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Proof. Similarly to the proof of the previous Lemma, u i ∈ H 1 0 (D ∩ D i) is the weak solution of anew problem −div(a∇u i ) = g i in D ′ (D ∩ D i ) withg i := fχ i + a∇u · ∇χ i + div(au∇χ i ).As in Lemma A.4, since again div is linear and continuous from H s to H s−1 , we can establish thatg i ∈ H s−1 (D ∩ D i ) and‖g i ‖ H s−1 (D∩D i ) ‖a‖ C t (D)a min‖f‖ H s−1 (D).Now let Q = {(y ′ , y d ) ∈ R d−1 ×R : |y ′ | < 1 and |y d | < 1}, Q 0 = {(y ′ , y d ) ∈ R d−1 ×{0}| ‖y ′ ‖ < 1}and Q + = Q ∩ R d +. For 1 ≤ i ≤ p, let α i be a bijection from D i to Q such that α i ∈ C 2 (D i ),αi −1 ∈ C 2 (Q), α i (D i ∩ D) = Q + and α i (D i ∩ ∂D) = Q 0 .For all y ∈ Q + , we define w i (y) := u i (αi−1 (y)) ∈ H0 1(Q +) with ∇w i (y) = Ji−t (y)∇u i (αi−1 (y)),where J i (y) := Dα i (αi −1 (y))) is the Jacobian of α i . Furthermore, for x ∈ D i ∩ D and ϕ ∈ H0 1(Q +),we define v(x) := ϕ(α i (x)). Then v ∈ H0 1(D i ∩ D) and ∇v(αi −1 (y)) = Ji t(y)∇ϕ(y), for all y ∈ Q +,so that∫∫whereD i ∩Da(x)∇u i (x) · ∇v(x) dx =We define F i ∈ H s−1 (Q + ) by=∫a(αi−1Q +(y)) ∇u i (α −1iQ +A i (y) ∇w i (y) · ∇ϕ(y) dy ,A i (y) := a(αi−1 (y)) |detJ i (y))| −1 (J i Ji t )(y) ∈ S d (R).(y)) · ∇v(α −1 (y)) |detJ i (y))| −1 dy〈F i , ϕ〉 H s−1 (Q + ),H 1−s0 (Q + ):= 〈g i , ϕ ◦ α i 〉 H s−1 (D i ∩D),H 1−s0 (D i ∩D), for all ϕ ∈ H1−s 0 .Indeed, since we assumed that α i and αi−1 are in C 2 , we have ϕ ◦ α i ∈ H0 1−s (D i ∩ D) and moreover‖ϕ ◦ α i ‖ H 1−s (D i ∩D) ‖ϕ‖ H 1−s (Q + ) (cf. [18, Theorems 6.2.17 and 6.2.25(g)]), which implies thatF i ∈ H s−1 (Q + ) and‖F i ‖ H s−1 (Q + ) ‖g i ‖ H s−1 (D∩D i ).We finally get that v i ∈ H 1 0 (Q +) solves∫Q +A i ∇v i · ∇ϕ dy = 〈F i , ϕ〉 H s−1 (Q + ),H 1−s0 (Q + )for all ϕ ∈ H 1 0 (Q + ).In order to apply Lemma 3.3 we check first that A i ∈ C t (Q + , S d (R)) and that it is coercive, andthen define an extension of A i to R d +. Recalling that α i is a C 2 –diffeomorphism from D i ∩ D toQ + , with α −1i∈ C 2 (Q + ), we have for any y ∈ Q + and ξ ∈ R d :• Coercivity:• Boundedness:A i (y)ξ · ξ = a(α −1i(y))|detJ i (y)| −1 |J t i (y)ξ|2 a min |ξ| 2 . Hence A min a min .A max := ‖A i ‖ C 0 (Q + ,S d (R)) = max a(x) ‖|detJ i | −1 J i Ji t ‖ Cx∈D i ∩D0 (Q + ,S d (R)) a max .i• Regularity:A i ∈ C t (Q + , S d (R)) and‖A i ‖ C t (Q + ,S d (R))≤ a max ‖|detJ i | −1 J i J t i ‖ C t (Q + ,S d (R))+ |a| C t (D) ‖|detJ i| −1 J i J t i ‖ C 0 (Q + ,S d (R)) ‖a‖ C t (D) .27
We now extend A i to R d +. Since we assumed that supp(χ i ) is compact in D i , we can choose Q iand ˜Q i such that supp(v i ) ⊂ Q i ⊂ Q i ⊂ ˜Q i ⊂ ˜Q i ⊂ Q and consider ψ ∈ C ∞ (R d , [0, 1]) such thatψ = 0 on Q i and ψ = 1 on ˜Qci . We define the extension Ai of A i on R d + by{Ai (x)(1 − ψ(x)) + aA i (x) :=min ψ(x)I d if x ∈ Q +a min ψ(x)I d if x ∈ Q c +Analogously to the case of a in Lemma A.4, we can deduce that, for any y ∈ R d + and