SUPER-GEOMETRIC CONVERGENCE OF A SPECTRAL ELEMENT METHOD FOR EIGENVALUE PROBLEMS WITH JUMP COEFFICIENTS *

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1 Journal of Comutational Mathematics Vol.8, No.,, htt:// doi:.48/jcm.9.-m6 SUPER-GEOMETRIC CONVERGENCE OF A SPECTRAL ELEMENT METHOD FOR EIGENVALUE PROBLEMS WITH JUMP COEFFICIENTS * Lin Wang College of Mathematics and Comuter Science, Hunan Normal University, Changsha 48, China and College of Information Technical Science, Nankai University, Tianjin 7, China rinoya@mail.nankai.edu.cn Ziqing Xie College of Mathematics and Comuter Science, Hunan Normal University, Changsha 48, China ziqingxie@hunnu.edu.cn Zhimin Zhang Deartment of Mathematics, Wayne State University, Detroit, MI 48, USA and College of Mathematics and Comuter Science, Hunan Normal University, Changsha 48, China zzhang@math.wayne.edu Abstract We roose and analyze a C sectral element method for a model eigenvalue roblem with discontinuous coefficients in the one dimensional setting. A suer-geometric rate of convergence is roved for the iecewise constant coefficients case and verified by numerical tests. Furthermore, the asymtotical equivalence between a Gauss-Lobatto collocation method and a sectral Galerkin method is established for a simlified model. Mathematics subject classification: Primary 65N, Secondary 65N5, 65N5, 65N, 65D, 74S5, 4A, 4A5. Key words: Eigenvalue, Sectral method, Collocation, Galerkin finite element method.. Introduction We often encounter eigenvalue roblems with discontinuous coefficients in ractice. Examles of such alications may be found in []. In this aer, we consider the following one dimensional model roblem: Find, u) R + H π, π) such that u x) = cx)ux), u π) = uπ), u π) = u π)..) Here cx) c > is a iecewise constant, or iecewise analytic function. The hysics background of this model roblem comes from the source-free Maxwell equations describing the transverse-magnetic mode in the one-dimensional eriodic media, where the function u reresents the electric field attern, and the dielectric function cx) describes a unit cell from a multilayer structure with π-eriodicity. This model roblem was discussed by Min and * Received March 8, 9 / Revised version received May 5, 9 / Acceted June 6, 9 / Published online February, /

2 Suer-Geometric Convergence of a Sectral Element Method 49 Gottlieb in [] where C conforming sectral collocation methods were constructed on two elements over H er π, π) = { v H π, π) : v π) = vπ), v π) = v π) }, and error bounds of tye O m ) were established. Note that the solution of.) belongs to C. It would be interesting to discuss C sectral element methods over H er π, π) = { v H π, π) : v π) = vπ)big}, since the construction of a C sectral element method is much simler than that of the global C sectral collocation method roosed in []. The idea of the sectral element can be found, e.g., in an early work []. Note that the sectral element method is equivalent to the so-called -version finite element method, see e.g., []. Under the finite element variational framework, we are able to rove a suer-geometric error bound of tye Oe log γ) ). In some earlier works of the third author, the suer-geometric error bound of tye Oe log γ) ) has been established for some sectral/collocation aroximations of the two-oint boundary roblem [7, 8]. Our error bound for the eigenvalue aroximation doubles the error bound for the associated eigenfunction aroximation, the fact we have known for the h-version finite element method. It is worthy to oint out that in the literature of the sectral method, it is a common ractice to consider error bounds of tye O m ), see, e.g., [5 7,, 5, 6], and reference therein. To the best of our knowledge, this is the first time that a suer-geometric convergence rate is established for the eigenvalue aroximation by the sectral method.. Theoretical Setting The variational formulation of.) is to find, u) R + H er π, π) such that In this aer, we also consider the Dirichlet roblem u, v ) = cu, v), v H er π, π)..) u x) = cx)ux), u) = = u). Its variational formulation is to find, u) R + H, ) such that u, v ) = cu, v), v H, )..) By the general theory [, 8], both roblems.) and.) have countable infinite sequence of eigen-airs j, u j ) satisfying <, u i, u j) = j cu i, u j ) = j δ ij. Furthermore, eigenvalues can be characterized as extrema of the Rayleigh quotient Ru) = u, u )/cu, u) as follows = inf u S = Ru ), k = inf Ru) = Ru k), k =,,..., u S, u,u j )=,j=,...,k

