WWave Computation on the Poincar´e dodecahedralspace
Agn`es BACHELOT-MOTET
Universit´e de Bordeaux, Institut de Math´ematiques, UMR CNRS 5251, F-33405Talence CedexE-mail: [email protected]
Abstract.
We compute the waves propagating on a compact 3-manifold ofconstant positive curvature with a non trivial topology: the Poincar´e dodecahedralspace that is a plausible model of multi-connected universe. We transform theCauchy problem to a mixed problem posed on a fundamental domain determinedby the quaternionic calculus. We adopt a variational approach using a space offinite elements that is invariant under the action of the binary icosahedral group.The computation of the transient waves is validated with their spectral analysisby computing a lot of eigenvalues of the Laplace-Beltrami operator.AMS classification scheme numbers: 35Q75, 58J45, 65M60 a r X i v : . [ m a t h - ph ] S e p ave Computation on the Poincar´e dodecahedral space
1. Introduction.
Some fundamental open questions regarding the nature of the universe concern itsgeometry and topology. They are the subject of many articles ([11], [20], [21], [28],[31] for example) in which models are compared with observations which are moreaccurate over the years. About geometry, after nine years of investigation, the databy WMAP (Wilkinson Microwave Anisotropy Probe) mission have provided strongevidence suggesting that the universe is nearly flat, with the ratio of its total matter-energy density to the critical value very close to one [4], but without fixing the sign ofits curvature. The Poincar´e dodecahedral space (PDS) is a plausible multi-connecteduniverse model with positive spatial curvature. More precisely, PDS is the quotientof the unit 3-sphere S by the group I ˚ of covering isometries. We suppose that thelocal geometry of the Universe is described by a Friedmann-Lemaˆıtre metric, exceptpossibly during the virialisation (low redshift) epoch, when use of this metric hasbeen claimed to give rise to dark energy as an artefact of assuming a homogeneousmetric despite the strong growth of non-linear inhomogeneities ([5], [6], [16], [25],[33]). With this assumption, PDS (denoted by K in equations) is endowed with thespherical metric ds K induced by ds S . It is a three dimensional C compact manifold,without boundary. Such a manifold can conveniently be converted to a description asa fundamental domain. It will have the shape of a dodecahedron, with pairs of facesidentified.Many authors have studied PDS ([1], [7], [13], [20], [22], [24], [28], [32]). Astationary approach has been used in all these articles. The exact knowledge of theeigenvalues of the Laplace-Beltrami operator ∆ K on PDS allows them to predict someresults (for example the cosmic microwave background temperature anisotropies),and to compare with the WMAP observations; but we note these theoretical resultsdepend on assumptions about the statistical nature of the power spectrum of densityperturbations that may not be fully valid in PDS [26].In this paper we are interested in solving numerically the linear wave equation onPDS l g Ψ : “ B t Ψ ´ ∆ K Ψ “ . (1)This equation plays a fundamental role in General Relativity. We recall that givena lorenzian manifold p M , g µν dx µ dx ν q , the gravitational fluctuations are described, atthe first order, by the linearized Einstein equations in vacuum l g h µν “ , l g : “ a | g | B µ ´ g µν a | g |B ν . ¯ . Another situation in which this equation appears, is the construction of the harmoniccoordinates x µ that are solutions of l g x µ “
0. Hence we may consider (1) in particularas the equation of the gravitational waves in the PDS universe p R t ˆ K , dt ´ ds K q that is the multiconnected analogue of the Einstein universe p R t ˆ S , dt ´ ds S q :both these manifolds have the same metric, they differ only by the topology.To solve equation (1), taking advantage of the selfadjointness of the Laplaceoperator, we could expand any normalizable solution Ψ P C ` R t ; L p K q ˘ on ahilbertian basis of eigenfunctions Ψ q satisfying ´ ∆ K Ψ q “ q Ψ q , 0 ă q , } Ψ q } L “ p t, x q “ a ` bt ` ÿ q ‰ ` c q e iqt ` c q e ´ iqt ˘ Ψ q p x q , ave Computation on the Poincar´e dodecahedral space a, b, c q , c q are determined by the initial data. This methodis powerful since the eigenvalues are explicit (see [1], [15], [17], [19]) and the drawbackof this spectral method, namely the numerical computation of the eigenmodes Ψ q ,has been recently overcome in [2]. Moreover, we can prove that the accuracy ofthe approximate solution Ψ r N s p t, x q : “ a ´ bt ´ ř ă q ă N ` c q e iqt ` c q e ´ iqt ˘ Ψ q p x q isestimated