A Spectral Method for the Biharmonic Equation

Transcription

1 A Spectral Metho for the Biharmonic Equation Kenall Atkinson, Davi Chien, an Olaf Hansen Abstract Let Ω be an open, simply connecte, an boune region in Ê,, with a smooth bounary Ω that is homeomorphic to Ë. Consier solving u γu f over Ω with zero Dirichlet bounary conitions. A Galerkin metho base on a polynomial approximation space is propose, yieling an approximation u n. With sufficiently smooth problem parameters, the metho is shown to be rapily convergent. For u ¾ C Ω an assuming Ω is a C bounary, the convergence of u u n H Ωµ to zero is faster than any power of n. Numerical examples illustrate experimentally an exponential rate of convergence. Introuction Consier the biharmonic problem with the Dirichlet bounary conitions u sµ γ sµu sµ f sµ s ¾ Ω () u sµ u sµ n s s ¾ Ω () The region Ω Ê,, is to be boune an simply-connecte; an its bounary Ω is to be smooth an homeomorphic to Ë. Assume f ¾ L Ω µ an Kenall Atkinson ( ) The University of Iowa, B MacLean Hall, Iowa City, IA 54, USA Kenall-Atkinson@uiowa.eu Davi Chien Olaf Hansen California State University San Marcos, 333 S. Twin Oaks Valley Roa, San Marcos, CA 996, USA chien@csusm.eu; ohansen@csusm.eu

2 Kenall Atkinson, Davi Chien, an Olaf Hansen γ min minγ sµ (3) s¾ω This can be looke upon as a problem in the Sobolev space H 4 Ω µ. It can also be reformulate as a variational problem. For backgroun on the use of this problem in mechanics, see [], [, Chap. 8]. Introuce the bilinear functional uvµ u sµ v sµ γ sµu sµv sµ s Ω an the linear functional f vµ f vµ f sµv sµ s Ω v ¾ L Ω µ Introuce the Hilbert space H Ω µ v ¾ H Ω µ v v on Ω n For the norm, use v v H Ωµ D k v L Ωµ k where k k k µ k k k, an D k v k v s s µ sµ s k sk The variational formulation of ()-() is to fin u ¾ H Ω µ for which uvµ f vµ v ¾ H Ω µ (4) For a iscussion of this reformulation, see Ciarlet [, p. 8]. With the above assumptions an efinitions, is a strongly elliptic operator on H Ω µ, vvµ c e v v ¾ H Ω µ with c e. Also, is a boune bilinear operator, vwµ c v w vw ¾ H Ω µ for some finite c. Finally, f f L Ωµ

3 A Spectral Metho for the Biharmonic Equation 3 The Lax-Milgram Theorem (cf. [7, Ü8.3], [9, Ü.7]) implies the existence of a unique solution u to (4) with u c e f (5) In Ü3 we present a Galerkin metho for approximating (4), making use of multivariate orthonormal polynomial approximations. Numerical examples are given in Ü4. Preliminaries Assume the existence of an explicitly known continuously ifferentiable mapping Φ : onto Ω (6) an let Ψ Φ : Ω enote the inverse mapping. A very simple example onto of such a mapping is when Ω is the ellipse s a s b with ab. Choose Φ xµ ax bx µ x ¾ It is necessary to know Φ explicitly, but not Ψ.The creation of such a mapping Φ is examine at length in [5]. Let ¾ Φ xµ Φ xµ x x J xµ DΦµ xµ Φ xµ Φ x ¾ (7) xµ x x ¾ K sµ DΨ µ sµ Ψ sµ s. Ψ sµ s Ψ sµ s.... Ψ sµ s s ¾ Ω (8) enote the Jacobian matrix of the transformations Φ an Ψ, respectively. Assume J xµ is nonsingular on, etj xµ x ¾ ;

