Domain Decomposition Method Applied to Radiation Problems

results for error analysis of finite element method We consider the following four problems: 1: (E)w: Find u ∈ V such that a(u, v) = (f, v) for all v ∈ V . 2: (Eh)w: Find uh ∈ Vh such that a(uh, vh) = (f, vh) for all vh ∈ Vh. 3: (E)w: Find u ∈ V such that a (u , v) = (f, v) for all v ∈ V . DDM APPLIED TO RADIATION PROBLEMS 389 4: (Eh )w: Find u N h ∈ Vh such that a (uh , vh) = (f, vh) for all vh ∈ Vh. Then, we have the above four equations are equivalent to the following operator equations respectively: 1. (E)op : Au = f. 2. (Eh)op : Ahuh = fh with Ah = PhA, fh = Phf. 3. (E)op : Au = f. 4. (Eh )op : A N h u N h = fh with A N h = PhA N , fh = Phf. By Riesz’s representation theorem, two operators A and AN are defined as: a(u, v) = (Au, v) and a (u, v) = (Au, v) for all v ∈ V . Using the relations Au = Au = f and PhAuh = Ahuh = fh = Ah u N h = Phf = PhAu = PhA u , we can transform the expression of the error u− uh as follows: u− uh = u− vh + vh − uh = u− vh + (Ah )Ah vh − uh = u− vh + (Ah )Ah vh − (Ah )−1fh = u− vh + (Ah )Ah vh − (Ah )−1Phf = u− vh + (Ah )Ah vh − (Ah )−1PhAu = u− vh + (Ah ){Ah vh − PhAu} = u− vh + (Ah )−1{PhANvh − PhAu} = u− vh + (Ah )−1{PhAN (vh − u) + PhAu− PhAu} = {I − (Ah )−1PhAN )}(u− vh) + (Ah )−1Ph(AN −A)u. Hence we can estimate the above difference as: ‖u− uh ‖ ≤ (I + ‖(Ah )−1‖‖AN‖) inf vh∈Vh ‖u− vh‖+ ‖(Ah )−1‖‖(AN −A)u‖. Therefore, our next task is to prove the followings: 1. The uniform boundedness of ‖(Ah )−1‖: ‖(Ah )−1‖ ≤ M < +∞ with respect to h and N . 2. The truncation error estimate: ‖(AN − A)u‖ ≤ C Nα ‖u‖W under the regularity condition for u: u ∈ W ⊂ V . Actually, we have proved these conditions for the obstacle scattering case in [LK98a]. In the next section, we treat the case of wave-guide. 390 KAKO, KANO, LIU, YAMASHITA Application to the wave-guide problem We can apply the abstract error estimation based on the following observations: 1. The sesquilinear form b2,r(p, q) is bounded and nonnegative in V . Hence a0,DN (p, q) ≡ a0(p, q) + b2,r(p, q) is an inner product in V 2. The form b1(p, q) + b2,i(p, q) is compact with respect to a0,DN(p, q) in V . 3. We can then apply the results by Mikhlin [Mik64] (see also Kako [Kak81]) and we can prove the convergence of the finite element method under some additional condition on the non-existence of a positive eigenvalue. 4. The difference between a(p, q) and a(p, q) is written as: a(p, q)− a (p, q) = ∑ N<n ηn( L 2 )Cn(p)Cn(q) = ({Λ− ΛN}p, q). and ‖{Λ − Λ}p‖L2(0,L) tends to zero exponentially with respect to N or estimated from above by C Nα ‖u‖W with any α and a corresponding higher order Sobolev space W . Some Numerical Examples In this section, we show some numerical examples calculated by using the methods introduced in the previous sections. Obstacle scattering (by X.-J. Liu) Fig.1 shows a typical wave profile computed by the method introduced in [XJK96], [LK98a] and [LK98b]. Figure 1: Wave profile of 2D obstacle scattering DDM APPLIED TO RADIATION PROBLEMS 391 Seismic wave in 2D foundation (by T. Yamashita) We show two numerical results in Fig.2 where a single source is placed inside the foundation [YT97]). The left figure is the case of the artificial boundary with radius R = 1 and the right one is the case with R = 1, 25. There is a good coincidence between these two results and the Rayleigh wave is well captured.