For time-averaged outputs (mean drag, lift) of high-Reynolds-number incompressible turbulent flow, the general Galerkin (G2) framework of Hoffman and Johnson computes outputs to a few percent with duality-based a posteriori error estimates, yet its guarantees rest on an unexplained empirical fact: the stability factors of the linearized dual (adjoint) problem stay moderate because of cancellation in oscillatory dual coefficients that, in the source's words, seems impossible to account for in purely theoretical estimates. The open problem is to give this observation a rigorous foundation: derive quantitative bounds connecting the statistics of a turbulent flow to the growth of its dual problem, obtain a posteriori output-error and epsilon-h computability estimates free of ad hoc assumptions such as boundedness of the discrete maximum velocity, and delineate the flows, outputs, and perturbations for which certified accuracy is possible — or prove such guarantees impossible.

Background

A fluid's Reynolds number ReRe measures the ratio of inertial to viscous forces; turbulence sets in at large ReRe, where direct numerical simulation (DNS, resolving all scales down to viscous dissipation) becomes astronomically expensive, and large-eddy simulation (LES, resolving large scales and modeling subgrid scales) introduces closure errors. The mathematical substrate is also insecure: for the three-dimensional incompressible Navier–Stokes equations it is still unknown whether smooth initial data guarantee smooth solutions for all time, so rigorous numerical analysis of turbulent flow must work with weak solutions (distributional solutions obtained as limits of approximations, without guaranteed uniqueness).

Against this background, Hoffman and Johnson proposed what they call general Galerkin (G2) methods: stabilized weak Galerkin finite element discretizations of the Navier–Stokes equations with no explicit turbulence model, where the error in a chosen output functional — a scalar quantity of interest such as the time average of drag — is estimated a posteriori, i.e. using the computed solution itself. In this goal-oriented framework (systematized by Becker and Rannacher, and surveyed by Giles and Süli), the error in the output is represented as an inner product of the computable residual with the solution of a dual, or adjoint, problem: a linearized Navier–Stokes system run backward in time, whose data encode the output. The size of the dual solution — the stability factor — converts residual errors into the output error; a small stability factor means the output is insensitive to fine-scale error, so coarse-mesh computation can be accurate even when pointwise solution errors are large.

Computations in this program showed a striking regularity: for time-averaged outputs of turbulent bluff-body flow, stability factors stay moderate enough that mean drag can be predicted to a few percent with about 10510^5 mesh points, and the estimated subgrid modeling contribution is small. The reason is apparently strong cancellation in the rapidly oscillating coefficients of the dual problem over the long averaging time — cancellation that worst-case theoretical bounds cannot capture, and that the authors report seems impossible to account for in purely theoretical estimates. The rigorous results in the program (existence of discrete weak solutions and epsilon-h relations connecting output tolerance ϵ\epsilon to mesh size hh) hold only under auxiliary assumptions, notably boundedness — or growth slower than h−1/2h^{-1/2} — of the discrete maximum velocity M(U)M(U), which is not derived from first principles. The updated second-edition version of the chapter by the same group in 2017 presents the same framework with expanded demonstrations and does not close these gaps. A conceptually adjacent line of research on adjoint sensitivity of chaotic dynamical systems — least-squares shadowing (LSS) and its non-intrusive variant (NILSS) — likewise observes that instantaneous adjoints of chaotic systems oscillate and grow while time-averaged sensitivities remain computable, but its rigorous linear-response theory covers uniformly hyperbolic systems, not Navier–Stokes turbulence.

Thus the empirical success of dual-based error control for turbulent outputs stands on an unproven foundation: nobody has explained, in rigorous terms, why the dual problems of turbulent flows are as benign as computations show, nor delimited when they are.

Problem Statement

Establish a rigorous theory of output-based computability for high-Reynolds-number incompressible turbulent flow, in the setting of the Hoffman–Johnson G2 framework, by resolving three jointly supporting questions whose common objective is to convert the empirically observed accuracy of dual-based error control into certified guarantees. The theory comprises either the affirmative results of (1)–(3), or a rigorous demonstration in place of (2) that no such certified guarantees can be established, together with the results of (1) and (3):

  1. Dual stability factors. Explain, and rigorously bound, the growth of solutions of the linearized dual (adjoint) Navier–Stokes problem for time-averaged outputs of turbulent flow: give quantitative results connecting the statistics of a turbulent flow to its dual stability factors, capturing the cancellation in the oscillatory coefficients of the dual problem that keeps average-output stability factors moderate — or prove that no such bounds hold in general (an out-of-scope analogy only: computational evidence of dual blow-up in some compressible regimes hints that uniform bounds may fail, but does not settle the incompressible case).
  2. Assumption-free computability estimates. Derive rigorous a posteriori output-error bounds and ϵ\epsilon–hh relations (connecting output tolerance ϵ\epsilon to mesh size hh and computational work) for weak Galerkin/G2 approximations of turbulent flow without ad hoc assumptions such as boundedness, or slower-than-h−1/2h^{-1/2} growth, of the discrete maximum velocity M(U)M(U), so that a guarantee of the form 'the output is computed within tolerance ϵ\epsilon at resolution hh' rests only on stated properties of the equations and the discretization.
  3. Scope of certified computability. Delineate the class of flows, outputs, boundary conditions, and perturbations for which the dual-based a posteriori framework yields reliable computability assessments, distinguishing regimes where outputs are computable on feasible meshes from those (such as those exhibiting dual blow-up) where they are not.

References

  1. J. Hoffman and C. Johnson, 'Computability and Adaptivity in CFD', Encyclopedia of Computational Mechanics, Wiley (2004; online 2007), DOI 10.1002/0470091355.ecm057.pub2, https://doi.org/10.1002/0470091355.ecm057.pub2
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