4 4 Kenall Atkinson, Davi Chien, an Olaf Hansen an without loss of generality, assume etj xµ x ¾ Differentiating the ientity Φ Ψ sµµ s over Ω, or the ientity Ψ Φ xµµ x over, leas to J xµk sµ I x Ψ sµ (9) Thus the components of K sµ can be obtaine by using K sµ J xµ s Φ xµ () K Φ xµµ J xµ x ¾ () Let v enote a general function efine over Ω. For the transformation s Φ xµ, introuce the notation v xµ v Φ xµµ; or equivalently, v sµ v Ψ sµµ Consier the erivatives with respect to s of v sµ. Let s enote the graient with respect to the components of s; an o similarly for x. Then s v sµ K sµ T x v xµ x Ψ sµ () x v xµ J xµ T s v sµ s Φ xµ with x v xµ the graient of v xµ written as a column vector, an analogously for s v sµ. Further erivatives are consiere later in Ü3... Approximation space For Ω, introuce the approximation space n x p xµ p ¾ Π n where Πn enotes the space of all polynomials in variables an of egree n. For general Ω, use the approximation space Òχ Æ Φ χ ¾ Ó n L n Let Î k enote the space of all polynomials of egree k that are orthogonal in to all polynomials in Πk using the stanar inner prouct More precisely, f gµ f xµg xµ x

5 A Spectral Metho for the Biharmonic Equation 5 Ò Ó Î k p ¾ Πk pqµ q ¾ Π k k an Î is the set of all constant functions. Then Π n Î Î Î n is an orthogonal ecomposition of Πn within L. A basis for Πn is efine by first efining a basis for each subspace Î k, k n. Let ϕ k j be an orthonormal basis of Î k, let M k imî k, an efine χ k j xµ x ϕk j xµ j M k Denote the corresponing basis for n by χ xµ N n N n M M n Let ψ j j N n be the corresponing basis for n using ψ χ ÆΦ that for, M n n N n n µ n µ Orthonormal bases ϕ k j for Îk, k, are consiere in [], [3], [].. Note 3 The numerical metho The numerical metho is a Galerkin metho for approximating (4): fin u n ¾ n for which u n vµ f vµ v ¾ n This is the stanar variational framework use with finite element methos, with the approximating elements require to belong to H Ω µ, a significant requirement. Write N n u n sµ j α j ψ j sµ Then the coefficients α j must satisfy the linear system N n j α j ψ j ψ i µ f ψ i µ i N n (3) The Lax-Milgram Theorem (cf. [7, Ü8.3], [9, Ü.7], [, p. 8]) implies the existence of u n for all n, with u n c e f

6 6 Kenall Atkinson, Davi Chien, an Olaf Hansen For the error in this Galerkin metho, Cea s Lemma (cf. [7, p. 37], [9, p. 6], [, p. 4]) implies the convergence of u n to u, an moreover, u u n c c e inf u v v¾ (4) n It remains to boun the best approximation error on the right sie of this inequality. The error analysis is similar to that given in the earlier papers [], [4], [6]. To boun the right sie, make use of the following connection between norms in H k Ω µ an H k ; the proof is omitte. Lemma. Assume Φ ¾ C Ω µ. Let v ¾ H k Ω µ for some k, k ¾ Æ, an let v xµ v Φ xµµ. Then for some c kc k inepenent of v. c k v Hk µ v H k Ωµ c k v Hk µ (5) In orer to look at rates of convergence as a function of n, this lemma is use to convert the boun (4) to the equivalent boun u u n c inf v¾ n u v (6) c a generic constant epenent on Φ, but not on u. Assume u ¾ H k Ω µ, an equivalently u ¾ H k, k. Bouning the right sie of (6) using [7, Thm 4.3] leas to the error boun u u n Combine with Lemma an (4), u u n c n k u H k µ. (7) c n k u Hk Ωµ, (8) again with c a generic constant. To obtain convergence for k, it can be shown that inf v¾ n u v as n This follows because the polynomials n n are ense in H Ω µ [note the comments following [7, Thm 4.3] an the enseness of the polynomials n Πn in H k ]. 3. Evaluating the integrals The integrals

