-order continuous velocity-pressure interpolation for conciseness, although other conforming approximations could be accommodated with additional terms for discontinuous pressures.
Abstract
from arXiv · showhide
We analyse a non-residual variational multiscale finite element formulation with dynamic subscales for incompressible generalised Newtonian Navier--Stokes flows. The apparent viscosity is assumed to be bounded and Lipschitz continuous with respect to the shear rate, covering several regularised rheological laws. The method separates, through orthogonal projections, the unresolved pressure-divergence, convective, and pressure-gradient contributions. For the linearised semi-discrete problem, we prove well-posedness and an unconditional stability estimate in an anisotropic VMS norm. The estimate controls viscous dissipation, subscale energies, discrete divergence, and a coupled acceleration-convection-pressure balance. For the non-linear formulation, a fixed-point argument yields existence and optimal-order a priori bounds under suitable regularity and smallness assumptions. The analysis also provides weak-in-time control of the discrete pressure-convection balance. A key point is that the dynamic pressure-related subscale provides the time-derivative contribution needed to control the projected pressure-gradient error, a mechanism unavailable in the corresponding quasi-static setting.
1. Introduction.
The paper addresses the missing semi-discrete stability analysis for dynamic non-residual VMS approximations of generalised Newtonian flows. It separates unresolved mechanisms through orthogonal projections and establishes stability and error results, with dynamic pressure subscales controlling the projected pressure-gradient contribution.
- Model class: The analysis targets bounded, Lipschitz shear-dependent viscosities, including regularised Carreau, Cross, Quemada, and Eyring-type laws.These assumptions provide uniformly elliptic diffusion and Lipschitz control of nonlinear viscosity terms.
- Motivation: Dynamic subscales retain time derivatives, avoiding the reverse CFL restriction associated with quasi-static subscales in transient stability analysis.The reverse condition is ∆t ≥ Ch^2, and retaining the subscale derivative restores robustness with respect to anisotropic time-space discretisation.
- Research gap: Existing analyses do not show how dynamic non-residual VMS subscales control pressure gradients in nonlinear shear-dependent flows.
- Method: The formulation separates unresolved pressure-divergence, convective, and pressure-gradient mechanisms through orthogonal projections and dynamic subscales.It uses two dynamic velocity subscales and pressure stabilisation acting on discrete divergence.
- Contributions: The paper proves unconditional semi-discrete stability for the linearised problem and develops well-posedness, fixed-point existence, and optimal-order bounds for the nonlinear formulation.The stability estimate controls viscous dissipation, subscale energies, discrete divergence, and a coupled acceleration-convection-pressure balance.
2. Notation and preliminaries.
This section establishes the functional, mesh, projection, and finite-element notation used in the analysis. It also records approximation, stability, and skew-symmetric convection properties needed later.
- Functional setting: The analysis uses Lebesgue and Sobolev spaces W^{m,p}(ω), with H^m(ω) denoting W^{m,2}(ω) and H^1_0(ω) imposing vanishing boundary trace.H^-1(ω) is the dual of H^1_0(ω), with the corresponding duality pairing denoted by ⟨·,·⟩_ω.
- Functional setting: Norms and inner products are specified over Ω, including L^2, L^4, L^∞, H^k, and W^{ℓ,p} quantities.The notation also distinguishes W^{ℓ,p} seminorms and uses ||·|| for the L^2(Ω)-norm.
- Differential notation: The symmetric gradient is denoted by ∇^s v, while ∂t denotes the time derivative.
- Analytic inequalities: Poincaré–Friedrichs, Korn, and Sobolev embedding inequalities provide domain-dependent constants used in the subsequent estimates.
- Convection: The skew-symmetric convective form satisfies c(u,v,v)=0 and the bound |c(u,v,w)| ≤ (C_emb)^2 ||u||_1 ||v||_1 ||w||_1.This form is used because discrete velocity fields are not generally pointwise divergence-free.
- Meshes and finite elements: Quasi-uniform meshes use h = max{h_K}, with element diameters bounded above and below proportionally to h; polynomial degree k remains fixed under h-refinement.Inverse inequalities are applied to finite-element functions on each element.
- Projection tools: The L^2-projection approximation estimate is ||v − P[v]||_p ≤ C h^ℓ |v|_{ℓ,p} for 0 ≤ ℓ ≤ k + 1 and 1 ≤ p ≤ ∞.The constant C is independent of h.
- Projection tools: Orthogonal projections P_u and P_p, together with complementary projections P_u^⊥ and P_p^⊥, separate resolved and unresolved velocity and pressure components.
3. Continuous problem and finite element formulation.
The section formulates incompressible generalised Newtonian flow with bounded, Lipschitz apparent viscosity and develops a dynamic non-residual VMS finite element approximation. Orthogonal subscale projections separate pressure-divergence, convective, and pressure-gradient effects, including dynamic velocity subscales for stabilisation.
- 3. Continuous problem and finite element formulation.: The continuous problem couples unsteady transport, incompressibility, and nonlinear diffusion through shear-rate-dependent apparent viscosity.The model is posed on a bounded polyhedral domain for incompressible, isothermal flow.
- 3.1. Governing equations.: The viscosity is Lipschitz continuous in shear rate and bounded between the positive constants µ∞ and µ0.These assumptions provide uniform ellipticity and control of nonlinear viscosity terms.
- 3.1. Governing equations.: Carreau, Carreau–Yasuda, Cross, Quemada, and Eyring-type regularised laws satisfy the stated bounded-viscosity assumptions.The viscosity remains bounded even when the velocity gradient is unbounded.
