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 ).

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

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