New PDF release: Artificial Boundary Method

By Houde Han, Xiaonan Wu

ISBN-10: 3642354637

ISBN-13: 9783642354632

ISBN-10: 3642354645

ISBN-13: 9783642354649

"Artificial Boundary process" systematically introduces the unreal boundary process for the numerical recommendations of partial differential equations in unbounded domain names. certain discussions deal with sorts of difficulties, together with Laplace, Helmholtz, warmth, Schrödinger, and Navier and Stokes equations. either numerical tools and blunder research are mentioned. The publication 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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30) has a unique solution uN (x) ∈ H 1 (Ωi ). 30), respectively, and C > 0 is a constant not depending on N . Proof. Let a( N (x) = u(x) − uN (x). Then, N , v) + bN ( N N (x) ∈ V0 and satisfies , v) = bN (u, v) − b(u, v), ∀v ∈ V0 . 2, we obtain N 2 1,Ωi α , N ) + bN ( = bN (u, N ) − b(u, a( N N , N N ) ). 3) of the 3-D Poisson equation. For r R0 , u(x) satisfies the Laplace equation. Expand u(R0 , θ, ϕ) in an infinite series as follows: ∞ u(R0 , θ, ϕ) = an0 0 a00 + P (cos θ) 2 2 n n=1 n Pnm (cos θ)(anm cos mϕ + bnm sin mϕ) .

BR On BR , we consider the following boundary value problem: Δv ∗ (x) = 0, ∀x ∈ BR , v ∗ (x) = v(x), ∀x ∈ ΓR . 60) has a unique solution and |∇v ∗ |2 dx |∇˜ v |2 dx = BR BR Ωi |∇v|2 dx. 58) of v on ΓR , we obtain the following series form for v ∗ : v ∗ (r, θ) = ∞ r c0 + 2 R n=1 n (cn cos nθ + dn sin nθ). 61), we obtain |∇v ∗ |2 dx = BR 2π ∂v ∗ ∂r R 0 ∞ 0 2π R = n=1 0 0 ∞ R = 2π n=1 ∞ =π 0 + r−2 ∂v ∗ ∂θ 2 rdrdθ n2 r2n−2 2 (cn + d2n )rdrdθ R2n n2 r2n−1 2 (cn + d2n )dr R2n n(c2n + d2n ) n=1 2 = π|v ∗ |1/2,ΓR 24 2 2 = π|v|1/2,ΓR .

59), we obtain |a(v − u, v − uN,R ) + bN (v − u, v − uN,R )| h h 2|v − u|1,Ωi v − uN,R h 1,Ω . 67), we find v−uN,R h 1,Ωi R0 1 (N +1)k−1 R N +1 |u|k−1/2,Γ0 +2|v−u|1,Ωi , ∀v ∈ V0h . 68) Finally, using the triangle inequality, we get the following estimate: 25 Artificial Boundary Method u−uN,R h 1,Ωi |u − v|1,Ωi + v − uN,R h R0 1 (N +1)k−1 R N +1 1,Ωi |u|k−1/2,Γ0 +3|u−v|1,Ωi , ∀v ∈ V0h . 63) holds. 63), we see clearly how the error u−uN,R is dependent h on the finite element mesh size (h), the accuracy of the artificial boundary (N ), and the position of the artificial boundary (R).

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Artificial Boundary Method by Houde Han, Xiaonan Wu

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