Download e-book for iPad: Artificial Boundary Method by Houde Han, Xiaonan Wu

By Houde Han, Xiaonan Wu

ISBN-10: 3642354637

ISBN-13: 9783642354632

ISBN-10: 3642354645

ISBN-13: 9783642354649

"Artificial Boundary procedure" systematically introduces the synthetic boundary strategy for the numerical strategies of partial differential equations in unbounded domain names. targeted discussions deal with types of difficulties, together with Laplace, Helmholtz, warmth, Schrödinger, and Navier and Stokes equations. either numerical tools and blunder 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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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). Corollary. 53) satisfies u ∈ H 2 (Ωi ) ∩ H k−1/2 (Γ0 ). Then, inf |u − v|1,Ωi v∈V0h Ch|u|2,Ωi , where C > 0 is a constant not dependent on h and N . 63), we get the following error estimate for the finite element solution uN,R : h u − uN,R h 1,Ωi 3Ch|u|2,Ωi + R0 1 (N + 1)k−1 R N +1 |u|k−1/2,Γ0 . 3 Global ABCs for the Exterior Problems of 3-D Poisson Equation In this section, we discuss the global ABCs for the exterior problems (II) and (III) of the 3-D Poisson equation.

On the artificial boundary ΓR , ∂u ∂n (1) = ΓR ka0 H0 (kR) ∂u(R, θ) = ∂r 2 H (1) (kR) 0 ∞ +k n=1 (1) Hn (kR) (1) Hn (kR) {an cos nθ + bn sin nθ}. 12) into the above equation, we get ∂u ∂n (1) k H0 (kR) = (1) 2π H (kR) ΓR 0 · 2π 0 2π 0 ∞ (1) Hn (kR) k u(R, ϕ)dϕ + π n=1 Hn(1) (kR) u(R, ϕ) cos n(θ − ϕ)dϕ. 13) Chapter 1 ∂u ∂n (1) = ΓR 2π k H0 (kR) 2π H (1) (kR) 0 ∞ + ∂u ∂n Global ABCs for Second Order Elliptic Equations u(R, ϕ)dϕ 0 (1) (1) = ΓR 0 ∞ 0 ∂u(R, ϕ) sin n(θ − ϕ)dϕ. 14) 2π k H0 (kR) 2π H (1) (kR) − 2π Hn (kR) k π n=1 nHn(1) (kR) u(R, ϕ)dϕ 0 (1) 2π Hn (kR) k π n=1 n2 Hn(1) (kR) ∂ 2 u(R, ϕ) cos n(θ − ϕ)dϕ.

23) is the exact boundary condition on the artificial boundary for the exterior problem (I). Using any one of the boundary conditions, we can reduce the exterior problem (I) to a boundary value problem on Ωi . 16) using infinite series. All these introduce additional difficulties for numerical computations, as we need to deal with singular integrals or infinite series. 16). Then, we obtain a series of highly accurate approximate ABCs (N = 0, 1, 2, · · · ) as follows: ∂u ∂r 2π N ΓR =− n πR n=1 0 u(R, ϕ) cos n(θ − ϕ)dϕ := S1N (u|ΓR ).

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Artificial Boundary Method by Houde Han, Xiaonan Wu

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