3 4 L. WANG, Z.Q. XIE AND Z.M. ZHANG where S = Her π, π) or H, ). Next, we describe the framework of our numerical aroximation. We artition the solution interval into m sub-intervals element) such that cx) is analytic on each interval. Let h be the maximum length of all elements, we then define a finite dimensional subsace S h S, as a iecewise olynomial of degree on each element. Our sectral element method is to find an eigen-air ), w ) R + S h such that w, v ) = )cw, v), v S h..) Note that the artition arameter h is fixed and convergence is achieved by increasing olynomial degree. Therefore, we may suress the index h later. By the general theory [,8], the roblem.) has a finite sequence of eigen-airs j,, w j, ) satisfying { m H <,, N,, N =, ) m Her π, π) w i,w j,) = j, cw i,, w j, ) = j, δ ij ;, = min w S = Rw, ),.4) k, = min w S, w,w j, )=,j=,...,k Rw) = Rw k,), k =,,....5) One imortant observation from the above Minimum-Maximum rincile is that the secific eigenvalue aroximation is from above in the sense k k,+ k, k, k,.. A Galerkin Sectral -Version) Method Without loss of generality, we consider iecewise constant cx) as in [] with jum at the center of the solution domain. In articular, we take { x π, ), cx) = ω x, π). Instead of constructing C shae functions for eigenvalue roblem.), we seek for a C aroximation w H, ) with traditional exansion w x) = w N + N + ) + w j φ j+ x) + w Nx) + j= j=+ w j ψ j + x),.) where N x), Nx), and N + x) are linear nodal shae functions at the left end, middle oint, and right end of the solution interval, resectively; φ j and ψ j are bubble functions on the left and right intervals, resectively. The counterart of φ k+ in [, ] is defined as k + ξ ) ˆφ k+ ξ) = L k t)dt = L k+ ξ) L k ξ)..) k + ) and the counterart of ψ k+ in [, ] is defined similarly. Note that w = for the eigenvalue roblem.). With this setting, the resulting stiffness matrix is diagonal and the mass matrix is 5-diagonal, see Aendix.

4 Suer-Geometric Convergence of a Sectral Element Method 4 4. Suer-Geometric Convergence Rate Let k, u k ) be the kth eigen-air and k,h, u k,h ) R S h be its h-version finite element aroximation. According to [,.7], with C ɛ h k,h k C ɛ h, ɛ h = inf χ S h u k χ, for simle eigenvalue k see [, )]. Transferring this theory to our sectral element method language, we have, for any simle eigen-air, u), with C ɛ C ɛ, 4.) ɛ = inf χ S u χ inf χ S u χ u u, where u S such that u is the iecewise Legendre exansion of u not solution of.) or.)). Note that the first comes from the Poicaré inequality and the last is based on the fact that the Legendre exansion minimizes the L -norm. Lemma 4.. Let u satisfy the regularity assumtion max x [,] uk) x) cm k for fixed constants c and M, and let ũ be the Legendre exansion of u on [, ]. Then under the assumtion + ) + ) > M, u ũ L[,] C ) + em, 4.) where C is indeendent of and M. Proof. The error of )-term Legendre exansion is u ũ L [,] = Using the result [,.58, Theorem..6], we have k= k + b k. 4.) b k = k k! k)! uk+) η k ), η k, ). 4.4) Note that k k!)/k)! = /k )!!. Alying the regularity assumtion u k) x) cm k, we derive u ũ L [,] <cm + ) )!! + )!! + M + )!! + )!! + M 4 + cm + ) = )!! + )!! ) + )!! + 5)!! + ) M + ) + ) + M 4 + ) + ) + 5) + 4cM + ) < )!! + )!!, 4.5)

