ON APPROXIMATION OF LAPLACIAN EIGENPROBLEM OVER A REGULAR HEXAGON WITH ZERO BOUNDARY CONDITIONS 1) 1. Introduction

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1 Journal of Computational Mathematics, Vol., No., 4, ON APPROXIMATION OF LAPLACIAN EIGENPROBLEM OVER A REGULAR HEXAGON WITH ZERO BOUNDARY CONDITIONS ) Jia-chang Sun (Parallel Computing Division, Institute of Software, Chinese Academy of Sciences Beijing 8, China) ( sunjc@mail.rdcps.ac.cn) Dedicated to Professor Zhong-ci Shi on the occasion of his 7th birthday Abstract In my earlier paper [4], an eigen-decompositions of the Laplacian operator is given on a unit regular hexagon with periodic boundary conditions. Since an exact decomposition with Dirichlet boundary conditions has not been explored in terms of any elementary form. In this paper, we investigate an approximate eigen-decomposition. The function space, corresponding all eigenfunction, have been decomposed into four orthogonal subspaces. Estimations of the first eight smallest eigenvalues and related orthogonal functions are given. In particulary we obtain an approximate value of the smallest eigenvalue λ 9 4 π =7.555, the absolute error is less than.. Mathematics subject classification: 4B35, 65N5, 65T5 Key words: Laplacian eigen-decomposition, Regular hexagon domain, Dirichlet boundary conditions.. Introduction GivenanoriginpointO and two plane vectors e and e,e.g.e = {, }, e = { 3, }, we form a 3-direction -D partition as drawn in Fig.. To deal with symmetry along the three direction, we apply a 3-direction coordinates instead of the usual -D Cartesian coordinates and barycentry coordinates, which is a very useful within a triangle domain. Setting the origin point O =(,, ), each partition line is represented by t l =integer (l=,,3), and each -D point P is represented by P =(t,t,t 3 ), t + t + t 3 =, (.) and any function f(p ) defined on the plane can be written as f(p )=f(t,t,t 3 ). In particulary, P k is called an integer node if and only if P k =(k,k,k 3 ), k + k + k 3 =. Let Ω be the unit regular hexagon domain Ω={P P =(t,t,t 3 ) t + t + t 3 =, t,t,t 3 } (.) we consider the following eigenvalue problem u = λu, (.3) with zero Dirichlet boundary u Ω = (.4) In terms of the 3-direction partition form, the Laplacian operator can be written as L = 3 = ( ) ( ) ( ). (.5) t t t t 3 t 3 t Received January 3, 4. ) Project supported by the Major Basic Project of China (No.G99938) and National Natural Science Foundation of China (No. 673).

2 76 J.C. SUN (,,-) t (,-,) (3,,-3)(3,-,-)(3,-,-)(3,-3,) (,,-3) (,,-3) (,3,-3) (-,3,-) (-,3,-) (-3,3,) O (,,-) O (,-,) (,,) t t 3 (-,,) (-,,) Figure.: 3-direction partition Figure.: Parallel hexagon domain Ω Definition.. Afunctionf(P ), defined in the 3-direction coordinate, is called periodic, if for all P =(t,t,t 3 ) the equality f(p + Q) =f(p ) holds for any integer vector Q =(n,n,n 3 ) with n + n + n 3 = (mod 6) The following results have been proved in my earlier paper [4]. Theorem.. For all integer triple j =(j,j,j 3 ), the function system g j (P )=e i π 3 (jt+jt+j3t3) forms a complex eigen decomposition of the Laplacian operator (.3) with three direction periodic boundary conditions, where P =(t,t,t 3 ), j + j + j 3 =. The corresponding eigenvalues equal to λ j,j,j 3 =( π 3 ) ((j j ) +(j j 3 ) +(j 3 j ) ) (.6) Since λ j,j,j 3 = λ j,j 3,j = λ j3,j,j = λ j, j 3, j = λ j3, j, j = λ j, j, j 3 we have the following real eigen-decomposition. Corollary.. For all integer triple j =(j,j,j 3 ), two function system cos( π 3 (j t + j t + j 3 t 3 )) and sin( π 3 (j t + j t + j 3 t 3 )) form an eigen decomposition of the Laplacian operator (.3) with three direction periodic boundary conditions. Moreover, except the smallest eigenvalue is single, all other eigenvalues are six multiple. It is clear that the first eigenfunction is a trivial constant. Several figures of eigenfunctions, related from the second to the four without counting multiple, are drawn in Figure.3-.. Definition.. TSin j (P ):= i[ gj,j,j 3 (P )+g j,j 3,j (P )+g j3,j,j (P ) g j, j 3, j (P ) g j, j, j 3 (P ) g j3, j, j (P ) ] (.7) Definition.3. TCos j (P ):= [ gj,j,j 3 (P )+g j,j 3,j (P )+g j3,j,j (P ) +g j, j 3, j (P )+g j, j, j 3 (P )+g j3, j, j (P ) ] (.8)

