Discrete exterior geometry approach to structure-preserving ... - ITM

1524 M. Seslija et al. / Journal of Geometry and Physics 62 (2012) 1509–1531where ⃗e ∗ p and ⃗e∗ q are integral forms on the primal mesh and its circumcentric dual, while ˙⃗ Rp and ˙⃗ Rq are time derivatives ofthe residues of the Hodge operator approximations given by∗ p ė c = p Mp,k˙⃗ep,k + ˙⃗ Rp,kˆσ p kσ ql∗ q ė c q = M q,l˙⃗e q,l + ˙⃗ Rq,l ,with subscripts k and l acting as selectors for vector components.Define discrete energy errors as δ⃗e p = ⃗e ∗ − p ⃗e p and δ⃗e q = ⃗e ∗ − q ⃗e q, and the output error as δ⃗ fb = ⃗ f∗− ⃗b fb .Subtracting (6.4) from (6.5) leads to− Mp δ˙⃗e p − ˙⃗ Rp = (−1) pq+1 D n−qiδ⃗e q + D n−qbδ⃗e ∗ b−M q δ˙⃗e q − ˙⃗ Rq = D n−p δ⃗e pδ⃗ fb = (−1) p T n−p δ⃗e p ,since δ⃗e b = ⃗e ∗ − b ⃗e b = ( ⋆σ n−q e c − b ⟨e b, ⋆σ n−qb⟩) n−qb⋆σ ∈∂(⋆K) = 0.bMultiplying the first equation in (6.6) by δ⃗e p and the second by δ⃗e q gives−⟨δ⃗e p , Mp δ˙⃗e p ⟩ − ⟨δ⃗e p , ˙⃗ Rp ⟩ = (−1) pq+1 ⟨δ⃗e p , D n−qiδ⃗e q ⟩−⟨δ⃗e q , M q δ˙⃗e q ⟩ − ⟨δ⃗e q , ˙⃗ Rq ⟩ = ⟨δ⃗e q , D n−p δ⃗e p ⟩.Then we haveThat is−⟨δ⃗e p , Mp δ˙⃗e p ⟩ − ⟨δ⃗e p , ˙⃗ Rp ⟩ − ⟨δ⃗e q , M q δ˙⃗e q ⟩ − ⟨δ⃗e q , ˙⃗ Rq ⟩ = (−1) pq+1 ⟨δ⃗e p , D n−qiδ⃗e q ⟩ + ⟨δ⃗e q , D n−p δ⃗e p ⟩ = 0.⟨δ⃗e p , Mp δ˙⃗e p ⟩ + ⟨δ⃗e q , M q δ˙⃗e q ⟩ = −⟨δ⃗e p , ˙⃗ Rp ⟩ − ⟨δ⃗e q˙⃗Rq ⟩. (6.7)Integration of (6.7) from 0 to t f yields12 M 122p δ⃗e p (t f ) + 1 12 M 2q δ⃗e q (t f )Let t ∗ be such thatM 122p δ⃗e p (t ∗ 1) + M 2q δ⃗e q (t ∗ )22= −≤ tf0 tf0⟨δ⃗e p (τ), ˙⃗ Rp (τ)⟩ + ⟨δ⃗e q (τ), ˙⃗ Rq (τ)⟩dτ∥˙⃗ Rp (τ)∥ ∥δ⃗e p (τ)∥ + ∥˙⃗ Rq (τ)∥ ∥δ⃗e q (τ)∥dτ. 11= max M 2p0≤t≤t f ∥δ⃗e p(t)∥ + M 2q ∥δ⃗e q∥,then 1M 2p δ⃗e p (t ∗ ) + 12 2Mq δ⃗e q (t ∗ 1) ≤ 2 M 2p δ⃗e p (t ∗ )≤ 4≤ 4 tf0 tf021+ M 2q δ⃗e q (t ∗ )It follows that M 12p δ⃗e p (t ∗ ) + 1tf 2Mq δ⃗e q (t ∗ ) ≤ 4 M − 1 2p˙⃗R p (τ) + M − 1 2q˙⃗R q (τ) dτ.Thus∥δ⃗e p (t ∗ )∥ + ∥δ⃗e q (t ∗ )∥ ≤ 4 M− 1 2p0 + ∞2 ∥˙⃗ Rp (τ)∥ ∥δ⃗e p (τ)∥ + ∥˙⃗ Rq (τ)∥ ∥δ⃗e q (τ)∥dτ 1M 2p δ⃗e p (t) + 12 −Mq δ⃗e q (t) 1 M2p˙⃗R p (τ) + M − 1 2q˙⃗R q (τ) dτ.1 M− 2q tf M − 1 2p˙⃗R p (τ) + M − 1 2q˙⃗R q (τ) dτ.∞0Estimation of the residues ⃗ Rp and ⃗ Rq can be conducted by employing Bramble–Hilbert techniques in the case of a weakformulation, or using a Taylor’s expansion of the efforts under the standard smoothness assumptions [27]. For the results onthe estimates of the Hodge star in one, two and three dimension the reader is invited to consult [27] and references therein.(6.6)