5 4 L. WANG, Z.Q. XIE AND Z.M. ZHANG when + ) + ) > M. formula [, 4.48)] and [4] This last term can be readily estimated by Stirling tye n ) n n! π e n + ), 4.6) 6 n)! n )!!. 4.7) 4 πn + 4 ) Therefore, )!! + )!! )! + )! 4 π +.5) 4 π +.5) )! π which, combined with 4.5), leads to 4.) with C = 4 c. ), e Now we are ready to rove our main theorem. Theorem 4.. Let, u) R + S be an eigen-air of roblem.), where is a simle eigenvalue. Let ) be its aroximation in the sense of.4) or.5). Then where C is indeendent of and M. ) C e ) +, 4.8) 4 Proof. Recall that cx) is constants on, /) and /, ), we searate u u = u u L,/) + u u L /,). 4.9) Recall that u is the iecewise Legendre exansion of u. The estimate for the first term is as follows, u u L,/) = 4 û û L,), 4.) where ûξ) = u + ξ)/4). Now we aly Lemma 4. to have û û L,) C e ˆM ) + = C e ) +. 4 Note that where M comes from the condition ˆM = M 4 =, max x [,] uk x) cm k. In our situation, u contains terms like sin x, cos x,... M =, and M = 4 ˆM from du x) = 4dû dx dξ ξ).

6 Suer-Geometric Convergence of a Sectral Element Method 4 The second term in 4.9) can be estimated similarly, and therefore, we have u u C e ) +. 4 The error bound 4.8) follows from 4.). Theorem 4.. Let, u) R + S be an eigen-air of roblem.), where is a simle eigenvalue. Let ) be its aroximation in the sense of.4) or.5). Then eπ ) + ) C. 4.) Proof. The roof is the same by the scaling between, ) and π, π). Remark. The error bounds in Theorems 4. and Theorem 4. are suer-geometric tye Oe +)log γ) ) with γ = lne /4) and γ = lneπ /), resectively. We shall demonstrate in the next section, by numerical tests, that our error bounds are shar. Our estimates also indicate that we need higher for larger to realize the convergence. This is consistent with our numerical exeriences. 5. Numerical Tests In this section, we imlement the numerical scheme described in Section to solve.) with ω = for the first 4 eigenvalues. We observe convergence for reasonably smaller. Actually, the error goes to the machine ɛ for for the first few eigenvalues. To verify our error bounds, we lot the ratio ) )/.5eπ /) + 5.) with some different s. Here is a list of the square roots of eigenvalues by increasing order): π arccos ), π arccos ), π arccos +, π arccos +,, π arccos ) +, π arccos ) +, π arccos +, π arccos +, 4, π arccos ) + 4, π arccos ) + 4, π arccos + 5, π arccos + 5, 6, Figures -4 demonstrate the ratio 5.) associated with,,, 4, 5, 7, 9, 4, resectively. We lot the ratio with different range of. Since for a larger eigenvalue, we need relatively higher to get into the asymtotic range. On the other hand, when getting bigger and the error aroaching the machine ɛ, the round-off error kicks in. So we can only observe the ratio in a small range of. Nevertheless, it is sufficient to make our oint clear. We see that the ratio 5.) maintains in a reasonable range for different eigenvalues. 6. A Collocation Method for the Smooth Case As a secial case when cx) is sufficiently smooth, say a constant, we may use only one element. Without loss of generality, let us consider u = u in, ) u ) = = u). 6.)

