DtN ² *1) May, 2016 MATHEMATICA NUMERICA SINICA Vol.38, No.2. Ë± Helmholtz µå Å± Dirichlet-to-Neumann. u = g, Γ, (1.1) r iku = o(r 1 2 ), r,

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1 16 Ý 5 38 Ð May, 16 MATHEMATICA NUMERICA SINICA Vol.38, No. Helmholtz ± µ³ DtN ² *1) ( Ò Ì ¼, 1144) Ë± Helmholtz µå Å± Dirichlet-to-Neumann (MDtN) ¹, È±É ¾, MDtN ÎÂÐ MDtN Å ÉÔ H 1 Ö Ð L Ö. Ü ¼Ú Ù. ÖÚ : ; Helmholtz µ; Dirichlet-to-Neumann ; ¹ ; Ö MR () ÔÝ: 65N3 1. Æ «¾ µ ÞÊ ³Ñ µä Ì¾ ½. ÜÆ, Ñ ¾ Ì¾ Ñ, ± ¾ µ ½. µ 9 Ü, Ñ ¾ Ä Ï Ç, É ÆÌ ³ ¼ µä È ÞÊ [1,]. ÆÏ Helmholtz Æ u+k u =, Ω Ú, u = g, Γ, u r iku = o(r 1 ), r, Ä Ω ßÒ Γ ÌÚ ¾Æ µ, ¾ k >. Engquist Majda [3] Ï Ë ½ Ã ¾. Bayliss Turkel [4] Laplace Helmholtz ¾.» ² ß³» [5,6] Ê ÀÞ µ Laplace, ÈÁ, Stokes Helmholtz Neumann Æ Ï Ó² ÏÆ«, Æ Ì Dirichlet-to-Neumann (DtN), Í Ú ¾ ½, ¾. ¾ Á µ ½ [7 1]. Í [11] Im(k) > Helmholtz Æ, DtN-FEM, Ñ Õ. ÆÍ,, DtN ¾ Ä ¾ È, ÈÓ¾ N Â ÂÐ Ì Ð., ¾ ÅÙ, N Û, Û, N * 15 Ý 9 3 ¹. 1) Å Ô : Ø À Å Ô ÍÊ (No. 1114). (1.1)

2 Ð «: Helmholtz µç Å± DtN ¹ 1 Ô ½ Ë. Harari Hughes [9,1] Û N ka ½ N ÖÐ. Ì Ö¹ N ka Ã, Grote Keller [13] ºß Åº Å ¾ Đ DtN (MDtN) Ñ ¾ «ØÙ MDtN ¾ DtN ¾., Koyama [14] Ê Helmholtz Æ DtN-FEM, Schatz Goldstein [15], u u N h m,ω a (m =,1) Õ. Hsiao [16] Garding s ², ± Inf-Sup À Õ. Koyama [17]» Schatz Goldstein, u u N m,ωa u N u N h m,ω a Õ Í u u N h m,ω a (m =,1) Õ. ÆÏ Schatz Goldstein, Í µ± u u N h,ω a u u N h 1,Ω a Ñ Õ ² º u u N h 1,Ω a u u N h,ω a Õ Ð. ÆÏÊ Helmholtz Dirichlet Æ, ¹ÄÐ Đ DtN Î, 3 ¹Ä Ý ÓÑ Õ, 4 ¹Ä Û¾ Î ØÐÆÑ Ð, 5 ¹Ä».. MDtN (1.1) { u H 1 (Ω) u Γ = g, ± D(u,v) =, v H 1 (Ω), (.1) Ä H 1(Ω) = {v H1 (Ω) v Γ = }, D(u,v) = ( u v k u v)dξdη. Ì (.1) ½, Ð Ñ ¾ Γ a := {x R x = a}(a > ) «Ý Γ, ¼Í Ω Ì ¾ µ Ω a ¾ µ B a. º Ð D(u,v) = ( u v k u v)dξdη + ( u v k u v)dξdη Ω a B a D 1 (u,v)+d (u,v). Ó²Á D (u,v) = ( u v k u v)dξdη = Ku vds ˆD (u,v), (.) Γ a B a Ä K Ê Æ DtN Î, ÏÆ«( [5]): Ω Ä Ku(θ) = k π u n = Ku, Γ a, (.3) π [ + m= H m (ka)cosm(θ θ )]u(θ )dθ, H m (z) = dh m (1) (z) dz H (1) m (z) = H (1) m+1 (z) H (1) m (z) m z Î n Γ a ÍÆ Õ, H (1) m (z) É m Hankel Þ¾., m =,1,, (.4)