ξ ∈ R d ,A i (y)ξ ·ξ a min |ξ| 2 , A max = ‖A i ‖ C 0 (R d + ,S d(R)) a maxand ‖A i (y)‖ C t (R d + ,S d(R)) ‖a‖ C t (D) . (A.6)We now define an extension of F i on R d +. Note again that we can choose an open set G i such thatsupp(F i ) ⊂ G i ⊂ G i ⊂ Q and extend F i to all of R d + such that ‖F i ‖ H s−1 (R d + ) ‖F i‖ H s−1 (Q + ). Finallywe continue v i by 0 on R d +, which yields v i ∈ H 1 0 (Rd +). Moreover, since A i = A i on supp(v i ) ⊂ Q i ,v i is then the weak solution on R d + of−div(A i (x)∇v i (x)) = F i (x),which enables us to apply Lemma 3.3 and to obtain that v i ∈ H 1+s (R d +) and‖v i ‖ H 1+s (R d + ) A )max(|AA 2 i | C t (R d +min,S d(R)) |v i| H 1 (R d + ) + ‖F i‖ H s−1 (R d + )+ A maxA min‖v i ‖ H 1 (R d + ) .Recalling that u i (x) = v i (α i (x)) for any x ∈ D ∩ D i and using the bounds in (A.6), as well as thetransformation theorem [18, Theorem 6.2.17], we finally get‖u i ‖ H 1+s (D) a ()maxa 2 ‖a‖ C t (D) ‖u‖ H 1 (D∩D i ) + ‖g i ‖ H s−1 (D∩D i ) + a max‖u‖mina H 1 (D)mina max ‖a‖ C t (D)‖f‖ H s−1 (D) .a 3 min∑The result in Proposition 3.1 follows directly from Lemmas A.4 and A.5, if we recall that u = m u i .References[1] I. Babuška, F. Nobile, and R. Tempone. A stochastic collocation method for elliptic partialdifferential equations with random input data. SIAM J. Numer. Anal., 45(3):1005–1034, 2007.[2] I. Babuška, R. Tempone, and G. E. Zouraris. Galerkin finite element approximations ofstochastic elliptic partial differential equations. SIAM J. Numer. Anal., 42(2):800–825, 2004.[3] A. Barth, C. Schwab, and N. Zollinger. Multi–level Monte Carlo finite element method forelliptic PDE’s with stochastic coefficients. SAM Research Reports 2010-18, ETH Zürich, 2010.[4] S. C. Brenner and L. R. Scott. The mathematical theory of finite element methods, volume 15of Texts in Applied Mathematics. Springer, New York, third edition, 2008.[5] J. Charrier. Strong and weak error estimates for the solutions of elliptic partial differentialequations with random coefficients. Technical Report HAL:inria-00490045, Hyper Articles enLigne, INRIA, 2010. Available at http://hal.inria.fr/inria-00490045/en/.i=028
Page 1 and 2: BICSBath Institute for Complex Syst
Page 3 and 4: only Hölder continuous with expone
Page 5 and 6: for almost all ω ∈ Ω. The diffe
Page 7 and 8: 2.4 Truncated Karhunen-Loève Expan
Page 9 and 10: 3.1 Regularity of the SolutionPropo
Page 11 and 12: Theorem 3.4. Let Assumptions A1-A3
Page 13 and 14: Corollary 3.10. Let Assumptions A1-
Page 15 and 16: 3.4 Truncated FieldsCombining the r
Page 17 and 18: Since all the expectations E[Y l ]
Page 19 and 20: Figure 1: Let d = 1 and σ 2 = 1 in
Page 21 and 22: Figure 4: Left: Number of samples N
Page 23 and 24: therefore we getA min ‖(R i h )
Page 25 and 26: ∥∂which means that2 ˜w∂x i
Page 27: Proof. Since supp(u 0 ) ⊂ D 0 , w

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