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Proof. This result is classical (for parts of the proof see [18, Section 8.6] or [20]). Set D δ := D\D hwhere δ denotes the maximum width of D δ , and let first s = 1. Since v h = 0 on D δ it suffices toshow that|v| H 1 (D δ ) ‖v‖ H 2 (D) h . (3.7)The result then follows for s = 1 with Lemma 3.6. The result for s < 1 follows by interpolation,since trivially, |v| H 1 (D δ ) ≤ ‖v‖ H 1 (D),To show (3.7), let w ∈ H 1 (D). Using a trace result we get‖w‖ L 2 (D δ ) ≤ ‖w‖ L 2 (S δ ) δ 1/2 ‖w‖ H 1 (D),where S δ = {x ∈ D : dist(x, ∂D) ≤ δ} ⊂ D is the boundary layer of width δ. It follows fromAssumption A4 that diam(τ) h 2 wherever the boundary is not convex. In regions where D isconvex it follows from the smoothness assumption on ∂D that the width of D δ is O(h 2 ). Henceδ h 2 , which completes the proof of (3.7).Now, for almost all ω ∈ Ω, the finite element approximation to (2.3), denoted by u h (ω, ·), isthe unique function in V h that satisfiesb ω (u h (ω, ·), v h ) = L ω (v h ), for all v h ∈ V h . (3.8)Since b ω (·, ·) is coercive and bounded in H0 1 (D) (cf. Lemma 2.1), we have|u h (ω, ·)| H 1 (D) ‖f(ω, ·)‖ H t−1 (D)/a min (ω). (3.9)as well as the following classical quasi optimality result (cf. [4, 18]).Lemma 3.8 (Cea’s Lemma). Let Assumptions A1–A3 hold. Then, for almost all ω ∈ Ω,|u(ω, ·) − u h (ω, ·)| H 1 (D) ≤ C 3.8 (ω)inf |u(ω, ·) − v h | Hv h ∈V 1 (D), where C 3.8 (ω) :=h( )amax (ω) 1/2.a min (ω)Combining this with the interpolation result above we get the following error estimates.Theorem 3.9. Let Assumptions A1–A4 hold, for some 0 < t < 1 and p ∗ ∈ (0, ∞]. Then, for allp 0, we have‖u − u h ‖ L p (Ω,H0 1(D)) C 3.9 ‖f‖ L p∗(Ω,H s−1 (D)) h s a 3/2max‖a‖ C, where C 3.9 :=t (D)∥ a 7/2 ∥minwith q = p∗ pp ∗−p. If Assumptions A2 and A3 hold with t = 1, then‖u − u h ‖ L p (Ω,H 1 0 (D)) C 3.9 ‖f‖ L p∗(Ω,L 2 (D)) h .∥L q (Ω)Proof. Let 0 < t < 1 in Assumptions A2 and A3. It follows from Proposition 3.1 and Lemmas 3.7and 3.8 that, for almost all ω ∈ Ω,|u(ω, ·) − u h (ω, ·)| H 1 (D) C 3.1 (ω) C 3.8 (ω) ‖f(ω, ·)‖ H s−1 (D) h s . (3.10)The result now follows by Hölder’s inequality. The proof for t = 1 is analogous.The usual duality (or Aubin–Nitsche) “trick” leads to a bound on the L 2 –error.11
Corollary 3.10. Let Assumptions A1–A4 hold, for some 0 < t < 1 and p ∗ ∈ (0, ∞]. Then, for allp 0, we have‖u − u h ‖ L p (Ω,L 2 (D)) C 3.10 ‖f‖ L p∗(Ω,H s−1 (D)) h 2s amax‖a‖ 7/2 2 C, where C 3.10 :=t (D)∥∥with q = p∗ pp ∗−p. If Assumptions A2 and A3 hold with t = 1, then‖u − u h ‖ L p (Ω,L 2 (D)) C 3.10 ‖f‖ L p∗(Ω,L 2 (D)) h 2 .a 13/2min∥L q (Ω)Proof. We will use a duality argument. For almost all ω ∈ Ω, let e ω := u(ω, ·)−u h (ω, ·) and denoteby w ω the solution to the adjoint problemb ω (v, w ω ) = (e ω , v)for all v ∈ H 1 0 (D),which, by Proposition 3.1 with f(ω, ·) = e ω , is also in H 1+s (D). By Galerkin orthogonality‖e ω ‖ 2 L 2 (D) = (e ω, e ω ) L 2 (D) = b ω (e ω , w ω − z h ), for any z h ∈ V h .Using the boundedness of b ω (·, ·) and Lemma 3.7, we then get‖e ω ‖ 2 L 2 (D) a max(ω) |u(ω, ·) − u h (ω, ·)| H 1 (D) ‖w ω ‖ H 1+s (D) h s .Now it follows from Proposition 3.1 and Theorem 3.9 that‖e ω ‖ 2 L 2 (D) ≤ a max(ω) C 3.1 (ω) 2 C 3.8 (ω) ‖f(ω, ·)‖ H s−1 (D)‖e ω ‖ L 2 (D) h 2s .Dividing by ‖e ω ‖ L 2 (D), the result follows again by an application of Hölder’s inequality.Remark 3.11. As usual, the analysis can be extended in a straightforward way to other boundaryconditions, to tensor coefficients, or to more complicated PDEs including low–order terms, providedthe boundary data and the coefficients are sufficiently regular, see [18] for details.3.3 Quadrature ErrorThe integrals appearing in the bilinear form∑∫b ω (w h , v h ) =a(ω, x)∇w h · ∇v h dxττ∈T h : τ∈D hand in the linear functional L ω (v h ) involve realisations of random fields. It will in general beimpossible to evaluate these integrals exactly, and so we will use quadrature instead. We will onlyexplicitly analyse the quadrature error in b ω , but the quadrature error in approximating L ω (v h )can be analysed analogously.In our analysis, we use the midpoint rule, approximating the integrand by its value at themidpoint x τ of each simplex τ ∈ T h . The trapezoidal rule which we use in the numerical sectioncan be analysed analogously. Let us denote the resulting bilinear form that approximates b ω onthe grid T h by∑∫˜bω (w h , v h ) = a(ω, x τ ) ∇w h (x) · ∇v h (x) dx ,ττ∈T h : τ∈D hand let ũ h (ω, ·) denote the corresponding solution to˜bω (ũ h (ω, ·), v h ) = L ω (v h ) , for all v h ∈ V h .Clearly the bilinear form ˜b ω is bounded and coercive, with the same constants as the exactbilinear form b ω and so we can apply the following classical result [6] (with explicit dependence ofthe bound on the coefficients).12
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: Theorem 3.4. Let Assumptions A1-A3
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 and 28: Proof. Since supp(u 0 ) ⊂ D 0 , w
Page 29 and 30: We now extend A i to R d +. Since w

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