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Smooth-branch velocity error for backward Euler MINI

Statement

Assume Countable Choice. Assume the MINI mesh and time-partition hypotheses of The mixed mini-element backward Euler Navier–Stokes scheme. Suppose $(u, p)$ is a strong solution of the no-slip Navier–Stokes equations with forcing $f \in L^{2} (0, T; L^{2} (D)^{d})$ and $u \in W^{1, \infty} (0, T; H^{2} (D)^{d} \cap V_{σ})$ and $p \in W^{1, \infty} (0, T; H^{1} (D) \cap L_{0}^{2} (D))$ . Set $M : = 1 + ∥ u ∥_{W^{1, \infty} H^{2}} + ∥ p ∥_{W^{1, \infty} H^{1}}, M_{u} : = ∥ u ∥_{L^{\infty} V},$ $M_{\infty} : = (∥ u_{0} ∥_{2}^{2} + C_{P} (D)^{2} ν^{- 1} ∥ f ∥_{L^{2} L^{2}}^{2})^{1 / 2}, M_{E} : = ν^{- 1 / 2} ∥ u_{0} ∥_{2} + C_{P} (D) ν^{- 1} ∥ f ∥_{L^{2} L^{2}},$ and $Λ : = 1 + C_{D} ν^{- 3} M_{u}^{4}$ , where $C_{D}$ depends only on the domain, dimension and the trilinear-form constant. For sufficiently fine time partitions satisfying $τ_{\max} Λ \leq \frac{1}{2}$ , initialize by $u_{h}^{0} = P_{Z_{h}} u_{0}$ and choose any backward Euler MINI solution. Then $\max_{0 \leq n \leq N} ∥ u (t_{n}) - u_{h}^{n} ∥_{2} + ν^{1 / 2} {(\sum_{n = 1}^{N} τ_{n} ∥ u (t_{n}) - u_{h}^{n} ∥_{V}^{2})}^{1 / 2} \leq C_{*} (h M + τ_{\max} M^{2}),$ where one may take $C_{*} : = C_{D, s h a p e} (1 + ν)^{2} (1 + ν^{- 4}) (1 + T)^{2} (1 + M_{\infty} + M_{E} + M)^{4} \exp (2 T Λ) .$ Here $f$ is the forcing of the strong solution and $C_{D, s h a p e}$ is independent of $h$ , all time steps, and $ν$ . The polynomial factor displays the positive and inverse viscosity powers produced by Young's inequalities and static Stokes projection. The explicit factor $ν^{- 3}$ in $Λ$ and exponential dependence on $T Λ$ are retained; no Reynolds-uniform estimate is asserted. The separate instantaneous pressure assertion is addressed in Remarks.

Facts & Assumptions

Given: Countable Choice, a bounded connected Lipschitz polyhedral domain, a shape-regular MINI mesh family, a partition of $[0, T]$ , viscosity $ν > 0$ , and the strong solution and initialization in the Statement.

[F1]

The exact solution satisfies the time-integrated mixed weak equation; for tests $z_{h} \in Z_{h}$ its pressure contribution is $b (z_{h}, {\overset{ˉ}{p}}_{n})$ . (The mixed mini-element backward Euler Navier–Stokes scheme, Energy weak solutions of incompressible Navier–Stokes)

[F2]

The linear solenoidal energy projection $S_{h}$ satisfies $∥ u - S_{h} u ∥_{V} \leq C_{0} (1 + ν^{- 1}) h (∥ u ∥_{H^{2}} + ∥ p ∥_{H^{1}})$ and the same estimate applies to $(u_{t}, p_{t})$ . (Approximation by the MINI solenoidal energy projection)

[F3]

Mean-adjusted face-average interpolation maps $H^{1} (D)$ to $Q_{h}$ with $∥ p - r_{h} ∥_{2} \leq C h ∥ p ∥_{H^{1}}$ . (Boundary-preserving face-average P1 interpolation, The conforming P1-bubble/P1 mini element)

[F4]

The skew form is trilinear and continuous, obeys $∣ c (a; b, w) ∣ \leq C_{D} ∥ a ∥_{2}^{1 / 2} ∥ a ∥_{V}^{1 / 2} ∥ b ∥_{V} ∥ w ∥_{V}$ , and $c (a; w, w) = 0$ . (Continuity, energy cancellation, and Sobolev bounds for skew convection, The skew convective form for incompressible flow)

[F5]