as sup t P R , x P K ˇˇ Ψ p t, x q ´ Ψ r N s p t, x q ˇˇ À ˜ ÿ q ě N q ` | c q | ` | c q | ˘¸ . Since the sequences c q and c q go to zero as q Ñ 8 faster than any inverse power of q when the initial data are C , this scheme provides a very precise approximation forthe smooth solutions. Nevertheless, in this paper we adopt the finite element methodpreviously developed in [3], where we studied a toy model of hyperbolic universe.This method is less accurate than the spectral approach since the accuracy does notincrease with the regularity of the initial data: it is first order. But it has severaladvantages: there is no heavy precomputing such as the calculation of the eigenmodes,and above all, this algorithm could be extended much more easily than the spectralexpansion, to investigate numerically the non-linear dynamics, under the simplifying(standard) assumption that the Friedmann-Lemaˆıtre metric remains valid despite thenon-linearity.The construction of PDS is detailled in the next section. In the third part weget a practical description of the particular fundamental domain F that contains p , , , q of R , and also we get a numerical description of the projection F v of F in R . We mainly use the quaternionic calculus. Next we consider the scalar waveoperator B t ´ ∆ K on the dodecahedral universe R t ˆ K . Then, we compute thesolutions of the wave equation in the time domain, by using a variational method anda discretization with finite elements. The domain of calculus is F v , therefore the initialCauchy problem on the manifold without boundary K , becomes a mixed problem on F v with suitable periodic boundary conditions on B F v . These boundary constraintsare implemented in the choice of the basis of finite elements. We validate our resultsby performing a Fourier analysis of the transient waves that allows to find a lot ofeigenvalues of the Laplace-Beltrami operator ∆ K on PDS.
2. The Poincare Dodecahedral Space.
To be able to perform computations on PDS we need to describe it accurately. Thissection recalls the properties of S , I ˚ and S { I ˚ that we need to know. The 3-sphereis the submanifold of four-dimensional Euclidean space such that x ` x ` x ` x “ d between any two points x and y on S is given by: d p x, y q “ arccos r x i y j s We also use the parametrisation: x “ cos χx “ sin χ sin θ sin ϕx “ sin χ sin θ cos ϕx “ sin χ cos θ with 0 ď χ ď π , 0 ď θ ď π and 0 ď ϕ ď π . These parametrisation leads to aconvenient way to visualize S : two balls in R glued together by their boundary. The ave Computation on the Poincar´e dodecahedral space θ and ϕ coordinates are the standard ones for S . For the first ball χ runs from 0 atthe center through π at the surface. For the second ball χ runs from π at the surfacethrough π at the center. We use a projection p of S in R : p : S Ñ R p x , x , x , x q ÞÑ p x , x , x q That is to say, to represent a point of coordinates p x i q t i “ , ,..., u of S , we consideronly the coordinates p x i q t i “ ,..., u and discard x . The two points of coordinates p x , x , x , x q and p´ x , x , x , x q have the same coordinates p x , x , x q ; they arerepresented by the same point. In the sequel,for any A Ă S , we use A v to denote thevisualization in R of A , that is A v : “ p p A q .To define a discrete fixed-point free subgroup Γ Ă SO p q of isometries of S , onemakes use of the fact that the unit 3-sphere S is identified with the multiplicativegroup H of unit quaternions by x “ p x , x , x , x q ðñ q “ x ` x i ` x j ` x k . The four basic quaternions t , i , j , k u of the set H of all quaternions, satisfy themultiplication rules i “ j “ k “ ´
1, and k “ ij “ ´ ji . They commute withevery real number. The norm of q is defined by } q } : “ x ` x ` x ` x . Theconjugate ¯ q of q “ a ` b i ` c j ` d k is defined by ¯ q : “ a ´ b i ´ c j ´ d k .In this paper, we are interested in the subgroup I ˚ of SO p q called the binaryicosahedral group ([30], [34], [27], [13]). It is a two sheeted covering of the icosahedralgroup I Ă SO p q consisting of all orientation-preserving symmetries of a regularicosahedron. Indeed, for any q in the multiplicative group of unit quaternions H weconsider p q : H Ñ H defined by p q p q q : “ qq q ´ . It fixes the identity quaternion , soin effect its action is confined to the equatorial 2-sphere spanned by the remaining basisquaternions p i , j , k q . Thus, by viewing R as the space of pure imaginary quaternions,subspace of H with basis t i , j , k u , we get a rotation in R . So p q belongs to SO p q .The map π : H Ñ SO p q defined by π p q q “ p q is a homomorphism of H onto SO p q . π is a two to one homomorphism as π p q q “ π p´ q q . By definition I ˚ is the pre-image of I by π . The order of I is 60, so the order of I ˚ is 120. I ˚ contains only right-handedClifford translations γ , in other words γ acts on an arbitrary unit quaternion q P S by left multiplication and translates all points q , q P S by the same distance χ ,i.e. d p q , γq q “ d p q , γq q “ χ . The right-handed Clifford translations act as right-handed cork screw fixed-point free rotations of S .Let take γ “ a ` b i ` c j ` d k an element of I ˚ (with arccos a “ χ ), and q “ x ` x i ` x j ` x k in S . We can express γ as a matrix and we have: γ q “ ¨˚˚˝ a ´ b ´ c ´ db a ´ d cc d a ´ bd ´ c b a ˛‹‹‚¨˚˚˝ x x x x ˛‹‹‚ I ˚ is generated by two isometries s and γ . Denoting by σ “ p ` ? q{ I ˚ “ (cid:10) s, γ | p sγ q “ s “ γ (cid:11) “ " ˘ , ˘ i , ˘ j , ˘ k , p˘ ˘ i ˘ j ˘ k q , p ˘ i ˘ σ j ˘ σ k q p with even permutations q * . ave Computation on the Poincar´e dodecahedral space I ˚ is a right-handed Clifford translation with χ equal to 0, π , π , π , π , π , π , π , or π . We deduce that π is the smallest non-zero translation distance. For example I ˚ is generated by s “ p ` i ` j ` k q and γ “ σ ` σ j ´ k . s is a right-handed Clifford translation with χ equal to π . Also, p s is a rotation in R , its angle is π and its axis is directed by the vector ? p , , q . γ is a Clifford translation with χ equal to π . Moreover p γ is a rotation in R , its angleis π and its axis is directed by the vector ? ´ σ p , σ , ´ q .We now consider S { I ˚ the quotient of the 3-sphere S under the action ofthe discrete fixed-point free subgroup I ˚ of isometries of S , with I ˚ acting by leftmultiplication. This is the Poincar´e dodecahedral space (PDS) ([30], [34], [27], [13]).To perform the computations of the waves on S { I ˚ , it is very useful to represent itby a fundamental domain F Ă S and an equivalence relation „ such that S { I ˚ “ F { „ . (2) F is such that: S “ ď t P I ˚ t p F q , a nd @ t P I ˚ , @ t P I ˚ , ˝ t p F q X ˝ t p F q“ H . (3) F is a regular spherical dodecahedron (dual of a regular icosahedron), and „ isobtained by identifying any pentagonal face of F v with its opposite face, after rotatingby π in the clockwise direction around the outgoing axis orthogonal to this last face.120 such spherical dodecahedra tile the 3-sphere in the pattern of a regular 120-cell(see figure 3).
3. Fundamental domain F of PDS, and its visualization F v . We have choosen the unique fundamental domain that contains p , , , q to simplifycalculations and visualization. In the following it will be denoted F . In order toperform computations we need to know the coordinates of all points in F , and theequations of its edges and faces. We also deduce the characteristics of F v : “ p p F q ,which is our domain of calculus and visualization. Proposition 3.1
The unique fundamental domain that contains p , , , q isthe geodesic convex hull in S of these vertices: ? p σ , ˘ σ , , ˘ q , ? p σ , , ˘ , ˘ σ q , ? p σ , ˘ σ , ˘ σ , ˘ σ q and ? p σ , ˘ , ˘ σ , q .Each of the faces F i of F is a regular pentagon in S included in tp x , x, y, z q P S , a i x ` b i y ` c i z “ x σ u , with p a i , b i , c i q equal to p˘ σ , ˘ , q , up to an even permutation.The set of all barycenters in R of the vertices of F i , denoted by F bi , is includedin a -plane of equations x “ σ ? and a i x ` b i y ` c i z “ x σ .Proof : F is the geodesic convex hull in S of its 20 vertices, so we begin to search thecoordinates of the vertices. As the shortest translation distance of the elements of I ˚ is π , the shortest translation distance in PDS is π . Two opposite faces must be theimage from each other by a Clifford translation with χ equal to π . I ˚ has 12 suchelements denoted by g i in the following parts:12 p σ ` i ˘ σ j ˘ k q p with even permutations of the three last coordinates q . ave Computation on the Poincar´e dodecahedral space SO p q denoted by p g i leave invariant p p F q : “ F v . Furthermore p , , q belongs to F v because p , , , q belongs to F . So OB i : “ ? ´ σ p , ˘ σ , ˘ q ,with even permutations, are the orthogonal axes to each pair of opposite faces of F v . B i are the vertices of an icosahedron that is the dual of a regular dodecahedron whosevertices are the barycenters of three equidistant vertices of the icosahedron. We get apentagonal face having OB i as symetry axis by finding the five vertices that are at thesame minimal distance from B i . Among these five points, two vertices are adjacent iftheir distance is the smallest one. We deduce that, in R , the three last coordinatesof the