Appendix of the paper: “Numerical solution of Boussinesq systems of the Bona-Smith family” D. C. Antonopoulos a , V. A. Dougalis a,b,1 and D. E. Mitsotakis ?? a Department of Mathematics, University of Athens, 15784 Zographou, Greece of Applied and Computational Mathematics, FO.R.T.H., P.O. Box 1527, 70013 Heraklion, Greece c UMR de Math´ ematiques, Universit´ e de Paris-Sud, Bˆ atiment 425, P.O. Box, 91405 Orsay France

b Institute

Abstract In this Appendix we provide proofs of Lemma 2(ii) and (iii), and of the estimate (16) of the paper “Numerical solution of Boussinesq systems of the Bona-Smith family”, using notation and results of that paper.

1. Prood of lemma 2(ii) Let v ∈ H 2 . Since Rh v ∈ H 2 , J Z J Z xj+1 X X 2 [(Rh v)′′ − ψ ′′ ] dx ≤ Ch−2 k(Rh v)′′ k2 = j=0

xj

j=0

xj+1

2

[(Rh v)′ − ψ ′ ] dx,

xj

where ψ is the interpolant of v in the space of piecewise linear, continuous functions Sh (0, 2), and where the second inequality follows from an inverse property on the polynomial space Pr−2 ([xj , xj+1 ]) and the quasiuniformity of the mesh xj . Therefore, by Lemma 1 k(Rh v)′′ k ≤ Ch−1 k(Rh v)′ − ψ ′ k ≤ Ch−1 (k(Rh v)′ − v ′ k + kψ ′ − v ′ k) ≤ Ckvk2 , from which there follows that kRh vk2 ≤ Ckvk2 due to the H 1 -stability of Rh . The proof for Rh0 is entirely analogous. 2 2. Proof of Lemma 2(iii) 1 Let v ∈ W∞ ∩ H01 and let V ∈ Sh0 be such that

(V ′ , χ′ ) = (v ′ , χ′ )

∀χ ∈ Sh0 .

(A1)

Email addresses: [email protected] (D. C. Antonopoulos), [email protected] (V. A. Dougalis), [email protected] (D. E. Mitsotakis). 1 Corresponding author Preprint submitted to Elsevier

30 March 2009

Then, it is not hard to see that V ′ = Peh v ′ , where Peh is the L2 -projection operator onto

d 0 S (µ, r) ⊕ {1}. dx h Using now the the definition of aD (·, ·) and (A1) we conclude that Sh (µ − 1, r − 1) =

aD (Rh0 v − V, χ) = (v − V, χ)

∀χ ∈ Sh0 .

(A2)

Consider now the problem Θ − bΘ′′ = v − V

¯ in I,

(A3)

Θ(−L) = Θ(L) = 0. Since by (2) aD (Rh0 Θ, χ) = aD (Θ, χ) = (v − V, χ) = aD (Rh0 v − V, χ) ∀χ ∈ Sh0 , we obtain that Rh0 Θ = Rh0 v − V . Therefore, k(Rh0 v)′ kL∞ ≤ k(Rh0 Θ)′ kL∞ + kV ′ kL∞ = k(Rh0 Θ)′ kL∞ + kPeh v ′ kL∞ ≤ C(kRh0 Θk2 + kv ′ kL∞ ),

using the stability of the L2 -projection in L∞ , [1]. By Lemma 2(ii), the elliptic regularity of the 1 , and the second solution Θ of (A3) and the Poincar´e inequality, we see that k(Rh0 v)′ kL∞ ≤ CkvkW∞ inequality of emma 2(iii) follows from the result (i) in the same Lemma. Rx 1 To prove the analogous estimate for v ∈ W∞ , we let now V = −L (Peh v ′ ). Then, V ∈ Sh (µ, r) with V ′ = Peh v ′ and V (−L) = 0. As before, we have aN (Rh v − V, χ) = (v − V, χ)

∀χ ∈ Sh ,

which gives, if Θ is a solution of the problem Θ − bΘ′′ = v − V

¯ in I,

Θ′ (−L) = Θ′ (L) = 0, 1 ≤ CkvkW 1 , arguing as in the first part the identity Rh v = Rh Θ + V . We conclude that kRh vkW∞ ∞ of the proof. 2

3. Proof of (16) From (7) of Proposition 5 it follows that max (kηh k2 + kuh k1 ) ≤ C.

(A4)

0≤t≤T

From (6), the definitions of f and g, and Lemma 4(i) we obtain, using (A4), for 0 ≤ t ≤ T kηh t k2 = kf (ηh , uh )k2 ≤ kfˆ(uh )k2 + kfˆ(ηh uh )k2 ≤ C (kuh k1 + kηh uh k1 ) ≤ C (kuh k1 + kηh k1 kuh k1 ) ≤ C, and 2

(A5)

1 kuh t k1 = kg(ηh , uh )k1 ≤ |c|kfˆ(ηh x x)k1 + kfˆ(ηh )k1 + kfˆ(u2h )k1 2  ≤ C kηh xx k + kηh k + ku2h k ≤ C (kηh k2 + kuh k1 kuh k) ≤ C.

(A6)

Differentiating in (6) with respect to t we see that ηh tt = fˆ(uht ) + fˆ(ηh t uh ) + fˆ(ηh uht ), uh tt = cfˆ(ηh ) + fˆ(ηh ) + fˆ(uh uh t ). xxt

t

Therefore, using (A4)–(A6) we have as before, for 0 ≤ t ≤ T kηh tt k2 ≤ C (kuht k1 + kηh t uh k1 + kηh uht k1 ) ≤ C (kuht k1 + kηh t k1 kuh k1 + kηh k1 kuht k1 ) ≤ C, and kuhtt k2 ≤ C (kηh t k2 + kηh t k + kuht k1 kuh k1 ) ≤ C. Continuing inductively we see that (16) holds. 2 Additional reference [1] J. Douglas, T. Dupont, L. B. Wahlbin, Optimal L∞ error estimates for Galerkin approximations to solutions of two-point boundary value problems, Math. Comp. 29 (1975) 475–483.

3

Numerical solution of Boussinesq systems of the Bona ...

Appendix of the paper: “Numerical solution of Boussinesq systems of the Bona-Smith family”. D. C. Antonopoulosa, V. A. Dougalisa,b,1and D. E. Mitsotakis?? aDepartment of Mathematics, University of Athens, 15784 Zographou, Greece. bInstitute of Applied and Computational Mathematics, FO.R.T.H., P.O. Box 1527, 70013 ...

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