By Houde Han, Xiaonan Wu

ISBN-10: 3642354637

ISBN-13: 9783642354632

ISBN-10: 3642354645

ISBN-13: 9783642354649

"Artificial Boundary approach" systematically introduces the synthetic boundary strategy for the numerical recommendations of partial differential equations in unbounded domain names. precise discussions deal with forms of difficulties, together with Laplace, Helmholtz, warmth, Schrödinger, and Navier and Stokes equations. either numerical tools and mistake research are mentioned. The ebook is meant for researchers operating within the fields of computational arithmetic and mechanical engineering.

Prof. Houde Han works at Tsinghua collage, China; Prof. Xiaonan Wu works at Hong Kong Baptist collage, China.

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M=1 R0 R N +3/2 ∞ πR0 (n + 1)a2n0 n=N +1 n + 2(n + 1)(n + m)! 2 (anm + b2nm ) (2n + 1)(n − m)! m=1 ∞ · n πR (n + 1)e2n0 + n=N +1 R0 R 1/2 N +3/2 2(n + 1)(n + m)! 2 2 (enm + fnm ) (2n + 1)(n − m)! m=1 ||| u|Γ0 ||| ||| v|ΓR ||| C R0 R N +3/2 u 1/2,Γ0 v 1/2 1,Ωi , where C is a constant not dependent on N . 43), we get the result of the theorem. 30). For simplicity, we assume that Γ is the surface of a polyhedron and g = 0. 45) ∀v ∈ V0 . 46) ∀v ∈ V0h . 46), the following theorem holds. 46) 1 has a unique solution uN h ∈ H (Ωi ), and the following error estimate holds.

40) ∞ c0 + v(R, θ) = (cn cos nθ + dn sin nθ). 42) N bβ,N (u, v) = πz0 a 0 c0 + π zn (an cn + bn dn ). 2 n=1 For any v ∈ V0 , we deﬁne the function v˜ on BR = {x | |x| R} as follows: ∀x ∈ Ωi , ∀x ∈ Ω0 . 43) From the deﬁnition, v˜ ∈ H 1 (BR ) and v˜|ΓR = v|ΓR . 44) v1 |ΓR = v|ΓR . 45) v1 is uniquely determined by v|ΓR . , BR (|∇v1 |2 + β 2 v12 )dx (|∇˜ v |2 + β 2 v˜2 )dx BR (|∇v|2 + β 2 v 2 )dx = Ωi Mβ v 2 1,Ωi . 46) On the other hand, on ΓR , v1 (R, θ) = v(R, θ). 47) where cn = 1 π dn = 1 π 2π v(R, θ) cos nθdθ, 0 2π v(R, θ) sin nθdθ.

44) v1 |ΓR = v|ΓR . 45) v1 is uniquely determined by v|ΓR . , BR (|∇v1 |2 + β 2 v12 )dx (|∇˜ v |2 + β 2 v˜2 )dx BR (|∇v|2 + β 2 v 2 )dx = Ωi Mβ v 2 1,Ωi . 46) On the other hand, on ΓR , v1 (R, θ) = v(R, θ). 47) where cn = 1 π dn = 1 π 2π v(R, θ) cos nθdθ, 0 2π v(R, θ) sin nθdθ. 48) n > 0, n = 0. 48), we obtain ∞ πz0∗ c20 +π zn∗ (c2n + d2n ) 2 n=1 Mβ v 2 1,Ωi . 30), we have zn 2n + α0 zn∗ , n = 0, 1, 2, · · · . 50) Then, ∞ 0 bβ (v, v) = πz0 c20 +π zn (c2n + d2n ) 2 n=1 ∞ α0 C v ∞ πz0∗ c20 +π zn∗ (c2n + d2n ) + 2 n(c2n + d2n ) 2 n=1 n=1 2 1,Ωi .

### Artificial Boundary Method by Houde Han, Xiaonan Wu

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