vertices of F could be: p˘ σ, ˘ σ, ˘ σ q , or p , ˘ σ , ˘ q p with even permutations q . So the coordinates of the 20 vertices C i belonging to S and being the vertices of aregular dodecahedron that has the same symetry axes are of the form: b ´ ` λ σ ˘ ` λ p˘ σ i ˘ σ j ˘ σ k q ,or b ´ λ σ ` λ p i ˘ σ j ˘ k q p with even permutations q , with λ P R such that two opposite faces can be the image from each other by aClifford translation with χ equal to π . Consider a face F of F such that ` ´ σ i ´ j ˘ is a symetry axis of F v orthogonal to the induced face of F v . Its adjacent verticessatisfy: C “ b ´ ` λ σ ˘ ` λ p p´ σ i ´ σ j ` σ k qq , C “ b ´ ` λ σ ˘ ` λ p ` ´ σ i ´ j ˘ q ,C “ b ´ ` λ σ ˘ ` λ p p´ σ i ´ σ j ´ σ k qq , C “ b ´ ` λ σ ˘ ` λ p ` ´ σ j ´ k ˘ q ,C “ b ´ ` λ σ ˘ ` λ p ` ´ σ j ` k ˘ q . And the opposite face having the same symetry axis has the following adjacent vertices: C “ b ´ ` λ σ ˘ ` λ p p σ i ` σ j ´ σ k qq , C “ b ´ ` λ σ ˘ ` λ p ` σ i ` j ˘ q ,C “ b ´ ` λ σ ˘ ` λ p p σ i ` σ j ` σ k qq , C “ b ´ ` λ σ ˘ ` λ p ` σ j ` k ˘ q ,C “ b ´ ` λ σ ˘ ` λ p ` σ j ´ k ˘ q . We are searching λ such that C has C as image by the Clifford translation p σ ` σ i ` j q . So the spherical distance between C and C is equal to π . Weget λ “ p ´ σ q “
92 1 σ . Due to the symetry of F v , the sign of λ is indifferent. Sowe have found only one λ ą F .Now we construct the faces F i of F . Each of them has five edges which are theshortest geodesic G joining two adjacent vertices S i and S j . The geodesics can bedescribed as follows [27]. A path l on S is a geodesic if and only if there is a 2-dimensional plane Π in R passing through the origin such that l Ă Π X S . All thegeodesics are circles. q P G ðñ D α ě , D β ě q “ αOS i ` βOS j and } q } “ ðñ D α ě , D β ě q “ αOS i ` βOS j and “ α ` β ` σαβ. Then a face is the set of shortest geodesics joining two points of the edges. One mayalso say that a face is the set of the projection on S of all barycenters of its five ave Computation on the Poincar´e dodecahedral space R . If we note F bi the set of all barycenters in R of the vertices S i , ..., S i of F i , we have: F i “ $&% b x ` x ` y ` z p x , x , y , z q , p x , x , y , z q P F bi ,.- , (5)and p x , x , y , z q P F bi ô Dp λ , .., λ q P r , s , ÿ j “ ,.., λ j OS ji “ ř j “ ,.., λ j p x , x , y , z q As F bi is included in a 2-plane of equations x “ σ ? and ax ` by ` cz “ x σ witheven permutations of p a, b, c q “ p˘ σ , ˘ , q , we deduce from (5) : @p x , x, y, z q P F i , ax ` by ` cz “ x σ . (See Appendix A for a detailed description of the faces). The proof is complete. Proposition 3.2
The set of t p S i q for all t in I ˚ , and all S i of F has verticesgiven by:(i) a set of vertices given by ? p˘ , ˘ , , q and all its permutations,(ii) a set of vertices given by ? p˘? , ˘ , ˘ , ˘ q and all its permutations,(iii) a set of vertices given by ? p˘ σ, ˘ σ, ˘ σ, ˘ σ q and all its permutations,(iv) a set of vertices given by ? p˘ σ , ˘ σ , ˘ σ , ˘ σ q and all its permutations,(v) a set of vertices given by ? p σ , ˘ σ , , ˘ q and all its even permutations,(vi) a set of vertices given by ? p˘? , ˘ σ , , ˘ σ q and all its even permutations,(vii) a set of vertices given by ? p˘ , ˘ , ˘ σ , ˘ σ q and all its even permutations.They are the vertices of regular dodecahedra which tesselate S . This result is obtained by an explicit calculus of t p S i q for all t in I ˚ and all vertices S i of F . Anyone of these regular dodecahedra is a fundamental domain. Followingfigure 1 shows the three last coordinates of these 600 vertices, viewed from a face ofthe centered dodecahedron. Straight lines between two vertices symbolize edges ofpentagonal faces. We note that these coordinates look like those of Coxeter [12], upto an odd permutation of the three last coordinates. They are adapted to I ˚ , unlikethose of [12]. From proposition 3.1 we deduce the following: Proposition 3.3 p p F q , denoted F v , is a centered regular dodecahedron in R suchthat p x, y, z q P F v ðñ p` a ´ x ´ y ´ z , x, y, z q P F . It is included in the first ball of the visualization of S . Its vertices are ? p˘ σ , , ˘ q , ? p , ˘ , ˘ σ q , ? p˘ σ , ˘ σ , ˘ σ q and ? p˘ , ˘ σ , q .Each face F i,v of F v is a regular pentagon included in an ellipsoid F i,v Ă (cid:32) p x, y, z q P R , σ p ax ` by ` cz q “ ´ x ´ y ´ z ( . with p a i , b i , c i q equal to p˘ σ , ˘ , q , up to an even permutation.The set of all barycenters in R of the vertices of F i,v , denoted by F bi,v , is includedin a -plane of equation a i x ` b i y ` c i z “ ? .ave Computation on the Poincar´e dodecahedral space Figure 1.