The discrete solution obeys $\max_{n} ∥ u_{h}^{n} ∥_{2} \leq M_{\infty}$ and $\sum_{n} τ_{n} ∥ u_{h}^{n} ∥_{V}^{2} \leq M_{E}^{2}$ . (Backward Euler MINI step existence and discrete energy bound)

[F6]

The L2 projector $P_{Z_{h}}$ onto the weak MINI kernel defined in Weak and pointwise divergence-free discrete velocities exists and is a best approximation in $L^{2}$ . (Discrete Helmholtz projection and pressure Schur identities)

[F7]

On the Stokes velocity space $V = H_{0}^{1} (D)^{d}$ , Poincare, Korn, Holder and Young give $∥ z ∥_{2} \leq C_{P} ∥ z ∥_{V}$ and $a_{ν} (z, z) \geq ν ∥ z ∥_{V}^{2}$ . (Stokes velocity and pressure spaces and weak forms, The Poincare inequality for zero-boundary Sobolev closures on domains bounded in one direction, The homogeneous Dirichlet strain identity, Holder's inequality for integrals, including the endpoint cases, Young's inequality for conjugate real exponents)

[A1]

Countable Choice is assumed through the finite-dimensional projection and scheme-existence inputs. (The Axiom of Countable Choice ( ${A C}_{ω}$ ), Approximation by the MINI solenoidal energy projection, Backward Euler MINI step existence and discrete energy bound)

[F8]

The static mixed projection contributes at most $1 + ν^{- 1}$ , pressure and trilinear Young estimates contribute inverse powers no worse than $ν^{- 3}$ , and the viscous consistency term contributes one factor $ν$ before it is balanced by coercivity. Thus all residual constants are bounded by $C_{D, s h a p e} (1 + ν + ν^{- 3})$ ; all such constants depend otherwise only on the domain, mesh-shape bound and dimension. (Mixed Galerkin error bound retaining beta_h, Approximation by the MINI solenoidal energy projection, Young's inequality for conjugate real exponents)

Proof

1.1A1F2F6F7given

Let $S_{h}$ be the projection of [F2] and set $η (t) = u (t) - S_{h} u (t)$ and $M_{ν} = (1 + ν^{- 1}) M$ . Linearity and time independence of $S_{h}$ give $η_{t} = u_{t} - S_{h} u_{t}$ . Applying [F2] to $(u (t), p (t))$ and $(u_{t} (t), p_{t} (t))$ yields $∥ η ∥_{L^{\infty} V} + ∥ η_{t} ∥_{L^{\infty} V} + ∥ η ∥_{L^{\infty} L^{2}} \leq C h M_{ν},$ where the last term follows from Poincare [F7]. Define $θ^{n} = S_{h} u (t_{n}) - u_{h}^{n} \in Z_{h}$ . Since $P_{Z_{h}}$ is a best $L^{2}$ approximation and $S_{h} u_{0} \in Z_{h}$ , [F2], [F6] and Poincare give $∥ θ^{0} ∥_{2} \leq 2 ∥ u_{0} - S_{h} u_{0} ∥_{2} \leq C h M_{ν}$ .

2.1F1F2step 1.1algebra

Write $u_{n} = u (t_{n})$ and ${\overset{ˉ}{u}}_{n} = τ_{n}^{- 1} \int_{t_{n - 1}}^{t_{n}} u (t) d t$ , with analogous notation ${\overset{ˉ}{p}}_{n}$ and ${\overset{ˉ}{c}}_{n} = τ_{n}^{- 1} \int c (u (t); u (t), \cdot) d t$ . Average the exact equation in time and subtract the discrete equation. For every $z_{h} \in Z_{h}$ this gives $(\frac{θ^{n} - θ^{n - 1}}{τ_{n}}, z_{h}) + a_{ν} (θ^{n}, z_{h}) = - (\frac{η (t_{n}) - η (t_{n - 1})}{τ_{n}}, z_{h}) - a_{ν} (η (t_{n}) + {\overset{ˉ}{u}}_{n} - u_{n}, z_{h}) - b (z_{h}, {\overset{ˉ}{p}}_{n}) + c (u_{h}^{n}; u_{h}^{n}, z_{h}) - {\overset{ˉ}{c}}_{n} (z_{h}) .$ The identity uses $a_{ν} (S_{h} u_{n} - u_{n}, z_{h}) = 0$ from the defining projection; all forcing averages cancel exactly.