- 3.2. Weak formulation.: The weak formulation combines time evolution, convection, viscous stress, pressure-velocity coupling, incompressibility, and forcing.The finite element approximation uses conforming velocity and pressure spaces Vh and Qh.
- 3.3. Galerkin finite element approximation.: Galerkin discretisation controls viscous dissipation but requires discrete inf-sup stability for pressure and additional stabilisation in convection-dominated regimes.The proposed construction addresses unresolved velocity scales using dynamic orthogonal subscales.
- 3.3. Galerkin finite element approximation.: The VMS method separates unresolved pressure-divergence, convective, and pressure-gradient contributions through orthogonal projections.This separation identifies which unresolved component controls each part of the discrete stability analysis.
- 3.4. Non-residual VMS formulation with dynamic subscales.: The formulation introduces two dynamic velocity subscales for convection and pressure gradients, while the pressure subscale is condensed into pressure stabilisation.The method is posed for equal-order continuous interpolation spaces, with dynamic subscale initial data obtained by projection.
- 3.4. Non-residual VMS formulation with dynamic subscales.: The semi-discrete analysis isolates stabilisation mechanisms for the linearised problem and uses a stabilised norm with a fixed-point argument for the nonlinear problem.The stabilisation parameters are time-dependent; fully discrete use requires consistent temporal extrapolation of their nonlinear dependence.
4. Numerical analysis.
The semi-discrete analysis establishes well-posedness and unconditional anisotropic stability for the linearised formulation, then obtains nonlinear existence and optimal a priori bounds through a fixed-point argument. The dynamic pressure-related subscale supplies the time-derivative control needed for projected pressure-gradient errors and supports an additional weak-in-time balance estimate.
- Linearised problem: The linearised semi-discrete problem is reformulated on a constrained finite-dimensional space using the resolved velocity and pressure-gradient subscale.The pressure is recovered separately after solving an ordinary differential equation for the velocity variables.
- Linearised problem: The linearised problem has a unique solution with resolved velocity and subscales bounded in time and the resolved velocity controlled in H1.The stated existence result assumes f ∈ L2(0, T; H−1(Ω)d) and u0 ∈ L2(Ω)d.
- Linearised stability: The anisotropic VMS norm controls viscous and subscale dissipation together with the coupled acceleration–convection–pressure-gradient balance.The estimate is obtained before temporal discretisation, so no time-step parameter enters the semi-discrete argument.
- Linearised stability: The stability bound is uniform with respect to the mesh size h for the resolved velocity and both dynamic subscales.The controlled quantities include uh, ˜u1, and ˜u2 in L∞(0, T; L2(Ω)d).
- Nonlinear problem: The nonlinear formulation uses an iterative map that is bounded on bounded sets, has an invariant ball, and yields a discrete solution by Brouwer’s fixed-point theorem.The resulting existence and optimal a priori error estimate requires suitable assumptions, sufficiently large µ∞, and sufficiently small constants D1, …, D5.
- Nonlinear problem: The dynamic pressure-related subscale provides the time-derivative contribution that controls the projected pressure-gradient error, unlike quasi-static subscales.The analysis also identifies an additional weak-in-time stability property for the discrete pressure-gradient and convective contributions.
5. Weak-in-time control of the pressure-convection balance.
The analysis establishes weak-in-time control of the discrete pressure-convection balance for the nonlinear VMS formulation. The resolved component is controlled through the momentum equation and stability estimate, while dynamic subscale equations control the unresolved component.
- The resulting estimate extends weak-in-time pressure-convection control to the nonlinear generalised Newtonian setting.The paper identifies this as an additional stability result consistent with the term-by-term structure of the non-residual VMS formulation.
- Theorem 5.1 establishes a weak-in-time pressure-convection estimate for the nonlinear VMS solution with forcing f ∈ L2(0, T; H−1(Ω)d).The estimate applies in dimensions d = 2, 3, with a constant independent of h.
- The proof estimates the resolved component of ∇p_h+N(u_h,u_h) through the momentum equation and bounds convective, diffusive, pressure-divergence, and forcing terms.The argument combines estimates (5.5)–(5.9) with the stability estimate and projection inequalities.
- Adding the dynamic velocity subscale equations and integrating by parts in time yields control of the unresolved contribution.The argument uses the stability estimate, the definition of τ0, and compatibility assumption (3.25).
- The pressure-convection balance is controlled by combining resolved and unresolved component estimates.The resolved component uses the momentum equation and stability estimate, whereas the unresolved component is controlled directly by the dynamic subscale equations.
6. Conclusions.
The paper analyses a dynamic non-residual VMS formulation for incompressible generalised Newtonian flows and proves semi-discrete stability and nonlinear existence results. Its analysis identifies the dynamic pressure-related subscale as the mechanism controlling the projected pressure-gradient error, while fully discrete analysis remains future work.
- The linearised semi-discrete problem is well posed and unconditionally stable in an anisotropic VMS norm without restrictions on the time interval.The norm controls the principal stability quantities identified in the analysis.
- For the nonlinear formulation, a fixed-point argument yields a discrete solution and optimal-order a priori estimates under suitable regularity and smallness assumptions.
- The dynamic pressure-related subscale provides the control needed for the projected pressure-gradient error, avoiding the large-time restriction associated with quasi-static subscales.
- Fully discrete analysis under anisotropic time-space discretisations remains a natural continuation of the semi-discrete theory.The paper suggests combining implicit-explicit strategies with the dynamic non-residual VMS formulation as a possible direction.