By Houde Han, Xiaonan Wu
"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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Additional resources for Artificial Boundary Method
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).
Artificial Boundary Method by Houde Han, Xiaonan Wu