3 16 Ý º (.1) Á Û³ ß² u H 1 (Ω a ) u Γ = g, ± D 1 (u,v)+ ˆD (u,v) =, v H 1 (Ω a ), (.5) Ä H 1 (Ω a) = {v H 1 (Ω a ) v Γ = }. Æ, D 1 (, )+ ˆD (, ) Ê² ÐÐß². DtN Î K H 1 (Γ a ) H 1 (Γ a ) ¾ÐÐ Î ( [18]), ÄÊ s >, H s (Γ a ) Sobolev Ã, Â Ì H s (Γ a ) = {w L (Γ a ) w s,γa < }, Ä w s,γ a = πa + m= (1+ m ) s a m, a m Î ¾, a m = π w(θ)e imθ dθ. (.5) : u N H 1 (Ω a ) u N Γ = g, ± Ä D 1 (u N,v)+ ˆD N (un,v) =, v H 1 (Ω a), (.6) ˆD N (u,v) = KN u N,v H 1 (Γ a) H 1 (Γ a) = Γ a K N u N vds, K N u(θ) = k π π [ N m= N ] H m (ka)cosm(θ θ ) u(θ )dθ., ²Ä ¾ È Á Ì ¾, m > N, u n =. Ì Ñ È Đ, Grote Keller Đ DtN ¾ : u n = Bu, Γ a, (.7) Ä B Õ Im r=a vbvds > ( v ) ÐÐ Î, DtN ¾ (.3) Æ Å Bu, Æ È K B, ¼Í Đ DtN ¾ ( [13]): u n = (KN B N )u+bu, Γ a. (.8) ²Á B Ô¾Ó ( m > N), Ô¾ÓÌ ¾. ÆÏ Bu (ik 1 a )u, Γ a, (.9) Ø MDtN ÎÂ : u n = LN u K N u ς m N u m e imθ +ςu, Γ a, (.1) Ä ς = ik+ 1 a, u m u(θ) Î ¾, u m = π u(θ)e imθ dθ.

4 Ð «: Helmholtz µç Å± DtN ¹ 3 3. MDtN-FEM ÓÒØĐ ÕÙ Ê (.6) MDtN : u N H 1 (Ω a ) u N Γ = g, ± D 1 (u N,v)+ D N (un,v) =, v H 1 (Ω a), (3.1) Ä D N (u,v) = LN u,v H 1 (Γ a) H 1 (Γ a) = Γ a L N u vds. Æ, D1 N(, )+ ˆD N (, ) Ê² ÐÐß²., Ð Ê ß²: u N h V h(ω a ) u N h Γ = g, ± D 1 (u N h,v h )+ D N (u N h,v h ) =, v h V,h (Ω a ), (3.) Ä V,h (Ω a ) = {v V h (Ω a ) v Γ = }, V h (Ω a ) H 1 (Ω a ) Ã Ω a ½Ã., = θ < θ 1 < θ < < θ N 1 < θ N = π ³ Γ a. Ø ÐÐ ¾ QU = F Á Ê 3., Ä Q = Q 1 +Q, Q 1 ÆÏ ÐÐß ² D 1 (, ), Q ÆÏ D N (, ). Æ Q 1, Q ÏÆ ¾ Î [q ij ] N N. (.5) (.6) (3.1) ÆÊ (3.) ½½ Ë Ð Á Ñ Ï [, 13, 16, 17, 19]. Ò Û ËÄ Õ. Þ 1. Ê z > m R, Re(H m (z)) >. (3.3). [] Ë A.1. Þ. ½ Ç k, a ¾ C ± kh ς m (ka) 1+m C.. Hankel Þ¾ ¾ ( [11]): H (1) m (.4) Á (z) = i(m 1)! π (z) m (1+ 1 m 1 H m (ka) = m ka (1+O( 1 m )). (z) ( 1 ) ) +O m, m. º m, kh m (ka) m = 1+m a m (1+O( m )) a. kh m (ka) 1+m ¾, Ø ½ C >, ± kh ς m (ka) 1+m C. Þ 3. Ê u,v H 1 (Ω a ), D 1 (u,v)+ ˆD (u,v) D 1 (u,v)+ D N ÐÐ, ½ Ç Ω a ±¾ α, ± (u,v) Ä Ê² D 1 (u,v)+ ˆD (u,v) α u 1,Ωa v 1,Ωa, D 1 (u,v)+ D N (u,v) α u 1,Ωa v 1,Ωa ; Re{D 1 (u,u)+ ˆD (u,u))}+(k +1) u,ω a u 1,Ω a, Re{D 1 (u,u)+ D N (u,u)}+(k +1) u,ω a u 1,Ω a.

