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Mini-element H1 velocity and L2 pressure error estimate

Statement

Assume Countable Choice. Let $D$ be a bounded connected Lipschitz polygonal/polyhedral domain in dimension $d = 2$ or $3$ , with a conforming shape-regular simplicial mesh family of mesh size $h$ . Put $V = H_{0}^{1} (D; R^{d})$ and $Q = L_{0}^{2} (D)$ . For $ν > 0$ and $F \in V^{'}$ , suppose $(u, p) \in V \times Q$ satisfies $2 ν \int_{D} ε (u) : ε (v) - \int_{D} p div v = F (v), - \int_{D} q div u = 0$ for every $v \in V, q \in Q$ , with $u \in H^{2} (D; R^{d})$ and $p \in H^{1} (D)$ . Let $(u_{h}, p_{h}) \in V_{h} \times Q_{h}$ solve the same equations for every $v_{h} \in V_{h}, q_{h} \in Q_{h}$ , where $V_{h}, Q_{h}$ are the P1-bubble/P1 mini spaces. Then $∥ u - u_{h} ∥_{H^{1} (D)} + ∥ p - p_{h} ∥_{L^{2} (D)} \leq C h (∥ u ∥_{H^{2} (D)} + ∥ p ∥_{H^{1} (D)}) .$ The constant is independent of $h$ and depends on $d$ , $D$ , shape regularity, $ν$ , $C_{d i v} (D)$ , and the Poincare constant. The mixed error formula retains $M_{a} = 2 ν$ , $M_{b} = \sqrt{d}$ , discrete-kernel coercivity $α_{*} = ν$ , and $β_{h} \geq (C_{F} C_{d i v} (D))^{- 1}$ explicitly.

Facts & Assumptions

Given: Countable Choice; the continuous Stokes solution with the stated regularity, the mini-element spaces, a conforming shape-regular mesh, and positive viscosity $ν$ .

[A1]

Countable Choice is inherited through the Sobolev Hilbert and interpolation inputs. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

The mini pair is $V_{h} = [S_{h, 0}^{1}]^{d} \oplus \sum_{K} b_{K} R^{d}$ , $Q_{h} = S_{h}^{1} \cap L_{0}^{2}$ , with continuous piecewise P1 pressures. (The conforming P1-bubble/P1 mini element)

[F2]

The face-average map is defined on all of $H^{1} (D)$ with values in continuous piecewise P1, has the H1-to-L2 and H2-to-H1 estimates, and maps $H_{0}^{1} (D)$ into the zero-boundary subspace. (Boundary-preserving face-average P1 interpolation)

[F3]

The mini Fortin map gives $β_{h} \geq (C_{F} C_{d i v} (D))^{- 1}$ , and the mixed Galerkin theorem gives the explicit velocity/pressure bounds $U_{h}, P_{h}$ . (Uniform Fortin stability of the mini element, Mixed Galerkin error bound retaining beta_h)

[F4]

The Stokes conventions are $a_{ν} (u, v) = 2 ν \int ε (u) : ε (v)$ , $b (v, q) = - \int q div v$ , and the velocity gradient norm; the zero-boundary Poincare inequality controls the full H1 norm. (Stokes velocity and pressure spaces and weak forms, A bounded-domain Poincare inequality for H1-zero fields)

[F5]

The homogeneous Korn identity gives $2 ∥ ε (v) ∥_{2}^{2} = ∥ \nabla v ∥_{2}^{2} + ∥ div v ∥_{2}^{2}$ . (The homogeneous Dirichlet strain identity)

[F6]

Holder bounds the mean of an L2 error by $∣ D ∣^{1 / 2}$ times its L2 norm. (Holder's inequality for integrals, including the endpoint cases)

[F7]

The L2 Cauchy--Schwarz inequality gives $\int ∣ f g ∣ \leq ∥ f ∥_{2} ∥ g ∥_{2}$ . (Cauchy-Schwarz inequality for $L^{2}$ )

Proof

1.1A1F1F2F6given

Set $w_{h} = I_{h} u$ . By [F2], $w_{h} \in V_{h}$ and $δ_{u} : = ∥ u - w_{h} ∥_{V} \leq C_{I} h ∥ u ∥_{H^{2}}$ . Define ${\tilde{p}}_{h} = I_{h} p$ and subtract its mean, $r_{h} = {\tilde{p}}_{h} - ∣ D ∣^{- 1} \int_{D} {\tilde{p}}_{h}$ . Then $r_{h} \in Q_{h}$ . Since $p$ has zero mean, $∣ D ∣^{1 / 2} ∣ D ∣^{- 1} ∣ \int_{D} ({\tilde{p}}_{h} - p) ∣ \leq ∥ {\tilde{p}}_{h} - p ∥_{2}$ , and [F2] gives $δ_{p} : = ∥ p - r_{h} ∥_{2} \leq 2 C_{I} h ∥ p ∥_{H^{1}}$ .

2.1A1F3F4F5F7step 1.1

The strain bound $∣ ε (v) ∣ \leq ∣ \nabla v ∣_{F}$ and Cauchy–Schwarz give $M_{a} \leq 2 ν$ ; $∣ div v ∣ \leq \sqrt{d} ∣ \nabla v ∣_{F}$ gives $M_{b} \leq \sqrt{d}$ . By [F5], $a_{ν} (v, v) = 2 ν ∥ ε (v) ∥_{2}^{2} \geq ν ∥ \nabla v ∥_{2}^{2}$ for every $v \in V_{h}$ , so $α_{*} = ν$ . The uniform Fortin theorem in [F3] supplies the stated $β_{h}$ bound.

3.1A1F2F3step 1.1step 2.1

Apply [F3] with the approximants of step 1.1 and these constants. The explicit quantities $R_{h}, U_{h}, P_{h}$ are linear combinations of $δ_{u}, δ_{p}$ with coefficients depending only on $ν, d, β_{h}$ ; the lower bound on $β_{h}$ makes those coefficients uniform in $h$ . Thus $∥ u - u_{h} ∥_{V} + ∥ p - p_{h} ∥_{2} \leq C h (∥ u ∥_{H^{2}} + ∥ p ∥_{H^{1}})$ .

4.1A1F4step 3.1∎

Since $u - u_{h} \in H_{0}^{1} (D)^{d}$ , [F4] gives $∥ u - u_{h} ∥_{H^{1}} \leq (1 + C_{P} (D)^{2})^{1 / 2} ∥ u - u_{h} ∥_{V}$ . Absorb this fixed domain factor into $C$ ; it is independent of $h$ . This proves the asserted full H1 velocity estimate without asserting additional regularity of the Stokes solution.

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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)