120 Cell
See Appendix A for a detailed description of F v . The following figure is adiagramm, and not a visualization, of F (or F v ) because their faces are not in aplane of R (or R ). Figure 2.
Faces F i,b viewed from F ,b . The dashed lines are hidden. ave Computation on the Poincar´e dodecahedral space
4. Equivalence relations on F and F v In order to have relation 2, the equivalence relation must identify any pentagonal faceof F with its opposite face, after rotating by π in the clockwise direction around theoutgoing axis orthogonal to this last face. First specify our notations. We considerthe Clifford translations g i that have been used for the construction of F , with: g : “ σ `
12 1 σ i ` j , g : “ σ ` i ´
12 1 σ k , g : “ σ `
12 1 σ j ´ k ,g : “ σ ´
12 1 σ i ` j , g : “ σ `
12 1 σ j ` k , g : “ σ ` i `
12 1 σ k . They are such that: @ i P t , ..., u , g i p F i q “ F i ` . The inverses of these six firsttranslations are the six other translations: g : “ p g q ´ “ σ ´
12 1 σ i ´ j , g : “ p g q ´ “ σ ´ i `
12 1 σ k , (6) g : “ p g q ´ “ σ ´
12 1 σ j ` k , g : “ p g q ´ “ σ `
12 1 σ i ´ j ,g : “ p g q ´ “ σ ´
12 1 σ j ´ k , g : “ p g q ´ “ σ ´ i ´
12 1 σ k . They all have a translation distance χ equal to π . We have for the vertices (see figure2 for notations): g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S .We define the relation „ by specifying the equivalence classes q of any q P F : @ q P F , q : “ I ˚ pt q uq X F . F has been constructed such that π is the translation distance beetwen two oppositefaces. Otherwise π is the smallest translation distance of the elements of I ˚ . So: q P F ñ q “ ! g i p q q , i P t , ..., u ) X F . It follows that: ‚ If q belongs to ˝ F , then q has only one element, ‚ If q is a vertex of F , then q has four elements, ‚ If q belongs to an edge of a face, without beeing a vertex, then q has threeelements, ‚ If q belongs to a face and does not belong to an edge, then q has two elements.So we have: Proposition 4.1
We define the equivalence classes on F by: q P ˝ F ñ q “ t q u , Dp i, j, k q , q P F i X F j X F k ñ q “ t q, g i p q q , g j p q q , g k p q qu , pDp i, j q , q P F i X F j q and p@ k ‰ i, j, q R F k q ñ q “ t q, g i p q q , g j p q qu , pD i, q P F i q and p@ j ‰ i, q R F j q ñ q “ t q, g i p q qu . Here the integers i, j, k belong to t , ..., u .ave Computation on the Poincar´e dodecahedral space „ on F v is easily deduced from this one on F . Let usdenote g i,v the application R Ñ R induced by g i on F v . Hence for all i in t , ..., u , g i,v p F i,v q “ F i ` ,v and g ´ i,v “ g i ` ,v . We have: Proposition 4.2
We define the equivalence classes on F v by: X P ˝ F v ñ X “ t X u , Dp i, j, k q X P F i,v X F j,v X F k,v ñ X “ t X, g i,v p X q , g j,v p X q , g k,v p X qu , pDp i, j q , X P F i,v X F j,v q and p@ k ‰ i, j, X R F k,v q ñ X “ t X, g i,v p X q , g j,v p X qu , pD i, X P F i,v q and p@ j ‰ i, X R F j,v q ñ X “ t X, g i,v p X qu . Here the integers i, j, k belong to t , ..., u . The geometrical meaning of this equivalence relation is clear : we identify anypentagonal face of F v with its opposite face, after rotating by π in the clockwisedirection around the outgoing axis orthogonal to this last face (see Figure 2).