5 4 16 Ý. Æ, ³, º Ë½. Î Ð Ó. º, u 1,Ω a = D 1 (u,u)+(k +1) u,ω a = D 1 (u,u)+ D N (u,u)+(k +1) u,ω a D N (u,u). Ê ², Æ, u 1,Ω a = Re{D 1 (u,u)+ D N (u,u)}+(k +1) u,ω a Re{ D N (u,u)}.», ¾ u(θ) = + m= { } Re{ D N (u,u)} = Re ūl N uds Γ a a m e imθ, a m = 1 π π u(θ)e imθ dθ, Ë 1, Á { = Re ūk N uds ς Γ a = Re {[k =. m N m N m N a m +ς a m } a m H m (kr)+( ik + 1 R ) kre{h m (kr)} a m + 1 R a m } a m ] Re{D 1 (u,u)+ D N (u,u)}+(k +1) u,ω a u 1,Ω a. É, ÆÁ Re{D 1 (u,u)+ ˆD (u,u))}+(k +1) u,ω a u 1,Ω a. Ì Î I h : H (Ω a ) V h (Ω a ) v I h v 1,Ωa C 1 h v,ωa, v H (Ω a ), Ä C 1 ³ h u. ³ ÂËÑ ²: inf u v 1,Ω a u I h u 1,Ωa C 1 h u,ωa. (3.4) v V h (Ω a) v m 1,Γa C v m,ωa, v H m (Ω a )(m = 1,), (3.5) Ä C Ç Ω a, Å v ±¾. Ð.5 Å (i) Ê ρ L (Ω a ), Ë ½ ϕ ρ H (Ω a ) H 1 (Ω a); (ii) ½ ±¾ C 3, ± D 1 (v,ϕ ρ )+ ˆD (v,ϕ ρ ) = (ρ,v), v H 1 (Ω a), (3.6) Þ 4. u u N h (.5) (3.) ½, ϕ ρ,ωa C 3 ρ,ωa. (3.7) u u N h 1,Ω a (8k +4α +1) inf u v h 1,Ω v h V h (Ω a), v h a ( Γ=g ) + 4 sup ˆD (u,v h ) D N (u,v h) v h 1,Ωa +8(k +1) u u N h,ω a, v h V,h (Ω a) (3.8)

6 Ð «: Helmholtz µç Å± DtN ¹ 5 Ä α Ë 3 Ä ±¾.. (.5) Á ² Å (3.) D 1 (u,v h )+ D N (u,v h ) = D N (u,v h ) ˆD (u,v h ), v h V,h (Ω a ). (3.9) D 1 (u u N h,v h )+ D N (u u N h,v h ) = D N (u,v h ) ˆD (u,v h ), v h V,h (Ω a ). (3.1) Ê v h V h (Ω a ) v h Γ = g, Ì u N h v h V,h (Ω a ), Ë 3 (3.1), Á u N h v h 1,Ω a Re{D 1 (u N h v h,u N h v h)+ D N (un h v h,u N h v h)}+(k +1) u N h v h,ω a ¼Í = Re{D 1 (u v h,u N h v h )+ D N (u v h,u N h v h ) +D 1 (u N h u,un h v h)+ D N (un h u,un h v h)}+(k +1) u N h v h,ω a = Re{D 1 (u v h,u N h v h )+ D N (u v h,u N h v h ) D N (u,un h v h)+ ˆD (u,u N h v h)}+(k +1) u N h v h,ω a α u v h 1,Ωa u N h v h 1,Ωa + ˆD (u,u N h v h ) D N (u,u N h v h ) +(k +1) u N h v h,ω a ( α u v h 1,Ωa + sup v h V,h (Ω a) +(k +1) u N h v h,ω a 1 ( α u v h 1,Ωa + sup v h V,h (Ω a) + 1 un h v h 1,Ω a +(k +1) u N h v h,ω a, ( u N h v h 1,Ω a α u v h 1,Ω a + + (k +1) u N h v h,ω a. ˆD (u,v h ) D N (u,v h ) ) u N h v h 1,Ωa v h 1,Ωa ˆD (u,v h ) D N (u,v h ) ) v h 1,Ωa sup v h V,h (Ω a) ˆD (u,v h ) D N(u,v ) h) v h 1,Ωa ² ² u u N h 1,Ω a u v h 1,Ωa + u N h v h 1,Ωa Æ ², Ð u u N h 1,Ω a u v h 1,Ω a + u N h v h 1,Ω a ( (+4α ) u v h 1,Ω a +4 sup v h V,h (Ω a) + 8(k +1)( u u N h,ω a + u v h,ω a ), ˆD (u,v h ) D N (u,v h ) ) v h 1,Ωa º, (3.8). Þ 5. u H (Ω a ) H s 1 (Γ a ) u N h V h(ω a ) (.5) (3.) ½, s =,3,, (.5) Å, Ø ½ Ç N, h, s, u u N h, Ç Ω a, k a ±¾ C,C 1,C,C 3, ± u u N h,ωa C 3 { (α C 1 h+c C N 1 ) u u N h 1,Ωa +C C (C 1 h+n 1 ) ε(u,n,s),γa }, (3.11)