7 A Spectral Metho for the Biharmonic Equation 7 ψ i ψ j µ ψ i sµ ψ j sµ γ sµψ i sµψ j sµ s (9) Ω must be compute. Begin by converting to an integral over.using the transformation s Φ xµ: ψ i ψ j µ s ψ i sµ sφ xµ s ψ j sµ sφ xµ γ Φ xµµ χ i xµ χ j xµ etj xµ x The quantities s ψ i sµ, i N n, must be converte to functions involving erivatives with respect to x for χ i xµ. For the transformation x Ψ sµ, let v sµ v Ψ sµµ; or equivalently, v xµ v Φ xµµ Look at the erivatives with respect to s of v sµ. Then for i v sµ s i j This is a proof of (). Next, v sµ s i v xµ x j Ψ s i Ψ s i s i j Ψ j sµ xψ sµ s i x v xµ j v xµ x j Ψ j sµ j s i x Ψ sµ v xµ x j Ψ j sµ xψ sµ s i Ψ j sµ xψ sµ k s i v xµ Ψ k sµ x j x k s i Summing over i, s v sµ i j v xµ x j Ψ j sµ xψ sµ s i Ψ j sµ v xµ Ψ k sµ () i jk s i x j x k s i Look at the terms in (). First, i j v xµ x j Ψ j sµ xψ sµ s i j v xµ s Ψ j sµ x j s Ψ sµ s Ψ sµ x v xµ ()

8 8 Kenall Atkinson, Davi Chien, an Olaf Hansen Using the notation of (8), i jk Ψ j sµ v xµ Ψ k sµ s i x j x k s i Returning to () an combining terms, jk jk jk v xµ x j x k i Ψ j sµ Ψ k sµ s i s i v xµ T K sµ x j x j K sµ k () k v xµ K sµk sµ T x j x k jk s v sµ s Ψ sµ s Ψ sµ x v xµ v xµ T K sµ x j x j K sµ k (3) k jk Formula () can be evaluate from knowing J xµ; see () above. The formula (3) is to be evaluate with v xµ χ xµ N n so as to create the elements ψ i ψ j µ. To evaluate (), we nee s Ψ j sµ, j. The first erivatives of Ψ can be obtaine from D s Ψ sµ D x Φ xµ where s Φ xµ. How to obtain the functions s Ψ j sµ? Begin by ifferentiating s Φ Ψ sµµ s ¾ Ω or s j Φ j Ψ sµ Ψ sµµ The erivative with respect to s i yiels δ i j k Φ j xµ x k j Ψ k sµ i j (4) xψ sµ s i Differentiate the components of (9), given in (4), with respect to s : for i j, Φ j xµ Ψ k sµ Ψ k sµ x k s i s Φ j xµ Ψ m sµ s i x k x m s Let i k k m

9 A Spectral Metho for the Biharmonic Equation 9 k k Sum over i: for j, Introuce Φ j xµ k Φ j xµ Ψ k sµ x k s i Φ j xµ Ψ k sµ x k s i Φ j xµ k Φ j xµ k x k s Ψ k sµ x k s Ψ k sµ x k s Ψ k sµ k km km km km Ψ k sµ s i m Φ j xµ Ψ k sµ x k x m s i Φ j xµ x k x m i Φ j xµ x k x m K sµ k Φ j xµ Ψ m sµ x k x m s i Ψ m sµ s i Ψ k sµ Ψ m sµ s i s i Φ j xµ K sµk sµ T x k x m km s Ψ sµ s Ψ sµ s Ψ sµ T ¾ Φ j xµ Φ j xµ x x x x D Φ j xµ..... j Φ j xµ Φ j xµ x x x x Introuce the ot prouct of two arrays of the same imension: A B i j A i jb i j T K sµ m (5) Then (5) can be written as J xµ j s Ψ sµ D Φ j xµ K sµk sµ T From (5), ¾ D Φ xµ K sµk sµ T J xµ s Ψ. sµ. D Φ xµ K sµk sµ T J xµ s Ψ sµ Φ xµ K sµ T K sµ which contains an implicit efinition of Φ an an implicit notational extension of the operation. Then