7 44 L. WANG, Z.Q. XIE AND Z.M. ZHANG.5. err/[e*i*sqrtlambda)/) + ] err/[e*i*sqrtlambda)/) + ] Fig. 5.. Ratio of the comuted errors over the estimated errors for and In this case, we are seeking an eigen-air ), w ) with w ξ) = w j ˆφj ξ) to satisfy j= w, ˆφ k) = )w, ˆφ k ), k =,...,. 6.) Again, this result in an identity matrix on the left and a 5-diagonal matrix on the right. Based on the analysis in Section 4, we have in this case e ) + ) C. 6.) Let us consider a sectral collocation method w x j ) = )w x j ), j =,,...,, 6.4) where x j s are zeros of L and L is the Legendre olynomial of degree on [, ]. Theorem 6.. For the model roblem 6.), the sectral collocation method 6.4) is equivalent to relacing all integrations in 6.) by the + -oint Gauss-Lobatto quadrature. Furthermore, all numerical integrations are exact excet the last term with ˆφ, ˆφ ) = j= where w j s are weights of the Gauss-Lobatto quadrature. ˆφ x j )w j = ˆφ, ˆφ ) ), 6.5) ) Proof. We multily both sides of 6.4) by ˆφ k x j )w j and sum u w x j ) ˆφ k x j )w j = ) w x j ) ˆφ k x j )w j 6.6) j= Since the + -oint Gauss-Lobatto quadrature rule is exact for olynomials of degree u to, then we have w x j ) ˆφ k x j )w j = w, ˆφ k ) = w, ˆφ k), k =,..., ; j= j= w x j ) ˆφ k x j )w j = w, ˆφ k ), k =,...,. j=

8 Suer-Geometric Convergence of a Sectral Element Method 45 x. err/[e*i*sqrtlambda)/) + ] err4/[e*i*sqrtlambda4)/) + ] Fig. 5.. Ratio of the comuted errors over the estimated errors for and 4 err5/[e*π*sqrtlambda5)/) + ] 5 x err7/[e*π*sqrtlambda7)/) + ] x Fig. 5.. Ratio of the comuted errors over the estimated errors for 5 and 7 x 4.8 x.5.6 err9/[e*π*sqrtlambda9)/) + ] err4/[e*π*sqrtlambda4)/) + ] Fig Ratio of the comuted errors over the estimated errors for 9 and 4 We see that the collocation method 6.4) is almost identical to the sectral method 6.) excet one term and their difference is w, ˆφ ) w x j ) ˆφ x j )w j = w, ˆφ ). j= w, ˆφ ) w, ˆφ ) = w [ ˆφ, ˆφ ) ˆφ, ˆφ ) ].

9 46 L. WANG, Z.Q. XIE AND Z.M. ZHANG Using the fact L, L ) = /, a direct calculation yields, and Therefore, ˆφ, ˆφ ) = ) L L, L L ) = ) L, L ) + L, L )) = ) + + ), 6.7) ˆφ, ˆφ ) = ) L, L ) + L, L )) = ) + ). 6.8) ˆφ, ˆφ ) ˆφ, ˆφ ) + ) ) =, ) and 6.5) follows. We see that the + -oints Gauss-Lobatto quadrature has a 5% over-shoot asymtotically in calculating ˆφ, ˆφ ). Nevertheless, the last coefficient w decays fast in general and the collocation method 6.4) is asymtotically equivalent to the sectral method 6.). As a consequence, the sectral collocation method 6.4) also enjoys the suer-geometric convergence rate 6.). Remark. It is feasible that a arallel result may be develoed for the Chebysheve sectral/collocation methods. It is also feasible that the error bounds in [] may be imroved to the similar suer-geometric rate as in this aer. 7. Aendix. The roerty of stiffiness matrix Since ˆφ i, ˆφ j ) = δ ij, it is straightforward to verify that Furthermore, observe that φ i, φ j) = 4δ ij = ψ i, ψ j), N, φ j) = = N, ψ j), N, N ) = 4. 4φ i, φ j ) = ˆφ i, ˆφ j ) = 4ψ i, ψ j ), φ k+, N) = = ψ k+, N) k >, N, N) = ) ) + ξ ξ + = ) = 4 6, φ, N) = 4 ˆφ, ˆN ) = 4 6 L L, ˆN ) = 4 6 φ, N) = 4 ˆφ, ˆN ) = 4 L L, ˆN ) = 4 ψ, N) = 4 ˆφ, ˆN ) = 4 6 L L, ˆN ) = 4 6 ψ, N) = 4 ˆφ, ˆN ) = 4 L L, ˆN ) = 4 + ξ ξ + ξ ξ = 4 6, 4 6, ξ ξ =, =.