7 6 16 Ý Ä α Ë 3 Ä ±¾, ε(u,n,s) = [ (1+m ) s 1 a m ] 1 + [ (1+m ) s 1 a m ] 1 m=, a m = 1 π π u(θ)e imθ dθ.. (.3), (.1), u v ¾, Ë ÂË (3.5), Á ˆD (u,v) D ( N (u,v) = π ς = ( π π N s 1 [ (ς kh m (ka))a m b m ) a m b m k H m (ka)a m b m ) ς kh m (ka) [ (m +1) s 1 am ]1 1+m ς kh m (ka) (m +1) 1 bm ]1 1+m ε(u,n,s) C,Γa v 1,Γa ε(u,n,s) C C,Γa v 1,Ω a, (3.1) Ä a m = 1 π π u(θ)e imθ dθ b m = 1 π π v(θ)e imθ dθ. É, ÐÁ ˆD (u,v) D N (u,v) C ε(u,n,s) N s u s 1,Γa v 3,Γa C C ε(u,n,s) N s u s 1,Γa v,ω a (3.13) ˆD (u,v) D N (u,v) C N 1 u 1,Γa v 3,Γa C C N 1 u 1,Ωa v,ωa. (3.14) Ê ρ = u N h u, ϕ ρ (3.6) ½, (3.1), Ë 3, (3.1), (3.13), (3.14), (3.4), (3.5) (3.7), (u u N h,ρ) = D 1 (u u N h,ϕ ρ )+ ˆD (u u N h,ϕ ρ ) = D 1 (u u N h,ϕ ρ I h ϕ ρ )+ D N (u un h,ϕ ρ I h ϕ ρ ) +D 1 (u u N h,i hϕ ρ )+ D N (u un h,i hϕ ρ )+ ˆD (u u N h,ϕ ρ) D N (u un h,ϕ ρ) = D 1 (u u N h,ϕ ρ I h ϕ ρ )+ D N (u un h,ϕ ρ I h ϕ ρ )+ D N (u,i hϕ ρ ) ˆD (u,i h ϕ ρ ) +ˆD (u u N h,ϕ ρ ) D N (u u N h,ϕ ρ ) = D 1 (u u N h,ϕ ρ I h ϕ ρ )+ D N (u u N h,ϕ ρ I h ϕ ρ ) D N (u,ϕ ρ I h ϕ ρ ) +ˆD (u,ϕ ρ I h ϕ ρ )+ D N (u,ϕ ρ) ˆD (u,ϕ ρ )+ ˆD (u u N h,ϕ ρ) D N (u un h,ϕ ρ) α u u N h 1,Ω a ϕ ρ I h ϕ ρ 1,Ωa + D N (u,ϕ ρ I h ϕ ρ ) ˆD (u,ϕ ρ I h ϕ ρ ) + ˆD (u,ϕ ρ ) D N (u,ϕ ρ ) + ˆD (u u N h,ϕ ρ ) D N (u u N h,ϕ ρ ) α C 1 h u u N h 1,Ωa ϕ ρ,ωa +C C 1 C h ε(u,n,s),γa ϕ ρ,ωa +C C ε(u,n,s) N s u s 1,Γa ϕ ρ,ωa +C C N 1 u u N h 1,Ωa ϕ ρ,ωa C 3 {(α C 1 h+c C N 1 ) u u N h 1,Ω a +C C (C 1 h+n 1 ) ε(u,n,s),γa} ρ,ω a.