10 Kenall Atkinson, Davi Chien, an Olaf Hansen Ò s Ψ sµ J xµ Φ xµ K sµk sµ TÓ an recall () to compute K sµ. This allows computing the coefficients ψ i ψ j µ of (3) by means of the change of variables s Φ xµ. Rewrite (3) as s v sµ s Ψ sµ T x v xµ D v xµ K sµk sµ T (6) Returning to ψ i ψ j µ, apply (6) with v xµ χ n j xµ x ϕn j xµ x x ϕn j xµ for j M k, k n. We nee to fin the first an secon orer erivatives of χ n j xµ, an thus also of ϕ n j xµ. χ n j xµ 4x k x x x ϕn j xµ x ϕ n j xµ k x k For k, χ n j xµ x k x χ n j xµ x k 4 x x 8x k ϕn j xµ 8x k 8x k x ϕ n j xµ x ϕ x n j xµ x k x ϕ n j xµ x k 4 x ϕ x n j xµ x k x x ϕ n j xµ x k x ϕ n j xµ x x k These can be combine with (6) to compute s χ j an thus to compute ψ i ψ j µ for i j N n. The next step is to look at particular orthonormal polynomials ϕ n j xµ an to compute

11 A Spectral Metho for the Biharmonic Equation ϕ n j ϕ n j xµ x k j M n j M n k ϕ n j xµ j M n x k x k The best choice as regars spee of calculation is to use the polynomials iscusse in [], as they satisfy a triple recursion that allows for a rapi calculation. For the planar case, these are given by ϕ nk xµ Cn k k h x µ x k C k kn ¼ x Õ x ½ x ¾ (7) for k n n The quantity Cm λ tµ m, enotes the Gegenbauer polynomial of egree m an inex λ. For the three imensional case, we use the polynomials ϕ n jk xµ h jk C j k 3 n j k C k j x Õ x x µ x µ j µ x x µk C k x 3 Õ x x x ¾ 3 j k n n (8) which uses again the Gegenbauer polynomials. The numbers h jk are normalization constants, see [3], an see [] for the triple recursion. µ 4 Numerical Examples 4. Planar Examples Our first examples are for Ω a planar region, an thus Φ : Ω. Example. Begin with the elliptical region Ω efine by the mapping s Φ xµ, x ¾, s x x s 3x 4x (9) Choose f sµ cos s µ sin s µ (3)

12 Kenall Atkinson, Davi Chien, an Olaf Hansen s -4 - s Fig. Solution u with f given by (3), γ sµ, an the region (9) n Fig. Compute error in u n with f given by (3) an γ sµ.

13 A Spectral Metho for the Biharmonic Equation Fig. 3 logn vs. log conµ, with con the conition number of (3), with the region (9) an γ sµ over Ω. The solution is shown in Figure. The true solution is unknown, so the error is estimate by using u n as the true solution with n much larger than n being use. In the present case, n was use. The maximum errors are shown in Figure, an it appears to be an exponential ecrease in the error. Figure 3 is a graph of logn vs. log conµ, with con the conition number of (3). It inicates that the conition number is Ç n p µ for some power p; experimentally an roughly, p 45, an p 4 seems most likely to be the theoretical power. That woul be consistent with the conition number being Ç Nn, as was observe earlier with the spectral metho for the Neumann bounary value problem for secon orer equations. Example. Consier the bounary mapping ϕ θ µ ρ θ µ cosθ sinθµ ρ θ µ 3 cosθ sinθ θ π (3) This can be extene to a polynomial mapping of egree in various ways, as iscusse in [5], an one such mapping is illustrate in Figure 4. This mapping Φ is obtaine using () the interpolation/quarature metho of Ü3 in [5], followe by () computing the least squares polynomial approximation over of egree in each component. The equation () is solve with the same choices for γ an f as in Example. Figure 5 illustrates the solution, using u. The errors are shown in Figure 6.

