Corrigendum to: Finite element approximation of the FENE-P model (IMA Journal of Numerical Analysis (2017) DOI: 10.1093/imanum/drx061)

We state corrections to the proofs of Proposition 4.2 and Proposition 6.2, which were correct in the original submitted version; see Barrett & Boyaval (2017). The proofs are correct as stated, except we replace = by "≤" in the definitions of B_γ and B^h_γ, and we do not appeal to Proposition 2.8 in Zeidler (1986), due to the choice of test functions. To correct the latter, we make the following additions. Proof of Proposition 4.2. On assuming, for any γ ℝ _>0, that the continuous mapping has no zero in B_γ, we define the continuous mapping G_γ : B_γ → B_γ such that (Formula Presented) By the Brouwer fixed point theorem G_γ has at least one fixed point (w,ψ) in B_γ, and so (Formula Presented) On the other hand, as (w,ψ) is a fixed point of G_γ, we have that (Formula Presented) It follows from (3.6), (3.7b), (3.5) and similarly to (4.29), on noting (4.28) and (1), that (Formula Presented) Therefore, on combining (2) and (3), we have for all γ sufficiently large that (Formula Presented) which obviously contradicts (4.30). Hence the mapping has a zero in B_γ for γ sufficiently large, and so there exists a solution (uⁿ_δ,h σⁿ_δ,h) to (4.13a,b). Proof of Proposition 6.2. On assuming, for any γ ϵ ℝ_>0, that the continuous mapping ^h has no zero in B^h_γ, we define the continuous mapping G^h_γ : B^h_γ → B^h_γ such that (Formula Presented) By the Brouwer fixed point theorem G^h_γ has at least one fixed point (w, ψ, ξ) in B^h_γ, and so (Formula Presented) On the other hand, as (w, ψ, ξ) is a fixed point of G^h_γ, we have that (Formula Presented) It follows from (6.40), (3.5), (3.7b), and similarly to (6.42), on noting (6.41) and (4), that (Formula presented) Therefore on combining (5) and (6), we have for all γ sufficiently large that (Formula Presented) which obviously contradicts (6.43). Hence the mapping ^h has a zero in B^h_γ for γ sufficiently large, and so there exists a solution (uⁿ_α,δ,h, σⁿ_α,δ,h,Qⁿ_α,δ,h) to (6.34a-c). The authors apologise for these errors.