5. The Wave Propagation on the Dodecahedral Space.
We consider the lorentzian manifold R t ˆ S { I ˚ endowed with the metric g µν dx µ dx ν “ dt ´ ds K , and we study the scalar covariant wave equation associated to this metric : B t Ψ ´ ∆ K Ψ “ . (7)Here ∆ K is the Laplace Beltrami operator on K , that is defined by∆ K : “ a | g | B µ g µν a | g |B ν , g ´ “ p g µν q , | g | “ | det g µν | . Since K is a smooth compact manifold without boundary, ´ ∆ K endowed with itsnatural domain t u P L p K q ; ∆ K u P L p K qu is a densely defined, positive, self-adjoint operator on L p K q and the global Cauchy problem is well posed by the spectralfunctional calculus :Ψ p t q “ cos ´ t a ´ ∆ K ¯ Ψ p q ` sin ` t ?´ ∆ K ˘ ?´ ∆ K B t Ψ p q . We shall use the functional framework of the finite energy spaces. Given m P N , weintroduce the Sobolev space H m p K q : “ (cid:32) u P L p K q , ∇ α K u P L p K q , | α |ď m ( . where ∇ K are the covariant derivatives. We can also interpret this space as the set ofthe distributions u P H m p S q such that u ˝ g “ u for any g P I ˚ . Then the standardspectral theory assures that for all Ψ P H p K q , Ψ P L p K q , there exists a uniqueΨ P C ` R ` t ; H p K q ˘ X C ` R ` t ; L p K q ˘ solution of (7) satisfyingΨ p t “ q “ Ψ , B t Ψ p t “ q “ Ψ , (8)and we have ż K | B t Ψ p t q | ` | ∇ K Ψ p t q | dµ K “ Cst.
Also we have a result of regularity : when Ψ P H p K q , Ψ P H p K q , thenΨ P C ` R ` t ; H p K q ˘ X C ` R ` t ; H p K q ˘ X C ` R ` t ; L p K q ˘ . ave Computation on the Poincar´e dodecahedral space R t ˆ K by its restriction to R t ˆ F .To perform the numerical computation of this solution, we take the domain ofvisualization F v Ă R of the fundamental polygon F Ă S as the domain of calculus.Therefore we introduce the map f that is one-to-one from F v Ă R onto F Ă S defined by f p x, y, z q “ ´a ´ x ´ y ´ z , x, y, z ¯ “ p f p x, y, z q , f p x, y, z q , f p x, y, z q , f p x, y, z qq , and we put ψ p t, x, y, z q : “ Ψ p t, f p x, y, z qq . (9)We introduce the usual Sobolev space H m for the euclidean metric of R : H m p F v q “ t u P L p F v q , @ α P N , | α |ď m, B αx,y,z u P L p F v qu . Proposition 5.1 Ψ P C ` R ` t ; H p K q ˘ X C ` R ` t ; L p K q ˘ is solution of (7) iff ψ belongs to C ` R ` t ; H p F v q ˘ X C ` R ` t ; L p F v q ˘ and satisfies the equation B tt ψ ´ ∆ F v ψ “ , p t, x, y, z q P R ` ˆ F v , (10) where ∆ F v “ p ´ x ´ y ´ z q B ` p ´ y q B ` p ´ z q B ´ xy B ´ xz B ´ yz B ´ x B ´ y B ´ z B , and the boundary conditions @p t, X, X q P R ˆ B F v ˆ B F v , X „ X ñ ψ p t, X q “ ψ p t, X q . (11) Moreover Ψ P C ` R ` t ; H p K q ˘ X C ` R ` t ; H p K q ˘ X C ` R ` t ; L p K q ˘ iff ψ P C ` R ` t ; H p F v q ˘ X C ` R ` t ; H p F v q ˘ X C ` R ` t ; L p F v q ˘ .Proof. We denote g the metric induced on F v by the metric of S and the map f . We get that the coefficients of g ij are: g ii : “ ´ ÿ j “ f j p x, y, z q B ii f j p x, y, z q ,g ik : “ ´ ÿ j “ f j p x, y, z q B ik f j p x, y, z q . So g ij “ ¨˚˝ ´ y ´ z ´ x ´ y ´ z xy ´ x ´ y ´ z xz ´ x ´ y ´ z xy ´ x ´ y ´ z ´ x ´ z ´ x ´ y ´ z yz ´ x ´ y ´ z xz ´ x ´ y ´ z yz ´ x ´ y ´ z ´ x ´ y ´ x ´ y ´ z ˛‹‚ , g ij “ ¨˝ ´ x ´ xy ´ xz ´ xy ´ y ´ yz ´ xz ´ yz ´ z ˛‚ , and det g “ ´ x ´ y ´ z . As ∆ F v : “ ? | g | B µ g µν a | g |B ν we obtain the expression of ∆ F v .Given u P H p F v q , the trace of u on F v is well defined since the domain F v isLipschitz and C piecewise. ψ satisfies the boundary conditions (11) iff its pull-back ave Computation on the Poincar´e dodecahedral space
12Ψ on F can be extended in a solution defined on the whole Poincar´e dodecahedron K . The extension to the smooth solutions is straightforward. The proof is completed.To handle the boundary when applying the finite element method, it is veryconvenient to take into account the boundary condition (11) by a suitable choice ofthe functional space. We introduce the spaces W m p F v q that correspond to the spaces H m p K q : W p F v q : “ L p F v , p ´ x ´ y ´ z q ´ dxdydz q , ď m, W m p F v q : “ (cid:32) u P H m p F v q , @p X, X q P B F v , X „ X ñ u p X q “ u p X q ( , endowed with the norm } u } W m : “ ÿ | α |ď m }B α u } W p F v q . In particular, we have W p F v q “ (cid:32) u P H p F v q , X „ X ñ u p X q “ u p X q ( , and ψ P C k p R t , W m p F v qq ðñ Ψ P C k p R t ; H m p K qq . The numerical method to solve the Cauchy problem will be based on its variationalformulation.