3 On Approximation of Laplacian Eigenproblem over a Regular Hexagon with Zero Boundary Conditions 77 Figure.3: -nd eigen. function Figure.4: -nd eigen. function Figure.5: -nd eigen. function 3 Figure.6: -nd eigen. function 4 Theorem.. TSin j (P ) and TCos j (P ) form an eigen-decoposition system of the Laplacian operator (.3) on an equilateral triangle subdomain: t and t,t 3 with zero Dirichlet and Neumann boundary conditions, respectively. The range of integer triple j in eigenvalue expression (.6) corresponds j >, j,j 3 < and respectively. j, j,j 3. Two Kinds of Orthogonal Decomposition for a Function Defined on a Parallel Hexagon Definition.. Afunctionf, defined over a parallel hexagon domain Ω, is called partial symmetry with respect to direction t if f(t,t,t 3 )=f( t, t 3, t ) or is called partial asymmetry with respect to direction t if f(t,t,t 3 )= f( t, t 3, t )

4 78 J.C. SUN Figure.7: -nd eigen. function 5 Figure.8: -nd eigen. function 6 Figure.9: 3-nd eigen. function -3 Figure.: -nd eigen. function 4-6 Definition.. A function f, defined over a parallel hexagon domain Ω, is called partial symmetry with respect to axis t t3 if f (t, t, t3 ) = f (t, t3, t ) or is called partial asymmetry with respect to axis t t3 if f (t, t, t3 ) = f (t, t3, t ) Lemma.. If f is partial symmetry with respect to direction t and g is partial asymmetry Figure.: 4-th eigen. function -3 Figure.: 4-th eigen. function 4-6

5 On Approximation of Laplacian Eigenproblem over a Regular Hexagon with Zero Boundary Conditions 79 with respect to a direction t on the same parallel hexagon, then the two functions are orthogonal <f,g>:= f(p )ḡ(p )dp = Ω Lemma.. If f is partial symmetry with respect to axis t t 3 and g is partial asymmetry with respect to axis t t 3 on the same parallel hexagon, then the two functions are orthogonal. There are similar definitions and lemmas of partial symmetry and asymmetry with respect to direction t, t 3 and axes t 3 t and t t, respectively. By using the above lemmas, it leads to the following orthogonal decomposition theorem Theorem.. Any function f, defined over a parallel hexagon domain Ω, can be decomposed into following four orthogonal sub-functions f(t,t,t 3 )=f (t,t,t 3 )+f (t,t,t 3 )+f 3 (t,t,t 3 )+f 4 (t,t,t 3 ) (.) where f (t,t,t 3 )= 4 {f(t,t,t 3 )+f( t, t 3, t )+f(t,t 3,t )+f( t, t, t 3 )} is both partial symmetry with respect to direction t and the axis t t 3, f (t,t,t 3 )= 4 {f(t,t,t 3 ) f( t, t 3, t )+f(t,t 3,t ) f( t, t, t 3 )} is partial asymmetry with respect to direction t and partial symmetry with the axis t t 3 f 3 (t,t,t 3 )= 4 {f(t,t,t 3 )+f( t, t 3, t ) f(t,t 3,t ) f( t, t, t 3 )} is partial symmetry with respect to direction t and partial asymmetry with the axis t t 3 f 4 (t,t,t 3 )= 4 {f(t,t,t 3 ) f( t, t 3, t ) f(t,t 3,t )+f( t, t, t 3 )} is both partial asymmetry with respect to direction t and the axis t t 3. Now we turn to construct another orthogonal decomposition. Definition.3. Afunctionf, defined over a parallel hexagon domain Ω, iscalledcentered symmetry if f(t,t,t 3 )=f( t, t, t 3 ) or is called centered asymmetry if f(t,t,t 3 )= f( t, t, t 3 ) Lemma.3. If f is centered symmetry and g is centered asymmetry on the same parallel hexagon, then the two functions are orthogonal on the domain Definition.4. Afunctionf, defined over a parallel hexagon domain Ω, is called isotropic if f is centered symmetry and partial symmetry with respect to all three directions t,t and t 3. Denote H I := {f f isotropic} (.) H II := {f f centered asymmetry & partial asymmetry with respect to a direction} H III := {f f centered symmetry & partial asymmetry with respect to a direction} H IV := {f f centered asymmetry & partial symmetry with respect to three directions} Lemma.4. The above four function spaces H I, H II,H III and H IV are orthogonal each other.

