(, ) (SEM) [4] ,,,, , Legendre. [6] Gauss-Lobatto-Legendre (GLL) Legendre. Dubiner ,,,, (TSEM) Vol. 34 No. 4 Dec. 2017

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1 JOURNAL OF SHANGHAI POLYTECHNIC UNIVERSITY Vol. 34 No. 4 Dec. 17 : ( DOI: /j.cnki.jsspu (, 19 :,,,,,, : ; ; ; ; ; : O 41.8 : A, [1],,,,, Jung [] Legendre, [3] Chebyshev Fourier,,,,,, (SEM [4],,,,,,,,, SEM : [5] [6] Gauss-Lobatto-Legendre (GLL Chebyshev Lagrange, Legendre Dubiner [7], (QSEM (TSEM,,,, : : (1987,,,,, wtshao@sspu.edu.cn : (16ZR1417, (XXKZD134

2 , C,,,,,,,,, 1 1.1, (β u( + ru( = f(, Ω + Ω (1 u( =, Ω ( : r ; β Ω +, β = β + ; Ω, β = β : u H 1 (Ω, (β u, v + r(u, v = (f, v, v H 1 (Ω Ω T, Ω e N el Nel, Ω = e=1 Ωe, e=1 Ωe = Ω e : N e (β u, v = (β e u e, v e e=1 N e N e (u, v = (u e, v e, (f, v = (f e, v e e=1 e=1 Ω e Ω st = [ 1, 1], Ω e Ω st Ω st, N : N u st (ξ = û st k ψ k (ξ, 1 ξ 1 : û st k k= ; ψ k(ξ, : ψ (ξ = 1 ξ, ψ N(ξ = 1 + ξ ψ k (ξ = 1 k k + 1 φ k 1(ξ, 1 k N 1 : φ k (ξ = L k (ξ L k+ (ξ; L k (ξ Legendre, Mu = F : M, u, u = [û e k ], k =, 1,, N, e = 1,,, N el; F f e ( 1. (QSEM,, : Ω = Nel e=1 Ωe, Ω e = [ e 1, e ] [y e 1, y e ],, Ω st = [ 1, 1], Ω e Ω st Ω st, N : N N u st (ξ, η = û st i,jψ i,j (ξ, η, û st i,j i= j=,, ψ i,j (ξ, η = ψ i (ξψ j (η, [3] : ( 1 ξ ( 1 η ψ, (ξ, η = ( 1 ξ ( 1 + η ψ,n (ξ, η = ( 1 + ξ ( 1 η ψ N, (ξ, η = ( 1 + ξ ( 1 + η ψ N,N (ξ, η = ( 1 i ( 1 η ψ i, (ξ, η = i + 1 φ i 1(ξ ( 1 i ( 1 + η ψ i,n (ξ, η = i + 1 φ i 1(ξ 1 i N 1 ( 1 ξ ( 1 j ψ,j (ξ, η = j + 1 φ j 1(η ( 1 + ξ ( 1 j ψ N,j (ξ, η = j + 1 φ j 1(η 1 j N 1 (3 (4 (5 (6

3 4 : 85 ( 1 ψ i,j (ξ, η= i 1 i 1(ξ( i+1 φ j j+1 φ j 1(η 1 i, j N 1 (7 Legendre, Laplacian 1 ( y = y A ξ1 ξ ( 1 + y C ξ ( y B ξ1 +, Duffy [8] (9 η 1 = 1 + ξ 1 1 ξ 1, η = ξ, 1 η 1, η 1 (1 ξ 1 = (1 + η 1(1 η 1, ξ = η (11, (8 (9 Ω e τ, Duffy, τ Ω st, ( y ξ A C 1 1 (a The element Laplacian matri 1 B Eqs.(8,(9 1 1 ξ 1 C A 1 B (a (b C η 1 C' the Duffy s transform 1 1 η 1 1 Fig. 1 1 (b The element mass matri 1 QSEM Laplacian, N = 9 Sparse structure of the elemental Laplacian and mass matrices of QSEM with the modal bases of order N = (TSEM Ω e 3 : {( A, y A, ( B, y B, ( C, y C }, Ω e τ = {(ξ 1, ξ 1 ξ 1, ξ ; ξ 1 + ξ } : ( = A ξ1 ξ ( B ξ1 + ( 1 + C ξ (8 Fig. A 1 (a τ(b, Ω st (c (c Each triangular element (a is mapped into the reference triangular τ; (b then mapped into the standard element Ω st ; (c finally all local operations are evaluated in Ω st (b τ, [3] : ψ A (η 1, η = 1 η 1 1 η ψ B (η 1, η = 1 + η 1 1 η (1 ψ C (η 1, η = 1 + η ψi AB (η 1, η = 1 i ( 1 i + 1 φ η i+1 i 1(η 1 (13 1 i N B

