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Penalties below and above the Nitsche coercivity threshold

Example

Assume Countable Choice and take any nondegenerate bounded simplex with its one-cell matching mesh and degree- $k$ space, $k \geq 1$ . For either Nitsche sign $θ \in {0, 1}$ , the algebraic penalty-free boundary form, obtained by evaluating the displayed formula at zero penalty, is not coercive: the constant function has zero gradient and zero form value but positive boundary part in $∥ \cdot ∥_{E, h}$ . In contrast, any $γ_{0} > (1 + θ)^{2} C_{\partial}^{2} / 4$ satisfies the sufficient coercivity threshold of Nitsche coercivity above the inverse-trace threshold. The example shows failure at zero and success above the proved bound; it does not claim that every positive penalty below that sufficient bound fails on every mesh.

Facts & Assumptions

Given: Countable Choice; one bounded simplex with all facets boundary; $k \geq 1$ ; a constant trial function; the form and energy seminorm of Symmetric and nonsymmetric Nitsche boundary forms; and the coercivity result Nitsche coercivity above the inverse-trace threshold.

[F1]

At $γ_{0} = 0$ , $a_{h}^{θ} (1, 1) = 0$ because $\nabla 1 = 0$ and $\partial_{n} 1 = 0$ ; the energy seminorm has $∥ 1 ∥_{E, h}^{2} = \sum_{F} h_{F}^{- 1} ∣ F ∣ > 0$ (Symmetric and nonsymmetric Nitsche boundary forms).

[F2]

If $γ_{0} > (1 + θ)^{2} C_{\partial}^{2} / 4$ , then $a_{h}^{θ} (v_{h}, v_{h}) \geq λ_{θ} ∥ v_{h} ∥_{E, h}^{2}$ with $λ_{θ} > 0$ (Nitsche coercivity above the inverse-trace threshold).

Verification

Given: The one-cell simplex setup and the data in [F1]–[F2].

1.1F1given

For $v_{h} = 1$ , each volume-gradient and normal-flux term in the definition vanishes. At $γ_{0} = 0$ the penalty term vanishes too, so $a_{h}^{θ} (1, 1) = 0$ . Since each facet has positive measure and $h_{F} > 0$ , the finite sum $\sum_{F} h_{F}^{- 1} ∣ F ∣$ is positive; hence no positive coercivity constant can bound this form below by $∥ 1 ∥_{E, h}^{2}$ .

2.1F2step 1.1algebra∎

For any $γ_{0}$ strictly above the displayed threshold, [F2] gives the positive constant $λ_{θ}$ and proves coercivity on the entire finite-element space, including the constant function. This establishes the stated below/above comparison without making a necessity claim for intermediate positive penalties.

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Sources

Jean-Luc Guermond, MATH 661 Fall 2025, Chapter 37, Nitsche boundary penalty method (standard reference, not scraped)
Mika Juntunen and Rolf Stenberg, Nitsche's method for general boundary conditions, Mathematics of Computation 78 (2009), 1353–1374 (standard reference, not scraped)