6 8 J.C. SUN Lemma.5. For a regular hexagon domain, Laplacian operator preserves the above symmetry and asymmetry of the original function. Therefore, we lay down our foundation for find approximate eigenvalues and related orthogonal function in succeed sections. Lemma.6. Both of four spaces, f,f,f 3,f 4 defined in (.) and H I,H II,H III,H IV in (.), are orthogonal decomposition of any eigen-function of (.3) with (.4). defined 3. Approximation to the Smallest Eight Eigenvalues and Related Functions 3.. The smallest eigenvalue estimation As is well know, the smallest eigenvalue of (.3) with zero boundary conditions (.4) is the minimum of the following quadratic functional < u, u > λ = inf u H (Ω) <u,u> Denote Ω S and Ω B to be an interior circle with radius R = 3 and an exterior equilateral triangle with side length H = 3, respectively. It is obvious Ω S Ω Ω B, hence, by using comparison principle, we have the following inequalities Lemma π = 6 3H π = λ min (Ω B ) <λ := λ min (Ω) <λ min (Ω S ) (π +4) 4(π 4) π R =.784π The lower bound comes from eigenvalues expression of Laplacian in an equilateral triangle with side length 3, e.g. see [] and [4]. To get the righthand side we rewrite the Laplacian eigen-problem in terms of polar coordinates u = u (r r r r ) u r θ, u(r, θ) r=r = (3.) Hence, denote ξ,k to be zero points of the Bessel function J (r), then all isotropic eigenvalues of (3.) equal to ξ,k /R.SinceJ (ξ, ) ξ,=.45 =.9558, J (ξ, ) ξ,=5.5 = Hence, the first two isotropic eigenvalues on the circle equal to.45 4/3 = 7.73 and 5.5 4/3 =47 approximately, and the related Bessel functions J (ξ,k r), (k =, ) are the strict eigenfunctions. In general, all zero points ξ n,k of Bessel functions J n (r) correspond eigenvalues of the Laplacian operator on the unit circle with zero boundary condition λ n,k = ξn,k. The related double eigenfunctions are J n(ξ n,k r)cosnθ and J n (ξ n,k r)sinnθ. Because three areas of the circle, the regular hexagon and the equilateral triangle equal to S ΩS =3π/4, S Ω = 3 3, S Ω B = or log( SΩ S Ω = S t Ω S SΩ t S ΩS ) B, t = log( SΩ B S ΩS ) =.94 Corollary 3.. We may take an approximation for the first eigenvalue as a generalized geometry means between the upper bound and the lower bound of (3.) (3.) λ λ min (Ω S ) ( t) λ min (Ω B ) t π =.7499π, where t =.94. (3.3)

7 On Approximation of Laplacian Eigenproblem over a Regular Hexagon with Zero Boundary Conditions 8 To get an elementary approximation for the first eigenfunction, we substitute u(r) =cos rπ R into the following expression λ = inf u(r)=,u()bounded R R ru dr ru dr then λ (4 + π )/(6R ) (π 4)/(4π ) = (4 + π ) 4R (π 4) π = Hence, in this sense the function is a good approximate of the first eigenfunction. To obtain a further approximation for an approximation of the first eigenfunction, we denote F, =4cos t π cos t π cos t 3π =cost π +cost π +cost 3 π + ; (3.5) F, =( t )( t )( t 3 ). (3.6) These two functions are isotropic and belong to space H I. The graphics of the function F, is shown in Figure 3.. A straight computation leads to < F,, F, > <F,,F, > = π 3 3(5π + 6)/π = 4(8 + 3π )π 3(5π + 6) =.7674π < F,, F, > <F,,F, > = =.7358π 3 Further, if test is done in the space spanned of F, and F,. Therefore function F can be taken as the first eigen-function. Next, the computation can be done in the space spanned of F and F. In fact, set a stiffness matrix K = < F,, F, > < F,, F, > = < F,, F, > < F,, F, > (3.4) and mass matrix M = <F,,F, > < F,,F, > = <F,,F, > < F,,F, > The smallest eigenvalue of the related generalized matrix eigenvalue problem equals to =.78π. The estimation can be improve further if we take the minimum in more wide space F (t,t,t 3 )=Span ν=,...,6 {F,ν } where F 3 = W + W + W 3 ; W =4cos πt sin πt sin πt 3 ; W =4sinπt cos πt sin πt 3; W 3 =4sinπt sin πt cos πt 3 ; F 4 =4cos 3πt F 6 =4cos πt The minimum value cos πt cos πt 3 ; F 5 =4cos πt cos 3πt cos πt 3 ; cos πt cos 3πt 3. λ 7.65 =.76π reaches when F = 946F, F,.5F (F 4 + F 5 + F 6 )