4 ψ AC j (η 1, η = ψ BC j (η 1, η = ( 1 i,j (η 1, η = ( 1 + η ψ Inter ( 1 η1 ( 1 1 j N ( 1 + η1 ( 1 1 j N i i + 1 φ i 1(η 1 j j + 1 φ j 1(η j j + 1 φ j 1(η ( 1 η (14 (15 i+1 P i+1,1 j 1 (η (16 1 i, j N, i + j N, Duffy Ω st 1 1, CC C, C, η GLL C η 1 GLL, η Gauss-Radau,, 3 Laplacian,, Matlab 7, Matlab PDE 1,, Dirac [] (βu + u = f + µδ( α, (, L (17 u( = u(l = (18 C 1 cos(γ + C sin(γ + 1 (, α u( = (19 C 3 cos(γ + + C 4 sin(γ (α, L : γ = 1/1; γ + = 1/ 1; α = 1π/6; L = 1π/; β = 1; β + = 1; f = 1; C i Dirichlet u( C, = α : [βu ] =α = β + u (α + β u (α = µ, µ = 1, 1 1 L N N el, [] Legendre (LCM,,, SEM, 4 5 µ = 1, 1,, N el =, N = (a The element Laplacian matri (b The element mass matri 3 TSEM Laplacian, N = 14 Fig. 3 Sparse structure of the elemental Laplacian and mass matrices of TSEM with the modal bases of order N = 14

5 4 : 87 1 LCM SEM L, µ = 1, 1 Tab. 1 Comparison of the L norm errors for LCM and SEM with µ = 1, Eample 1 LCM [] N SEM N el = N el = 4 N el = LCM SEM L, µ = 1, 1 Tab. Comparison of the L norm errors for LCM and SEM with µ = 1, Eample 1 LCM [] N SEM N el = N el = 4 N el = Plot of the numerical solution u( Pointwise errors Plot of the numerical solution u( Pointwise errors SEM, µ = 1, N el =, N = 1, 1 Fig. 4 The numerical solution and pointwise errors of SEM with µ = 1, N el =, N = 1, Eample 1 5 SEM, µ = 1, N el =, N = 1 1 Fig. 5 The numerical solution and pointwise errors of SEM with µ = 1, N el =, N = 1, Eample 1 Ω = (, L (, 1 [] (βu + u yy + (1 + π u = sin(πy(1 µδ( α ( u =, (, y Ω (1 : u(, y = sin(πy C 1 cos(γ + C sin(γ + 1, (, α C 3 cos(γ + + C 4 sin(γ + + 1, (α, L : β β + γ γ + α L C i 1 µ = 1, Ω