14 4 Kenall Atkinson, Davi Chien, an Olaf Hansen s s Fig. 4 Mappings for limacon bounary mapping (3) s s 3 4 Fig. 5 Solution u with f given by (3), γ sµ, an Ω given by (3)

15 A Spectral Metho for the Biharmonic Equation n Fig. 6 Compute error in u n with f given by (3), γ sµ, an the bounary mapping (3) The conition numbers, shown in Figure 7, appear to increase like Ç N n, as with Example. Example 3. Consier the mapping Φ xµ x x ax x T x x ¾ (3) for a given a, with the image efining Ω. In aition, use the interpolation/quarature metho of Ü3 in [5] to create another mapping Φ that agrees with Φ on the bounary of. These mappings are illustrate in Figure 8. Clearly Φ is a better behave mapping as compare to Φ. We solve u γu f as before, but now let f sµ cos stµsin t µ (33) The solution is shown in Figure 9. The maximum errors are shown in Figure, an there appears to be an exponential ecrease in the error. The conition numbers are shown in Figure, an again they appear to increase like Ç N n. 4. A three imensional example Example 4. Here we consier the case of an ellipsoi

16 6 Kenall Atkinson, Davi Chien, an Olaf Hansen log n Fig. 7 logn vs. log conµ, with con the conition number of (3), with the bounary mapping (3). s t s s (a) Φ (b) Φ Fig. 8 The mapping Φ for bounary (3) with a=.95

17 A Spectral Metho for the Biharmonic Equation s s Fig. 9 Solution u with f given by (33), γ sµ, an Ω given by (3) - Φ Φ n Fig. Compute error in u n with f given by (33), for the mappings Φ an Φ with the bounary specifie by (3).

18 8 Kenall Atkinson, Davi Chien, an Olaf Hansen 3 Φ Φ Fig. logn vs. log conµ, with con the conition number of (3), for the mappings Φ an Φ with the bounary specifie by (3). with the obvious mapping Ω s s s 3 µ s s s3 (34) 3 Φ x x x 3 µ x 3x x 3 x x x 3 ¾ 3 (35) We solve equation () with γ sµ an calculate the right han sie f in such a way that the solution of () is given by u s s s 3 µ s s s3 e 3 s s 3 s 3 µ (36) 3 To stuy the influence of faster growing erivatives we use a secon right han sie f on the same omain Ω, such that the solution is given v s s s 3 µ s s 3 s3 e 7 s s 3 s 3 µ (37) We expect slower but still exponential convergence for the secon example. This is confirme in the numerical calculation, see Figure, where the maximum errors are plotte versus n. The error graph for the solution u shows some saturation aroun n, because we reach the precision limit of the Gauß quaratures that we use for the evaluation of the integrals in equation (3). The graph of log conµ versus

19 A Spectral Metho for the Biharmonic Equation 9 5 u v Fig. Compute error in u n an v n, for the solutions u an v given in (36) an (37). log n in Figure 3 shows again a polynomial behavior. From the numerical results we estimate a conition number of Ç N n, where we remember that Nn Ç n 3 µ. 5 Nonhomogeneous bounary conitions Consier the Dirichlet biharmonic problem u sµ γ sµu sµ f sµ u sµ g sµ s ¾ Ω u sµ n s g sµ s ¾ Ω (38) This can be reuce to two simpler problems. Consier first the stanar Dirichlet biharmonic problem w sµ s ¾ Ω w sµ w sµ g sµ g sµ s ¾ Ω n s (39) Define v u w. Then v satisfies