8 8 J.C. SUN Finally, if we add the following extra five high frequency functions F 7 =4cos πt cos 3πt cos 3πt 3 ; F 8 =4cos 3πt cos πt cos 3πt 3 ; F 9 =4cos 3πt cos 3πt cos πt 3 ; F, =4cos 3πt cos 3πt cos 3πt 3 ; and F, = F, into the above space, we may obtain more accuracy upper bound for the smallest eigenvalue λ =.75π. As is well known that the numerical results of first smallest eigen-values are less than their true values. Therefore, we obtain a very close pair of upper bound and lower bound for the first smallest eigenvalue of (.3) with zero boundary conditions (.4). Proposition π <λ <.75π (3.7) 3.. The nd to 5th eigenvalue approximation According to the eigenvalue max-min property, we know λ =sup u inf u H (Ω) <u,u>= < u, u > <u,u> Since the first eigen-function is invariant under rotation 6 o with non zero within the hexagon domain, the second eigen function must be partial asymmetry along a bisection line, e.g. t 3 = or t = t, and the line in the only zero line within the domain. Therefore, the second eigenvalue is double multiple. Denote G =4sinπt cos πt cos πt 3 ; G =4cos πt sin πt cos πt 3 ; G 3 =4cos πt cos πt sin πt 3; The three functions are partial asymmetry along direction t, t and t 3, respectively. They have no zero lines except on their asymmetry axis. F = G + G 3 G ; F = 3(G G 3 ); It is obvious that function F and F are partial asymmetry with respect to direction t and t t 3, respectively. Hence, both of them belong to H II, orthogonal to F inh I,andtheyare orthogonal each other. By symmetry and asymmetry, it is obvious <G,F >=< G,F >=< G 3,F >=, <G,G >=< G,G 3 >=< G 3,G >, (3.8) <G,G >=< G,G >=< G 3,G 3 >. Hence Moreover <F,F >=< F,F >=< F,F >= <F,F >=< F,F >= 3 75π ( π + 675π )=6.764 < F,F >=< F,F >=( π +36π ) 3 = 39.6 λ = λ 3 = 39.6 =8.777 =.9π 6.764

9 On Approximation of Laplacian Eigenproblem over a Regular Hexagon with Zero Boundary Conditions Figure 3.3: Minimum surface approximation F, Similarly, the 4th and 5th eigenvalue are also double multiple. Their corresponding functions have two across bisection zero lines. Denote W =4cos πt sin πt sin πt 3 ; W =4sinπt cos πt sin πt 3; W 3 =4sinπt sin πtcos πt 3 ; F 3 = 3(W W 3 ); F 3 = W + W 3 W ; The two functions are orthogonal each other <F 3,F 3 >=, because of permutation symmetry, { <W,W >=< W,W 3 >=< W 3,W >, (3.9) <W,W >=< W,W >=< W 3,W 3 >. Note that both of F 3 and F 3 belong to H III, they are orthogonal to F H I and F,F H II. Moreover the two Rayleigh quotations are the same, because <F 3,F 3 >=< F 3,F 3 >= π ( π π )= < F 3,F 3 >=< F 3,F 3 >=( π +54π ) 3 = 39.9 λ 4 = λ 5 = 39.9 =33.33 = 3.377π The 6th to 8th eigenvalue approximation As another isotropic function in H I we may take F 6=W + W + W 3 ; By using (3.9) and isotropic property of F 6 H I, it is orthogonal to the above four functions F,F H II and F 3,F 3 H III. <F6,F >=< F6,F >=< F6,F 3 >=< F6,F 3 >=. To get an approximation to the second isotropic eigenfunction, we find a linear combination of two isotropic functions F, in (3.5) and the above F 6 such that F 6 =.78F +3F 6 It corresponding the larger generalized eigenvalues of the sub-space spanned of two functions F andf 6. And <F 6,F 6 >=.458, < F 6,F 6 >=94.353

10 84 J.C. SUN Figure 3.4: Two orthogonal surfaces F andf related to λ = λ Figure 3.5: Two orthogonal surfaces F3 and F3 related to λ4 = λ Figure 3.6: Isotropic surface F6 related to λ6

