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The dual mortar basis is biorthogonal and uniformly stable on the trace space

Statement

For the spaces and basis of Dual P1 mortar multipliers for boundary contact, put $d_{i} : = \sum_{e_{k} ∋ a_{i}} h_{k}$ . Then $\int_{Γ_{C}} χ_{i} ϕ_{j} d s = d_{i} δ_{i j} (0 \leq i, j \leq N) .$ The pair $(Λ_{h}, W_{h})$ satisfies the mesh-independent trace-space inf-sup bound $\inf_{0 \neq λ_{h} \in Λ_{h}} \sup_{0 \neq g_{h} \in W_{h}} \frac{\int_{Γ_{C}} λ_{h} g_{h} d s}{∥ λ_{h} ∥_{L^{2} (Γ_{C})} ∥ g_{h} ∥_{L^{2} (Γ_{C})}} \geq \frac{1}{\sqrt{3}} .$ Moreover, $λ_{h} \in Λ_{h}^{+}$ defines a nonnegative functional on the cone of nonnegative functions in $W_{h}$ , and $⟨ λ_{h}, g_{h} ⟩ = \sum_{i} d_{i} λ_{i} g_{h} (a_{i})$ .

Facts & Assumptions

Given: A finite contact partition with edge lengths $h_{k} > 0$ , continuous P1 nodal basis $ϕ_{i}$ , and local dual functions $χ_{i}$ as in Dual P1 mortar multipliers for boundary contact.

[F1]

On each edge, the local endpoint basis is $(ϕ_{L}, ϕ_{R}) = (1 - t, t)$ for $0 \leq t \leq 1$ , with arclength measure $h_{k} d t$ . (Dual P1 mortar multipliers for boundary contact)

[F2]

The local dual functions are $χ_{L} = 4 ϕ_{L} - 2 ϕ_{R}$ and $χ_{R} = - 2 ϕ_{L} + 4 ϕ_{R}$ , assembled by their nodal indices. (Dual P1 mortar multipliers for boundary contact)

Proof

technique · direct

Given: A contact edge partition, its continuous P1 trace basis, and the broken dual basis.

1.1F1F2givenalgebra

On one edge, direct integration of [F1]–[F2] gives $\int_{e} χ_{L} ϕ_{L} = h_{k}$ , $\int_{e} χ_{L} ϕ_{R} = 0$ , $\int_{e} χ_{R} ϕ_{L} = 0$ , and $\int_{e} χ_{R} ϕ_{R} = h_{k}$ . Summing over edges, only the edges incident to node $a_{i}$ contribute to $\int χ_{i} ϕ_{i}$ , and no edge contributes to $\int χ_{i} ϕ_{j}$ for $i \neq j$ ; hence the global pairing is $d_{i} δ_{i j}$ . This also proves linear independence of the $χ_{i}$ .

2.1F1F2step 1.1algebra

On an edge, the consistent P1 mass matrix is $h_{k} (\begin{array}{cc} 1 / 3 & 1 / 6 \\ 1 / 6 & 1 / 3 \end{array})$ , whose eigenvalues are $h_{k} / 6, h_{k} / 2$ . The local dual coefficient map is $(\begin{array}{cc} 4 & - 2 \\ - 2 & 4 \end{array})$ , with eigenvalues $2, 6$ on the same sum/difference vectors. Therefore, for local nodal coefficients $(λ_{L}, λ_{R})$ , $2 h_{k} (λ_{L}^{2} + λ_{R}^{2}) \leq ∥ λ_{L} χ_{L} + λ_{R} χ_{R} ∥_{L^{2} (e)}^{2} \leq 6 h_{k} (λ_{L}^{2} + λ_{R}^{2})$ ; for primal nodal coefficients $(g_{L}, g_{R})$ , $\frac{h_{k}}{6} (g_{L}^{2} + g_{R}^{2}) \leq ∥ g_{L} ϕ_{L} + g_{R} ϕ_{R} ∥_{L^{2} (e)}^{2} \leq \frac{h_{k}}{2} (g_{L}^{2} + g_{R}^{2})$ .

3.1step 1.1step 2.1F1F2algebra∎

Summing the local inequalities in step 2.1 gives $2 S_{λ} \leq ∥ λ_{h} ∥_{2}^{2} \leq 6 S_{λ}$ and $S_{g} / 6 \leq ∥ g_{h} ∥_{2}^{2} \leq S_{g} / 2$ , where $S_{λ} = \sum_{i} d_{i} λ_{i}^{2}$ and $S_{g} = \sum_{i} d_{i} g_{i}^{2}$ . By step 1.1, $⟨ λ_{h}, g_{h} ⟩ = \sum_{i} d_{i} λ_{i} g_{i}$ . Taking $g_{h}$ with nodal coefficients $g_{i} = λ_{i}$ makes the numerator $S_{λ}$ and the denominator at most $\sqrt{6 S_{λ}} \sqrt{S_{λ} / 2} = \sqrt{3} S_{λ}$ , proving the inf-sup bound. If $λ_{i} \geq 0$ and $g_{h} \geq 0$ , then every $g_{i} \geq 0$ and the same pairing is nonnegative; conversely testing against each nonnegative hat $ϕ_{i}$ shows any nonnegative functional has $d_{i} λ_{i} \geq 0$ , hence $λ_{i} \geq 0$ .

Depends on

Used by

The dual mortar contact system is stable and exactly complementary Theorem

Cited to discharge well-definedness by Dual P1 mortar multipliers for boundary contact.

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Rolf H. Krause and Barbara I. Wohlmuth, Multigrid Methods for Mortar Finite Elements, complete 7-page paper (standard reference, not scraped)