8 Ð «: Helmholtz µç Å± DtN ¹ 7 (3.11) ². ÐÞ. u H (Ω a ) H s 1 (Γ a ) u N h V h(ω a ) (.5) (3.) ½, s =,3,, h Ô N Ô, ½ Ç N, h, s, u u N h, Ç Ω a, k a ±¾ C,C 1,C,C 3 C 4, ± Æ u u N h 1,Ω a [C 4 (8k +4α +1)]1 C 1 h u,ωa +C C [C 4 +4(1+k )C 3 C 4(C 1 h+n 1 ) ] 1 ε(u,n,s),γa, (3.15) u u N h,ω a C 1 C 3 [C 4 (8k +4α +1)]1 (α C 1 h+c C N 1 )h u,ωa +C C C 3 {[C 4 +4(1+k )C 3C 4 (C 1 h+n 1 ) ] 1 +(C1 h+n 1 )} ε(u,n,s),γa. (3.16).» (3.8), (3.4), (3.1) (3.11) Á ( ε(u,n,s) ) u u u N h 1,Ω a (8k +4α +1)C 1 h u,ω a +4C C N s 1 s 1 +8(k +1)C,Γa 3 { (α C 1 h+c CN 1 ) u u N h 1,Ωa +C C (C 1 h+n 1 ) ε(u,n,s) ( ε(u,n,s) ) u N s 1 s 1 +16(k +1)C,Γa 3,Γa } (8k +4α +1)C 1 h u,ω a +4C C { (α C 1 h+c CN 1 ) u u N h 1,Ω a +CC (C 1 h+n 1 ) ( ε(u,n,s) ) u N s 1 s }, 1,Γa 1 h Ô N Ô, Æ h h, N N C 4 = > 1 16(k +1)C3 (αc1h+cc N 1 ), u u N h 1,Ω a C 4 (8k +4α +1)C 1 h u,ω a ( ε(u,n,s) ) u +[4C 4 C C +16(1+k )C3 C 4C C (C 1h+N 1 ) ] N s 1 s 1,Γa, (3.15) ²³. (3.11) (3.15) ²Á { u u N h,ω a C 3 (α C 1 h+c C N 1 ) [C 4 (8k +4α +1)]1 C1 h u,ωa +C C [C 4 +4(k +1)C3 C 4(C 1 h+n 1 ) ] 1 ε(u,n,s) },Γa +C 3 C C (C 1 h+n 1 ) ε(u,n,s),γa (α C 1 h+c C N 1 )[C 4 (8k +4α +1)]1 C1 C 3 h u,ωa +C C C 3 {[C 4 +4(1+k )C3 C 4(C 1 h+n 1 ) ] 1 +(C1 h+n 1 )} ε(u,n,s),γa.

9 8 16 Ý 4. ß. Ω = {(x,y) x 1, y 1} Γ = Ω, Helmholtz Æ u+k u =, Ω Ú, u = g, Γ, u r iku = o(r 1 ), r, ½: u(r,θ) = H (1) (kr), r 1, θ π. Γ a = {(x,y) x +y = 9} Ω a È 6 16 Ì Õ P=6, Õ M=16 ²ß, Â Ã É Ä É Á Æ È 3 8, 1 3, ÑÊ u u N h L Re (Ωa) 1, Rate = u un h u u N, Order = log Rate. h/ 1 k =, DtN-FEM Ñ MDtN-FEM Ñ DtN-FEM MDtN-FEM M P N = 3 N=4 N = 5 Rate N = 3 Rate N = 5 Rate Order

10 Ð «: Helmholtz µç Å± DtN ¹ È Ê À N, DtN-FEM ÑÊ MDtN-FEM ÑÊ Â. N = 3, Ê ÀÈ M P, DtN-FEM ÑÊ MDtN-FEM ÑÊ Â DtN FEM MDtN FEM u u N h L Re (Ω a ) N Ã 3 8 É, Ë± Á N, DtN-FEM MDtN-FEM ÒË 1 DtN FEM MDtN FEM 1 ln( u u N h L Re (Ω a ) ln(m P) Ã 3 N = 3, Ë± Á ln(m P), DtN-FEM MDtN-FEM Ç ln( u u N h L Re (Ω a)) Â Á, N < 4, DtN-FEM, ÑÊ Ó MDtN-FEM ÑÊ ; N 4, Æ ÑÊ Æ Á. ºÁ Ó, MDtN-FEM ÛÔÂ Ê È¾ N Ã, N Á Đ ¾. ¼Â 3 Á, N = 3, Ê ÀÈ M P, MDtN-FEM DtN-FEM.