11 On Approximation of Laplacian Eigenproblem over a Regular Hexagon with Zero Boundary Conditions Figure 3.7: Two orthogonal surfaces F 7 and F 8 related to λ 7 and λ 8 (λ 7 <λ 8 ) the Rayleigh quotation λ = = 3.883π.458 The functions corresponding the 7th and 8th belong to space H IV. They are centered asymmetry and have zero lines on three direction t ν = integer or on the three bisection t µ t ν = integer, µ ν, respectively. F 7 =4sin(t t )π sin(t t 3 )π sin(t 3 t )πf ; F 8 =4sinπt sin πt sin πt 3 ; (3.) Since F 7 is partial symmetry to axis t t 3 and F 8 is partial asymmetry to t t 3, hence they are orthogonal both in ordinary sense and in energy sense. <F 7,F 8 >=< F 7,F 8 >= As F 7 and F 8 belong to space H IV, by Lemma.4, they are orthogonal to F,F 6 H I, F,F H II and F 3,F 3 H III. Moreover, a straight computation leads to < F 7,F 7 >=( π π ) 3= <F 7,F 7 >= 45π ( π + 5π )=.546 The Rayleigh quotation < F 7,F 7 > <F 7,F 7 > = 4.74 =48.3 = π.546 < F 8,F 8 > <F 8,F 8 > = 5.35 = = π Lower Bound of Eigenvalues Via Difference Scheme To get a lower bound of eigenvalues, we may use difference scheme of (.3) with zero boundary conditions (.4). Let mesh size h = /(N + ), using two variable Taylor expanding follows L h u j = 3h {6u(j,j,j 3 ) u(j,j + h, j 3 h) u(j,j h, j 3 + h) u(j + h, j,j 3 h) u(j h, j,j 3 + h) u(j h, j + h, j 3 ) u(j h, j,j 3 + h)} = u j h 6 u j h4 576 {u(6,) +5u (4,) +45u (,4) +9u (,6) } u=uj + O(h 6 ) (4.)

12 86 J.C. SUN Since the biharmonic operator is positive, hence, for sufficient smooth function and suitable small mesh size h, we can take eigenvalues of the partial difference equation as a lower bound of the related Laplacian eigenvalues. The number of total inner knots equals to 3N(N + ) +. The related stiffness matrix has tri-diagonal block form A = T ridiag{a,a ; A,A,A 3 ; ; A N,N,A N,N,A N,N+ ; ; A N,N,A N,N } As an example, for h=/, the related stiffness matrix becomes A = 3h For N = 48, the first eight approximate smallest eigenvalues are equal to {.749,.8363,.8363, 3.853, 3.853, 3.795, 4.899, 5.36}π (4.3) respectively. 5. Conclusion TSin j (P ) in Theorem., as eigenfunctions of Laplacian operator over an equilateral triangle with zero boundary conditions, still are a part of eigenfunctions in the regular hexagon case. The function space, corresponding all eigenfunction, have been decomposed into four orthogonal subspaces. For the smallest eight eigenvalues of (.3) over the regular hexagon with zero boundary condition (.4), we get an upper bound {.75,.9,.9, 3.377, 3.377, 3.883, , }π and a lower bound {.749,.8363,.8363, 3.853, 3.853, 3.795, 4.899, 5.36}π Their maximum relative gaps are less than 4%. The pictures of eigenfunction, corresponding from the smallest to 8th, are shown in Fig. 3. to Fig. 3.7, where λ = λ 3 and λ 4 = λ 5 are two double roots, other four eigenvalues are single roots. Moreover, so far only λ 8 and related eigenfunction (3.) are exact. Approximations to other eigenvalues and related functions can be constructed in a similar way. How to use these approximation of partial eigen-decomposition for constructing a fast solver of the related discrete system is our future work. References [] Mark. A. Pinsky, Completeness of the eigenfunctions of the equilateral triangle, SIAM J. Math. Anal., 6 (985), [] Milan Prager, An investigation of eigenfunctions over the equilateral triangle and square, Applications of mathematics, 43 (998), 3 3. [3] Jiachang Sun, Orthogonal piecewise polynomials basis on an arbitrary triangular domain and its applications, Journal of Computational Mathematics, 9 (), [4] Jiachang Sun, Multivariate Fourier Series over a Class of non Tensor-product Partition Domains, Journal of Computational Mathematics, (3), [5] Jiachang Sun and Huiyuan Li, Generalized Fourier transform on an arbitrary tirangular domain, Advances in Computational Mathematics, toappear. [6] Zhijie Yang and Jiachang Sun, The construction and classification of a set of complete orthogonal bases on an arbitrary triangular domain, Mathematica Numerica Sinica, 5: (3), 9 3. (4.)