10 Suer-Geometric Convergence of a Sectral Element Method 47 The secial arrangement of ordering has advantage of symmetric structure in the resulting matrix. It is straightforward to verify that the weak formulation u, v ) = cu, v), v S yields a diagonal stiffness matrix 4I and a 5-diagonal mass matrix 4 A, where the uer half of A is of the form +) ) ) ) 5) ) 5) ) ) 5) ) 7) and the lower half of A is obtained by taking the above exression u-side-down and multilied by ω. The bottom right entry / of the uer art and the to left entry ω / of the lower art is the only overla of the two arts. Acknowledgement. The authors would like to thank an anonymous referee for the constructive comments esecially the hint to simlify the roof of Theorem 6.) that significantly imroves the resentation of this article. The second authors work was suorted in art by the Programme for New Century Excellent Talents in University NCET-6-7), the National Natural Science Foundation of China NSFC 8766, 575), and Key Laboratory of High Performance Comuting and Stochastic Information Processing, Ministry of Education of China, Hunan Normal University The third authors work was suorted in art by the US National Science Foundation grant DMS-698. References [] J. Arndt and C. Haenel, π - Unleashed, Sringer, Berlin,. [] I. Babuška and J. Osborn, Egenvalue Problems, in: Handbook of Numerical Analysis, Vol.II: Finite Element Methods, Eds. P.G. Ciarlet and J.L. Lions, Elsevier Science Publishers B.V. North- Holland), 99. [] I. Babuška and M. Suri, The and h versions of the finite element method, basic rinciles and roerties, SIAM Rev., 6 994), [4] F.L. Bauer, Decryted Secrets, Methods and Maxims of Crytology, nd, revised, and extended edition, Sringer-Verlag, Heidelberg,. [5] C. Bernardi and Y. Maday, Sectral Methods, In Handbook of Numerical Analysis, Vol. V, P.G. Ciarlet and J.-L. Lions eds., North-Holland 997), [6] J.P. Boyd, Chebyshev and Fourier Sectral Methods, nd edition, Dover, New York,. [7] C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang, Sectral Methods: Fundamentals in Single Domains, Sringer Series: Scientific Comutation, Berlin, 6.,

11 48 L. WANG, Z.Q. XIE AND Z.M. ZHANG [8] F. Chatelin, Sectral Aroximations of Linear Oerators, Academic Press, New York, 98. [9] P.J. Davis and P. Rabinowitz, Methods of Numerical Integration, nd Ed., Academic Press, Boston, 984. [] D. Gottlieb and T.A. Orszag, Numerical Analysis of Sectral Methods: Theory and Alications, SIAM, Philadelhia, 977. [] M.S. Min and D. Gottlieb, Domain decomosition sectral aroximations for an eigenvalue roblem with a iecewise constant coefficient, SIAM J. Numer. Anal., 4-5), 5-5. [] A.T. Patera, A sectral element method for fluid dynamics: laminar flow in channel exansion, J. Comut. Phys., ), [] G.M. Phillis, Interolation and Aroximation by Polynomials, Sringer, New York,. [4] G. Sansone, Orthogonal Functions, Dover, New York, 99. [5] J. Shen and T. Tang, Sectral and High-Order Methods with Alications, Science Press, Beijing, 6. [6] L.N. Trefethen, Sectral Methods in Matlab, SIAM, Philadehia,. [7] Z. Zhang, Suerconvergence of Sectral collocation and -version methods in one dimensional roblems, Math. Comut., 74 5), [8] Z. Zhang, Suerconvergence of a Chebyshev sectral collocation method, J. Sci. Comut., 4 8) 7-46.