3.1F2F3F7step 2.1

Test step 2.1 with $z_{h} = θ^{n}$ and multiply by $τ_{n}$ . For the pressure choose the mean-adjusted interpolant $r_{h} (t)$ from [F3] and set ${\overset{ˉ}{r}}_{h}^{n} = τ_{n}^{- 1} \int_{t_{n - 1}}^{t_{n}} r_{h} (t) d t \in Q_{h}$ ; then $b (θ^{n}, {\overset{ˉ}{r}}_{h}^{n}) = 0$ . Linearity and the triangle inequality for the Bochner average give $∥ {\overset{ˉ}{p}}_{n} - {\overset{ˉ}{r}}_{h}^{n} ∥_{2} \leq C h M$ , and $∣ div θ^{n} ∣ \leq \sqrt{d} ∥ θ^{n} ∥_{V}$ , so $∣ b (θ^{n}, {\overset{ˉ}{p}}_{n}) ∣ \leq C_{D} h M ∥ θ^{n} ∥_{V}$ . Also $∥ {\overset{ˉ}{u}}_{n} - u_{n} ∥_{V} \leq τ_{n} M$ . The time derivative estimate [F2] gives $∥ η (t_{n}) - η (t_{n - 1}) ∥_{2} \leq C τ_{n} h M_{ν}$ . Cauchy–Schwarz and Young, with viscous fractions $ν τ_{n} / 16$ for each of the viscous and pressure terms, yield $∣ (η (t_{n}) - η (t_{n - 1}), θ^{n}) ∣ + τ_{n} ∣ a_{ν} (η (t_{n}) + {\overset{ˉ}{u}}_{n} - u_{n}, θ^{n}) ∣ + τ_{n} ∣ b (θ^{n}, {\overset{ˉ}{p}}_{n} - {\overset{ˉ}{r}}_{h}^{n}) ∣ \leq \frac{τ_{n}}{8} ∥ θ^{n} ∥_{2}^{2} + \frac{ν τ_{n}}{8} ∥ θ^{n} ∥_{V}^{2} + C_{D, ν} τ_{n} (h^{2} M_{ν}^{2} + τ_{n}^{2} M^{2}) .$ Here $C_{D, ν}$ is bounded by $C_{D, s h a p e} (1 + ν + ν^{- 1})$ for these linear residuals.

3.2F4F5F7step 1.1step 2.1

The identity $c (u_{h}^{n}; u_{h}^{n}, θ^{n}) - c (u_{n}; u_{n}, θ^{n}) = - c (u_{h}^{n}; η (t_{n}), θ^{n}) - c (η (t_{n}); u_{n}, θ^{n}) - c (θ^{n}; u_{n}, θ^{n})$ follows by bilinearity in the first two slots and skew cancellation $c (u_{h}^{n}; θ^{n}, θ^{n}) = 0$ from [F4]. Apply [F4], Poincare and Young after multiplication by $τ_{n}$ . Each of these three terms is bounded by $ν τ_{n} ∥ θ^{n} ∥_{V}^{2} / 64$ plus respectively $C_{D, ν} τ_{n} M_{\infty} ∥ u_{h}^{n} ∥_{V} h^{2} M_{ν}^{2}, C_{D, ν} τ_{n} ∥ u_{n} ∥_{V}^{2} h^{2} M_{ν}^{2}, C_{D} ν^{- 3} τ_{n} M_{u}^{4} ∥ θ^{n} ∥_{2}^{2} .$ For the time-averaging remainder, [F4] and $∥ u (t) - u_{n} ∥_{V} \leq τ_{n} M$ give $τ_{n} ∣ c (u_{n}; u_{n}, θ^{n}) - {\overset{ˉ}{c}}_{n} (θ^{n}) ∣ \leq C_{D} τ_{n}^{2} M^{2} ∥ θ^{n} ∥_{V} \leq ν τ_{n} ∥ θ^{n} ∥_{V}^{2} / 64 + C_{D, ν} τ_{n}^{3} M^{4}$ .

4.1F4F5F8step 2.1step 3.1step 3.2algebra

The Hilbert identity $2 (x - y, x) = ∥ x ∥^{2} - ∥ y ∥^{2} + ∥ x - y ∥^{2}$ applied to the time term in step 2.1 converts the tested equation to an energy recurrence. The viscous Young fractions in steps 3.1–3.2 sum to $3 / 16$ of $ν τ_{n} ∥ θ^{n} ∥_{V}^{2}$ before doubling, so at least $5 / 8$ remains and in particular the displayed half remains. Doubling also changes the time residual in step 3.1 into a coefficient at most $τ_{n} / 4$ of $E_{n}$ ; the convection coefficient is at most $C_{D} ν^{- 3} τ_{n} M_{u}^{4}$ . Thus, enlarging $C_{D}$ if necessary, $E_{n} - E_{n - 1} + \frac{ν}{2} τ_{n} ∥ θ^{n} ∥_{V}^{2} \leq τ_{n} Λ E_{n} + C_{D, ν} τ_{n} {h^{2} M_{ν}^{2} (1 + M_{\infty} ∥ u_{h}^{n} ∥_{V} + M_{u}^{2}) + τ_{n}^{2} M^{4}}, E_{n} : = ∥ θ^{n} ∥_{2}^{2},$ with $Λ = 1 + C_{D} ν^{- 3} M_{u}^{4}$ . The residual coefficient satisfies $C_{D, ν} \leq C_{D, s h a p e} (1 + ν + ν^{- 3})$ by [F8].