Theorem 5.2
Given ψ P W p F v q , ψ P W p F v q , there exists a unique ψ P C ` R ` t ; W p F v q ˘ X C ` R ` t ; W p F v q ˘ X C ` R ` t ; W p F v q ˘ solution of the equation (10),and satisfying ψ p , . q “ ψ p . q , B t ψ p , . q “ ψ p . q . (12) ψ is the unique function in C ` R ` t ; W p F v q ˘ X C ` R ` t ; W p F v q ˘ X C ` R ` t ; W p F v q ˘ satisfying (12) and such that for any φ P W p F v q , we have : “ d dt ż F v p ´ x ´ y ´ z q ´ ψ p t, x, y, z q φ p x, y, z q dx dy dz (13) ` ż F v p ´ x ´ y ´ z q ´ ∇ ψ p t, x, y, z q ¨ ∇ φ p x, y, z q dx dy dz ´ ż F v p ´ x ´ y ´ z q ´ rp x, y, z q ¨ ∇ ψ p t, x, y, z qsrp x, y, z q ¨ ∇ φ p x, y, z qs dx dy dz. Proof:
The existence and the uniqueness of the solution of the Cauchy problemare given by the previous proposition since the boundary conditions are imposed byour choice of space W p F v q . Now the mixed problem can be expressed as a variationalproblem. ψ is solution iff for all φ P W p F v q , we have :0 “ (cid:10) B t ψ ´ ∆ F v ψ ; φ (cid:11) W p F v q “ d dt ż F v a ´ x ´ y ´ z ψ p t, x, y, z q φ p x, y, z q dx dy dz ´ ż F v a ´ x ´ y ´ z p ∆ F v ψ q p t, x, y, z q φ p x, y, z q dx dy dz. (14) ave Computation on the Poincar´e dodecahedral space ν p x, y, z q the unit outgoing normal at p x, y, z q belonging to face F i,v . We know explicit form of g i,v , the application R Ñ R induced by g i on F v .Thanks to relations written in Appendix A, we have: g ,v p x, y, z q “ ¨˝ σ ´ σ ´ σ ´ ´ σ ´ σ σ ˛‚¨˝ xyz ˛‚ , g ,v p x, y, z q “ ¨˝ ´ σ σ ´ σ σ ´ ´ σ σ ˛‚¨˝ xyz ˛‚ ,g ,v p x, y, z q “ ¨˝ σ σ ´ σ σ ´ σ σ ´ ˛‚¨˝ xyz ˛‚ , g ,v p x, y, z q “ ¨˝ σ σ σ ´ σ ´ ´ σ σ ˛‚¨˝ xyz ˛‚ ,g ,v p x, y, z q “ ¨˝ σ ´ σ σ ´ σ ´ σ ´ σ ´ ˛‚¨˝ xyz ˛‚ , g ,v p x, y, z q “ ¨˝ ´ ´ σ ´ σ σ σ ´ ´ σ σ ˛‚¨˝ xyz ˛‚ . Moreover we know ν p x, y, z q thanks to the equation of the ellipsoids (Appendix A).We verify that: g i,v p ν p x, y, z qq “ ´ ν p g i,v p x, y, z qq , Since p u ˝ g i,v q | F i,v “ u | F i,v , we have for u P W p F v qB ν p X q u p x, y, z q “ g i,v r ν p x, y, z qs . ∇ u p g i,v p x, y, z qq “ ´B ν p g i,v p x,y,z qq u p g i,v p x, y, z qq . We deduce that for u P W p F v q , v P W p F v q , we have ż F i,v v p x, y, z qB ν p x,y,z q u dσ p x, y, z q “ ´ ż g i,v p F i,v q v p x, y, z qB ν p x,y,z q u dσ p x, y, z q“ ´ ż F i ` ,v v p x, y, z qB ν p x,y,z q u dσ p x, y, z q , (15)and therefore ż B F v v p x, y, z qB ν p x,y,z q u dσ p x, y, z q “ , where dσ denotes the volume measure on pB F v , g q . We conclude that : ż F v p ∆ F v ψ q p t, x, y, z q φ p x, y, z q a det g dx dy dz “ ´ ż F v g p grad ψ, grad φ q a det g dx dy dz. To simplify the writing we shall note in the following X instead of p x, y, z q and | X | instead of x ` y ` z for p x, y, z q