11 1 16 Ý 5. Û Ð ÆÏÊ ¾ µ Helmholtz Æ, Ñ ¾ Đ DtN, ¼ ÍÐ MDtN-FEM, Æ µ Ó Ý Õ. Í Û Ê DtN- FEM MDtN-FEM Æ, «ØÐ Õ Ð, Ó MDtN-FEM ÑÊ DtN-FEM ¾Ð. Å. À Ü ± ±. Ü [1] Yu D H. Natural Boundary Integral Method and Its Applications[M], Beijing/Dordrecht/New York/London: Kluwer Academic Publisher/Science Press,. [] Han H D, Wu X N. Artificial Boundary Method[M], Beijing: Tsinghua University Press, 1. [3] Enquist B, Majda A. Absorbing boundary conditions for numerical simulation of waves[j]. Math. Comput., 1977, 31(139): [4] Bayliss A, Gunzburger M, Turkel E. Boundary conditions for the numerical solution of elliptic equations in exterior regions[j]. SIAM J. Appl. Math., 198, 4(): [5] Feng K. Finite element method and natural boundary reduction[c]. In: Proc. Inter. Cong. Math., Warszawa, 1983, [6] Feng K. Asymptotic radiation conditions for reduced wave equation[j]. J. Comput. Math., 1984, (): [7] Feng K, Yu D. Canonical integral equations of elliptic boundary value problems and their numerical solutions[c]. In: (K. Feng, J.L. Lions, eds.) Proc. China-France Symp. on the Finite Element Method (April 198), pp Beijing: Science Press, [8] Keller J B, Givoli D. Exact nonreflecting boundary condition[j]. J. Comput. Phys., 1989, 8(1): [9] Harari I and Hughes T J R. Galerkin/least squares finite element methods for the reduced wave equation with non-reflecting boundary conditions in unbounded domains[j]. Comput. Methods Appl. Mech. Engrg., 199, 98(3): [1] Bao G. Finite element approximation of time harmonic waves in periodic structures[j]. SIAM J. Numer. Anal., 1995, 3(4): [11] Li R. On The coupling of BEM and FEM for exterior problems for the Helmholtz equation[j]. Math. Comput., 1999, 68(7): [1] Harari I. A survey of finite element methods for time-harmonic acoustics[j]. Comput. Methods Appl. Mech. Engrg., 6, 195(13): [13] Grote M G, Keller J B. On nonreflecting boundary conditions[j]. J. Comput. Phys., 1995, 1(): [14] Koyama D. Error estimates of the DtN finite element method for the exterior Helmholtz problem[j]. J. Comput. Appl. Math., 7, (1): [15] Goldstein C l. A finite element method for solving Helmholtz type equations in waveguides and other unbounded domains[j], Math. Comput., 198, 39(16): [16] Hsiao G C, Nigamb N, Pasciak J E, Xu L. Error analysis of the DtN-FEM for the scattering problem in acoustics via Fourier analysis[j]. J. Comput. Appl. Math., 11, 35(17):

12 Ð «: Helmholtz µç Å± DtN ¹ 11 [17] Koyama D. Error estimates of the finite element method for the exterior Helmholtz problem with a modified DtN boundary condition[j]. J. Comput. Appl. Math., 9, 3(1): [18] Masmoudi M. Numerical solution for exterior problems[j]. Numer. Math., 1987, 51(1): [19] Jiang X, Li P J, Wei Y. Numerical solution of acoustic scattering by an adaptive DtN finite element method[j], Commun. Comput. Phys., 13 13(5): [] Koyama D. A controllability method with an artificial boundary condition for the exterior Helmholtz problem[j]. Japan J. Indust. Appl. Math., 3, (1): THE FINITE ELEMENT METHOD WITH A MODIFIED DtN BOUNDARY CONDITION FOR EXTERIOR PROBLEMS OF THE HELMHOLTZ EQUATION Zheng Quan Gao Yue Qin Feng (College of Sciences, North China University of Technology, Beijing 1144, China) Abstract In this paper, we investigate a finite element method with a modified Dirichlet-to- Neumann boundary condition (MDtN-FEM) for the Helmholtz equation on unbounded domains in R. The a priori error estimates depending on the mesh size, the location of MDtN boundary and the truncation of the series in MDtN are established in the H 1 - and L -norms. Numerical examples demonstrate the advantage in accuracy and efficiency for the method. Keywords: unbounded domain; Helmholtz equation; finite element method; modified Dirichlet-to-Neumann boundary condition; error estimate Mathematics Subject Classification: 65N3