By Pierluigi Crescenzi, Viggo Kann.

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J j i i ··· ah (φNloc , φNloc ) ah (φ1 , φNloc ) ⎡ ⎢ ⎢ bi = ⎢ ⎣ ⎡ ⎤ bh (uh , φ1i ) bh (uh , φ2i ) .. ⎥ ⎥ ⎥, ⎦ U1i U2i .. ⎤ ⎥ ⎢ ⎥ ⎢ Ui = ⎢ ⎥ ⎦ ⎣ i UNloc ⎤ lh (φ1i ) lh (φ2i ) .. 1. 6) is locally Lipschitz with respect to U. Proof. 5), we have by deﬁnition r(u1 − u2 )vdx. 2b), we get bh (u1 − u2 , v) ≤ r(u1 − u2 ) ≤ LS u − u 1 2 L2 (Ω ) v L2 (Ω ) L2 (Ω ) v L2 (Ω ) , which means that the non-linear form bh (u, v) is locally Lipschitz continuous in the ﬁrst argument. 2 Adaptivity 31 non-linear forms bh (u1 − u2 , φli ), l = 1, 2, .

We prove the a posteriori bounds with respect to the energy norm induced by the SIPG discretization. 1): ✟ ☛ Begin ✠ ✡ ❄ Initialization:mesh, 0 < tol, θ ✲ ❄ SOLVE ❄ ESTIMATE: compute η ☛ ❄ η < tol ✡ No ❄ ✟ ✠ MARK: ﬁnd subset MK ❄ Yes REFINE: reﬁne triangles K ∈ MK ☛ ❄ ✟ ✛ ✡End ✠ Fig. 4) on a given triangulation ξh . The ESTIMATE step is the key part of the adaptive procedure, by which the elements with large error are selected to be reﬁned. As an estimator, we use the modiﬁed version of the residual-based error indicator and of the error estimator in Sch¨otzau and Zhu [80].

2b), we get bh (u1 − u2 , v) ≤ r(u1 − u2 ) ≤ LS u − u 1 2 L2 (Ω ) v L2 (Ω ) L2 (Ω ) v L2 (Ω ) , which means that the non-linear form bh (u, v) is locally Lipschitz continuous in the ﬁrst argument. 2 Adaptivity 31 non-linear forms bh (u1 − u2 , φli ), l = 1, 2, . . , Nloc, i = 1, 2, . . , Nel, each component of the vector b(U) is locally Lipschitz continuous, which yields that the vector b(U) is locally Lipschitz with respect to U. 2 (Existence and uniqueness of the discrete solutions). 4) has a unique solution.

### A Compendium of NP Optimization Problems by Pierluigi Crescenzi, Viggo Kann.

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