By Robert Beauwens, Martin Berzins
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Proof. We can see the ﬁrst inequality in . We prove only the second inequality. 5) . 6). S ∈Γh S L2 (K ) 1 h− K (η p − qh ) −1/2 L2 ( S ) ✷ hS ¯ ¯J S (t ) − t ∂ J (t ) (η p − qh ) ds dt ∂t 1 h K2 ¯J S ( T ) + S ∈Γh L2 (K ) (η p − qh ) ηh ( T ) − qh ( T ) L2 ( S ) dt L2 ( S ) dt . 6) C. Xiong, Y. Li / Applied Numerical Mathematics 61 (2011) 181–200 199 Lemma 6. We have a posteriori error estimates of the form | η y |V 1 | η p |V 2 ξ, ρ, where 2 L2 ( Q T ) ξ 2 = T η y (0) 2 H 1 (Ω) + T (c 1 + 1) η y (0) + T C∗ K ∈Th T +C 2 dt L2 (K ) h K r K − ( T − t )˙r K (t ) K ∈Th ρ 2 = T η p (T ) 2 L2 ( Q T ) ∗ 2 H 1 (Ω) + T (c 1 + 1) η p ( T ) hK 0 +C ∗∗ h S2 J S (0) S ∈Γh L 2 (Ω) K ∈Th T L2 (K ) 2 L2 (K ) h K r¯ K ( T ) K ∈Th dt + C ∗∗ S ∈Γh 1 h S2 ¯J S ( T ) + T C ∗∗ L2 ( S ) S ∈Γh ¯ ¯J − t ∂ J S (t ) ∂t 1/2 hS 0 2 dt , L2 ( S ) S ∈Γh + T C∗ 2 ∂ r¯ K (t ) r¯ K − t ∂t J − ( T − t ) ˙J S (t ) 1/2 hS 0 T +C 1 + T C ∗∗ T ∗ 0 2 L2 (K ) h K r K (0) 2 dt .
Applied Numerical Mathematics 61 (February 2011) by Robert Beauwens, Martin Berzins