20 Kenall Atkinson, Davi Chien, an Olaf Hansen Fig. 3 logn vs. log conµ, with con the conition number of (3), with γ sµ, an the mapping (35) for the omain (34). v sµ γ sµv sµ f sµ γ sµw sµ f sµ v sµ v sµ n s s ¾ Ω s ¾ Ω (4) Begin by solving (39) numerically, obtaining an approximating solution w sµ w sµ Then solve (4) with w sµ replacing w sµ in the efinition of f sµ. The problem (4) can be solve using the methos given earlier in this paper. Solve for an approximating solution v n sµ v sµ, an then efine u sµ v n sµ w sµ s ¾ Ω as the approximating solution of (38). To solve (39), a number of methos have been propose, often using bounary integral equation reformulations. For a review of some of these, see [4, Chaps. 9,5], [5], [6]. Remark. The eigenvalue problem for the biharmonic equation ()-() is iscusse an illustrate in the book [3, Chap. 9]. Traitional spectral methos use univariate approximations with a ecomposition of the partial ifferential equation into univariate problems. Consier, for example, using a polar coorinates ecomposition of the unit isk. But this leas to problems when treating the solution u at the center of the isk. The present spectral metho makes use of the recently evelope theory an tools for multivariate approximation

21 A Spectral Metho for the Biharmonic Equation over, avoiing artificial problems that can occur when using univariate approximations. References. K. Atkinson, D. Chien, an O. Hansen. A spectral metho for elliptic equations: The Dirichlet problem, Avances in Computational Mathematics, 33 (), pp K. Atkinson, D. Chien, an O. Hansen. Evaluating polynomials over the unit isk an the unit ball, Numerical Algorithms 67 (4), pp K. Atkinson, D. Chien, an O. Hansen. Spectral Methos Using Multivariate Polynomials On The Unit Ball, in preparation. 4. K. Atkinson an O. Hansen. A spectral metho for the eigenvalue problem for elliptic equations, Electronic Transactions on Numerical Analysis 37 (), pp K. Atkinson an O. Hansen. Creating omain mappings, Electronic Transactions on Numerical Analysis 39 (), pp K. Atkinson, O. Hansen, an D. Chien. A spectral metho for elliptic equations: The Neumann problem, Avances in Computational Mathematics 34 (), pp K. Atkinson an W. Han. Theoretical Numerical Analysis: A Functional Analysis Framework, 3 r e., Springer-Verlag, K. Atkinson an W. Han. An Introuction to Spherical Harmonics an Approximations on the Unit Sphere, Springer-Verlag,. 9. S. Brenner an L. Scott, The Mathematical Theory of Finite Element Methos, Springer- Verlag, New York, P. Ciarlet. The Finite Element Metho For Elliptic Problems, North-Hollan, F. Dai an Y. Xu. Approximation Theory an Harmonic Analysis on Spheres an Balls, Springer-Verlag, 3.. P. Destuyner an M. Salaun. Mathematical Analysis of Thin Plate Moels, Springer, C. Dunkl an Y. Xu. Orthogonal Polynomials of Several Variables, Cambrige Univ. Press,. 4. M. Jaswon an G. Symm (977) Integral Equation Methos in Potential Theory an Elastostatics, Acaemic Press. 5. Y.-M. Jeon. An inirect bounary integral equation for the biharmonic equation, SIAM J. Num. Anal. 3 99), pp Y.-M. Jeon. New bounary element formulas for the biharmonic equation, Avances in Comp. Maths 9 (998), pp Huiyuan Li an Yuan Xu. Spectral approximation on the unit ball, SIAM J. Num. Anal. 5 (4), pp B. Logan an L. Shepp. Optimal reconstruction of a function from its projections, Duke Mathematical Journal 4, (975), F. Olver, D. Lozier, R. Boisvert, an C. Clark (Eitors). NIST Hanbook of Mathematical Functions, Cambrige University Press,.. A.P.S. Selvaurai. Partial Differential Equations in Mechanics, The Biharmonic Equation The Poisson Equation, Springer,.. A. Strou. Approximate Calculation of Multiple Integrals, Prentice-Hall, Inc., 97.. Yuan Xu. Lecture notes on orthogonal polynomials of several variables, in Avances in the Theory of Special Functions an Orthogonal Polynomials, Nova Science Publishers, 4,