It then follows from the stability result in Proposition 2.3 that bigl|!bigl|!bigl|bigl{ y_{h}^- widetilde{y}_{h},sigma_{h}^-widetilde{ sigma}_{h} bigr} bigr|!bigr|!bigr|_{delta} leq Cbigl| u^-u_{h}^ bigr| _{L^{2}(Omega)}.
In view of the a priori stability estimate in Proposition 5.1 and the following result of local existence of a small solution to the transformed MHD problem, we immediately obtain Theorem 2.1.
Following the stability result in Proposition 2.5 and the conclusion (4.9), we derive bigl|!bigl|!bigl|bigl{ z_{h}^-widetilde{z}_{h}, omega_{h}^-widetilde{omega}_{h} bigr} bigr|!bigr|!bigr|_{delta} leq C bigl|!bigl|!bigl|bigl{ y_{h}^-widetilde{y}_{h}, sigma_{h}^-widetilde{sigma}_{h} bigr} bigr|!bigr|!bigr|_{delta} leq C bigl| u^-u_{h}^bigr| _{L^{2}(Omega)}.
Then the robust stability problem in Proposition 1 is reduced to check whether or not the existence of symmetric matrices P > 0 and Q k > 0, k = 1,..., m, in the following LMI (10) [ Ξ P A 1 ⋯ P A m A 1 T P B 1 T P B 1 − Q 1 0 0 ⋮ 0 ⋱ 0 A m T P 0 0 B m T P B m − Q m ] < 0 where Ξ = A 0 T P + P A 0 + B 0 T P B 0 + ∑ k = 1 m Q k.
□ Proof of Proposition 10.5 The stability (A1) follows as in Proposition 5.1 for (μ ^, v ^ ) ∈ X (T ^ ) and (μ, v ) ∈ X (T ) up to sums of squares of some additional terms h T ‖ μ ^ v ^ − μ v ‖ L 2 (T ) for T ∈ S ⊂ T. Those extra terms motivate the error measure and, because of h T ≤ ‖ h (T 0 ) ‖ L ∞, lead to the proof of (A1) without additional difficulty.
□ Proposition 4.4 Stability Let Assumption 4.1 hold.
Proposition 4.12 Stability (A1) and discrete reliability (A4) imply (i)–(ii).
Proposition 4.3 Stability & Convergence Let Assumption 4.1 hold and assume that the sets Φ R (F, w, γ, t ) are compact for every γ ∈ R, t ∈ R, and w ∈ Y.
Since the first term of η does not depend on the argument V, standard inverse estimates as for the linear case prove stability (B1) as in [56, Proposition 4.4] and Proposition 5.1.
The following proposition shows the stability in L ∞-norm.
