A discrete trace theory for non-conforming polytopal hybrid discretisation methods

Abstract

In this work we develop a discrete trace theory that spans non-conforming hybrid discretization methods and holds on polytopal meshes. A notion of a discrete trace seminorm is defined, and trace and lifting results with respect to a discrete $H^1$-seminorm on the hybrid fully discrete space are proven. Building on these results we also prove a truncation estimate for piecewise polynomials in the discrete trace seminorm. Finally, we conduct two numerical tests in which we compute the proposed discrete operators and investigate their spectrum to verify the theoretical analysis. The development of this theory is motivated by the design and analysis of preconditioners for hybrid methods, e.g., of substructuring domain decomposition type.

Figures (8)

• Figure 1: (1) Cells between two boundary faces at a certain height ($t_1,t_2,\ldots,t_J$ are the cells in $\mathcal{I}_{{f \! f' }}$). (B) Partitioning of these cells ($t$ is an example of a cell in $\mathcal{C}_{{f \! f' } s}$).
• Figure 2: Cells between a boundary face $f$ and the cell $t_{{f \! f' },f}$ at distance $|x_f-x_{f'}|$ above $f$.
• Figure 3: Cone $\mathfrak C$ of apex $x_f$ and opening defined by the trace of a cell $t$ on $\partial \Omega$.
• Figure 4: Illustration of $A_t$.
• Figure 5: Illustration of $\mathfrak{F}$.

Theorems & Definitions (39)

• Theorem 2.2: Trace inequality
• Theorem 2.3: Lifting
• Remark 2.4: Scaling
• Remark 2.5: Extension to more general hybrid spaces
• Remark 2.6: Discrete trace theory in non-Hilbertian spaces
• Lemma 3.0
• Lemma 3.0: Estimates on the cardinality of sets of faces
• Lemma 3.0: Discrete Hardy inequality
• Lemma 3.0
• Lemma 3.0