5.1F5F7F8step 1.1step 4.1algebra

Since $τ_{n} Λ \leq 1 / 2$ , rearrange step 4.1 and use $(1 - x)^{- 1} \leq e^{2 x}$ for $0 \leq x \leq 1 / 2$ . Iterating gives $\max_{n} E_{n} \leq e^{2 T Λ} (E_{0} + C_{D, ν} \sum_{n} τ_{n} {h^{2} M_{ν}^{2} (1 + M_{\infty} ∥ u_{h}^{n} ∥_{V} + M_{u}^{2}) + τ_{n}^{2} M^{4}}) .$ By Cauchy–Schwarz and [F5], $\sum_{n} τ_{n} ∥ u_{h}^{n} ∥_{V} \leq \sqrt{T} M_{E}$ and $\sum_{n} τ_{n}^{3} \leq T τ_{\max}^{}$ . Summing step 4.1 and dropping the nonnegative final $E_{N}$ also gives $\frac{ν}{2} \sum_{n} τ_{n} ∥ θ^{n} ∥_{V}^{2} \leq E_{0} + T Λ \max_{n} E_{n} + C_{D, ν} \sum_{n} τ_{n} {h^{2} M_{ν}^{2} (1 + M_{\infty} ∥ u_{h}^{n} ∥_{V} + M_{u}^{2}) + τ_{n}^{2} M^{4}} .$ Since $M_{u} \leq M$ , these bounds and $E_{0} \leq C h^{2} M_{ν}^{2}$ imply $\max_{n} ∥ θ^{n} ∥_{2} + ν^{1 / 2} (\sum_{n} τ_{n} ∥ θ^{n} ∥_{V}^{2})^{1 / 2} \leq C_{D, s h a p e} (1 + ν)^{2} (1 + ν^{- 4}) (1 + T)^{2} (1 + M_{\infty} + M_{E} + M)^{4} e^{2 T Λ} (h M + τ_{\max} M^{2})$ . Here $(1 + ν)^{2} (1 + ν^{- 4})$ dominates $1 + ν^{- 1}$ from $M_{ν}$ and the residual inverse powers through $ν^{- 3}$ from [F8]; the extra $\sqrt{Λ}$ in the energy norm combines with those factors to give no power worse than $ν^{- 4}$ .

6.1F2step 5.1algebra∎

Finally $u_{n} - u_{h}^{n} = η (t_{n}) + θ^{n}$ . The projection estimate [F2] bounds $\max_{n} ∥ η (t_{n}) ∥_{2}$ by $C h M_{ν}$ and $\sum_{n} τ_{n} ∥ η (t_{n}) ∥_{V}^{2}$ by $C T h^{2} M_{ν}^{2}$ . Combine with step 5.1; the explicit viscosity factor in $C_{*}$ absorbs $M_{ν} / M = 1 + ν^{- 1}$ . This proves the velocity estimate. Its $ν$ dependence is explicit and contains an exponential in $ν^{- 3} M_{u}^{4} T$ ; it makes no Reynolds-uniform claim.

Remarks

The positive velocity estimate above retains the original mesh, data, mass initialization, arbitrary-step and any-root scope. The originally paired unrestricted instantaneous $L_{t}^{4 / 3} L_{x}^{2}$ pressure-error rate is false: the contextual Graded MINI mass initialization can destroy the smooth instantaneous pressure rate ↗ constructs an exact-scope graded-mesh counterexample with smooth data and genuine backward Euler roots. That counterexample is not a logical premise of the velocity proof. Pressure-primitive estimates remain compatible with it.

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Sources

Jean-Luc Guermond, MATH 661 Fall 2025, Chapters 49–54 (complete course notes) (standard reference, not scraped)
Daniele Boffi, Franco Brezzi, Michel Fortin, Mixed Finite Element Methods and Applications (complete textbook) (standard reference, not scraped)