6 [, α] [, 1], [α, L] [, 1], QSEM 3, L L LCM [] LCM,, QSEM 3 LCM [] SEM L L, µ = 1, Tab. 3 Comparison of the L and L norm errors for LCM and SEM with µ = 1, Eample LCM [] N QSEM L L L TSEM [9], (β u = f, (, y Ω = [ 1, 1] ( [u] =, (, y Γ (3 [ u n] = µ, (, y Γ (4 u = g, (, y Ω (5 : Γ = Ω Ω +, Ω = {(, y r R, r = + y }, Ω + = [ 1, 1] /Ω, [ u n] = u n + u + n +, n ± Ω ± : γ α /β, r R u(, y = γ α ( 1 β + + β 1 γ α β +, r > R : α = 3, r = 1/, R = 1/ f, µ g β 4 : (1 β + = 1 3, β = 1; ( β + = 1 6, β = 1; (3 β + = 1, β = 1 3 ; (4 β + = 1, β = 1 6 β + /β 4 N = 4,, TSEM L L, h, 4, β + /β, TSEM, 6 β + /β = 1 6, 1 6 4, Ω = [ 1, 1], [1] (β(, y u = f(, y+ C δ( X(sδ(y Y (sds (6 Γ h 4 TSEM L L, 3 Tab. 4 The L and L errors of TSEM, Eample 3 β + = 1 3, β = 1 β + = 1 6, β = 1 L L L L 1/ / / / h β + = 1, β = 1 3 β + = 1, β = 1 6 L L L L 1/ / / /

7 4 : 89 u y 1. u (a β + /β = 1 6 (b β + /β = β + /β, TSEM, 3 Fig. 6 The numerical solutions of TSEM with the very small and big ratio β + /β, Eample 3 y.5 1., f(, y = 8r + 4 r + 1, r 1/ β(, y = b, r > 1/ : u(, y = r, r 1/ ( 1 1 8b 1 ( r 4 /4 + b + r /b+ C log(r/b, r > 1/ Dirichlet 5 b = 1, C =.1 b = 3, C =.1, TSEM L L, 7, TSEM,, b = 3 <,,, 5 TSEM L L, 4 Tab. 5 The L and L errors of TSEM, Eample 4 h b = 1, C =.1 b = 3, C =.1 L L L L 1/ / / / / u y.5 1. u y.5 1. (a b = 1, C =.1 (b b = 3, C =.1 7 TSEM, 4 Fig. 7 The numerical solutions of TSEM, Eample 4

8 ,,,, : [1] WU Y, TRUSCOTT S, OKADA M. A finite difference scheme for an interface problem [J]. Japan J Indust Appl Math, 1, 7(: [] SHIN B C, JUNG J H. Spectral collocation and radial basis function methods for one-dimensional interface problems [J]. Appl Numer Math, 11, 61(8: [3] SHEN J, TANG T. Spectral and high-order methods with applications [M]. nd ed. Beijing: Science Press, 6. [4] KARNIADAKIS G, SHERWIN S. Spectral/hp element methods for CFD [M]. nd ed. New York: Oford University Press, 13. [5] DEHGHAN M, SABOURI M. A spectral element method for solving the Pennesbioheat transfer equation by using triangular and quadrilateral elements [J]. Appl Math Model, 1, 36(1: [6] FAKHAR IZADI F, DEHGHAN M. A spectral element method using the modal basis and its application in solving second-order nonlinear partial differential equations [J]. Math Method Appl Sci, 15, 38(3: [7] DUBINER M. Spectral methods on triangles and other domains [J]. J Sci Comput, 1991, 6(4: [8] DUFFY M G. Quadrature over a pyramid or cube of integrands with a singularity at a verte [J]. Siam Journal on Numerical Analysis, 198, 19(6: [9] HOU S, LIU X D. A numerical method for solving variable coefficient elliptic equation with interfaces [J]. J Comput Phys, 5, (: [1] LEVEQUE R J, LI Z. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources [J]. Siam J Numer Anal, 1994, 31(4, A Numerical Study of The Modal Bases Spectral Element Method for Solving Interface Problems SHAO Wenting (School of Science, Shanghai Polytechnic University, Shanghai 19, China Abstract: The numerical solution of one and two dimensional interface problems using the modal bases spectral element method (SEM was studied. Equations with the discontinuous coefficient or singular source were concerned. For two dimensional cases, the implementations of the quadrilateral element (QSEM and the triangular element (TSEM were both investigated. Several numerical eamples with eact solutions were provided to illustrate the feasibility and effectiveness of the modal bases SEM. Compared with other numerical results, eponential accuracy was regained for the one dimensional case, and better accuracy was also obtained for the two dimensional case. Keywords: spectral element method; quadrilateral elements; triangular elements; interface problems; discontinuous coefficient; singular source