Constrained and unconstrained stable discrete minimizations for p-robust local reconstructions in vertex patches in the de Rham complex

TL;DR

The paper develops a complete, $p$-robust theory for constrained and unconstrained local minimizations in the De Rham complex on vertex patches in 3D. It introduces a unified framework for $H^1$, $H( ext{curl})$, and $H( ext{div})$ minimizations using Raviart–Thomas and Nédélec spaces, with constants depending only on patch shape-regularity and independent of the polynomial degree $p$. The core contribution is a new $H( ext{curl})$-constrained minimization result and an extended, boundary-patch theory that covers the full De Rham diagram, including a rigorous treatment via equivalent parachute and reference-tetrahedron patches. The findings underpin stable local commuting projectors and local-best/global-best equivalences, with direct implications for $p$-robust a posteriori error estimation and a posteriori estimator efficiency. Overall, the work completes the De Rham-local-minimization toolbox in 3D and strengthens the theoretical foundation for stable, $p$-robust finite element analysis.

Abstract

We analyze constrained and unconstrained minimization problems on patches of tetrahedra sharing a common vertex with discontinuous piecewise polynomial data of degree p. We show that the discrete minimizers in the spaces of piecewise polynomials of degree p conforming in the H1, H(curl), or H(div) spaces are as good as the minimizers in these entire (infinite-dimensional) Sobolev spaces, up to a constant that is independent of p. These results are useful in the analysis and design of finite element methods, namely for devising stable local commuting projectors and establishing local-best/global-best equivalences in a priori analysis and in the context of a posteriori error estimation. Unconstrained minimization in H1 and constrained minimization in H(div) have been previously treated in the literature. Along with improvement of the results in the H1 and H(div) cases, our key contribution is the treatment of the H(curl) framework. This enables us to cover the whole De Rham diagram in three space dimensions in a single setting.

Figures (23)

• Figure 1: Interior patch (left) and boundary patch (right, with $\Gamma_{\boldsymbol a} = \emptyset$ and consequently $\Gamma = \partial \omega_{\boldsymbol a}$)
• Figure 2: Boundary patches with $\Gamma_{\boldsymbol a} \neq \emptyset$; $\Gamma_{\boldsymbol a}$ corresponding to all faces $F \in \mathcal{F}_{\boldsymbol a}$ lying on the boundary of $\omega_{\boldsymbol a}$ and sharing the vertex $\boldsymbol a$ (left), $\Gamma_{\boldsymbol a}$ corresponding to some faces $F \in \mathcal{F}_{\boldsymbol a}$ lying on the boundary of $\omega_{\boldsymbol a}$ and sharing the vertex $\boldsymbol a$ (right)
• Figure 3: Patch enumeration of Proposition \ref{['prop_enumeration']}, where "loops" around edges give property \ref{['it_enum_i']} and where all elements except $K_1$ and $K_8$ have both already enumerated and not yet enumerated neighbors, i.e., property \ref{['it_enum_ii']} (left). Invalid enumeration (right).
• Figure 4: Two-color refinement (black and white) $\mathcal{T}_e'$ around an edge $e$ of Proposition \ref{['prop_two_color']} (one of the tetrahedra in $\mathcal{T}_e$, different from $K_j$, is cut into $K_{+}$ and $K_{-}$) (left). Three-color refinement (black, grey, and white) around a vertex $\boldsymbol a$ of Proposition \ref{['prop_three_color']} (trivial situation where $\mathcal{T}_{\boldsymbol a}'$ can be taken as $\mathcal{T}_{\boldsymbol a}$) (right).
• Figure 5: Minimizations \ref{['tmp_definition_bxipj']} on the elements $K_j$, $j=1,2,6,8$, of the patch $\mathcal{T}_{\boldsymbol a}$ as shown and enumerated in Figure \ref{['fig_ex_numbering']}, left. The boundary datum $\boldsymbol r_p^j$ is zero on the external faces $F_j^{\rm ext} \subset \partial \omega_{\boldsymbol a}$ (this is always imposed) (faces $F_j^{\rm ext}$ hatched) and induced by the tangential trace of $\boldsymbol \xi_p^i$ on the previously enumerated elements $K_i$ as defined in Step 1 (there is no such datum on $K_1$ and respectively 1,2, and 3 on $K_2$, $K_6$, and $K_8$) (interfaces in grey).

Theorems & Definitions (52)

• Proposition 3.1: Constrained minimization in $\boldsymbol H_{0,\Gamma}(\operatorname{div},\omega_{\boldsymbol a})$
• Corollary 3.2: Unconstrained minimization in $\boldsymbol H_{0,\Gamma}({\operatorname{\bf {curl}}},\omega_{\boldsymbol a})$
• proof
• Theorem 3.3: Constrained minimization in $\boldsymbol H_{0,\Gamma}({\operatorname{\bf {curl}}},\omega_{\boldsymbol a})$
• Corollary 3.4: Unconstrained minimization in $H^1_{0,\Gamma}(\omega_{\boldsymbol a})$
• proof
• Remark 3.5: Converse inequalities
• Remark 3.6: Unconstrained $L^2(\omega_{\boldsymbol a})$ and constrained $H^1(\omega_{\boldsymbol a})$ minimizations
• Remark 3.7: Stable broken polynomial extensions
• Corollary 4.1: Constrained minimization in $\boldsymbol H(\operatorname{div},\omega_{\boldsymbol a})$ with inhomogeneous boundary conditions