By Robert Beauwens, Martin Berzins

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Batra, M. Porﬁri, D. Spinello, Free and forced vibrations of a segmented bar by a meshless local Petrov–Galerkin (MLPG) formulation, Comput. Mech. 41 (2008) 473–491. R. Cannon, Y. Lin, An inverse problem of ﬁnding a parameter in a semi-linear heat equation, J. Math. Anal. Appl. 145 (2) (1990) 470–484. R. Cannon, Y. L. Matheson, The solution of the diffusion equation in two-space variables subject to the speciﬁcation of mass, Appl. Anal. 50 (1993) 1–19. R. Cannon, J. van der Hoek, Diffusion subject to speciﬁcation of mass, J.

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Proof. We can see the ﬁrst inequality in [3]. 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

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