P F v ; and dX will designate dx dy dz . Thereforewe compute : ż F v p ´ | X | q ´ ∆ F v ψ p t, X q φ p X q dX “´ ż F v p ´ | X | q ´ “ p ´ x q B ψ p t, X qB φ p t, X q`p ´ y q B ψ p t, X qB φ p t, X q ` p ´ z q B ψ p t, X qB φ p t, X q ‰ dX ` ż F v p ´ | X | q ´ r xy pB ψ p t, X qB φ p t, X q ` B ψ p t, X qB φ p t, X qq` xz pB ψ p t, X qB φ p t, X q ` B ψ p t, X qB φ p t, X qq` yz pB ψ p t, X qB φ p t, X q ` B ψ p t, X qB φ p t, X qqs dX. ave Computation on the Poincar´e dodecahedral space ż F v p ´ | X | q ´ ∆ F v ψ p t, X q φ p X q dX “´ ż F v p ´ | X | q ´ ∇ ψ p t, X q ¨ ∇ φ p t, X q dX ` ż F v p ´ | X | q ´ p X ¨ ∇ ψ p t, X qq p X ¨ ∇ φ p t, X qq dX. The proof of the theorem is complete.We solve this variational problem by the usual way. We take a family V h ,0 ă h ď h , of finite dimensional vector subspaces of W p F v q . We assume that Y ă h ď h V h “ W p F v q . We choose sequences ψ ,h , ψ ,h P V h such that ψ ,h Ñ ψ in W p F v q , ψ ,h Ñ ψ in L p F v q . We consider the solution ψ h P C p R t ; V h q of @ φ h P V h , d dt ż F v p ´ | X | q ´ ψ h p t, X q φ h p X q dX ` ż F v p ´ | X | q ´ ∇ ψ h p t, X q ¨ ∇ φ h p t, X q dX ´ ż F v p ´ | X | q ´ p X ¨ ∇ ψ h p t, X qq p X ¨ ∇ φ h p t, X qq dX “ , satisfying ψ h p , . q “ ψ ,h p . q , B t ψ h p , . q “ ψ ,h p . q . Thanks to the conservation of theenergy, ż F v ` | B t ψ p t, x, y, z q | ` | ∇ F v ψ p t, x, y, z q | ˘ a det g dx dy dz “ Cst, this scheme is stable : @ T ą , sup ă h ď h sup ď t ď T } ψ h p t q} W ` ›››› ddt ψ h p t q ›››› L ă 8 . Moreover, when ψ P C ` R ` t ; W p F v q ˘ , it is also converging : @ T ą , sup ď t ď T } ψ h p t q ´ ψ p t q} W ` ›››› ddt ψ h p t q ´ ddt ψ p t q ›››› L Ñ , h Ñ . If we take a basis ` e hj ˘ ď j ď N h of V h , we expand ψ h on this basis : ψ h p t q “ N h ÿ j “ ψ hj p t q e hj , and we introduce U p t q : “ t ` ψ h , ψ h , ¨ ¨ ¨ , ψ hN h ˘ M “ p M ij q ď i,j ď N h , D “ p D ij q ď i,j ď N h , K “ p K ij q ď i,j ď N h ave Computation on the Poincar´e dodecahedral space M ij : “ ż F v a ´ | X | e hi p X q e hj p X q dX,K ij : “ ż F v a ´ | X | ` B x e hi p X qB x e hj p X q ` B y e hi p X qB y e hj p X q ` B z e hi p X qB y e hj p X q ˘ dX.D ij : “ ´ ż F v a ´ | X | ` x B x e hi p X q ` y B y e hi p X q ` z B z e hi p X q ˘` x B x e hj p X q ` y B y e hj p X q ` z B z e hj p X q ˘ dX. Then the variational formulation is equivalent to M X ` p K ` D q X “ . (16)This differential system is solved very simply by iteration by solving M p U n ` ´ U n ` U n ´ q ` p ∆ T q p K ` D q U n “ . We know that this scheme is stable, and so convergent by a consequence of the Laxtheorem [18], whensup U ‰ ă p K ` D q U, U ąă M U, U ą ă T . Therefore if there exists K ą @ h Ps , h s , @ φ h P V h , ›››› ∇ x,y,z φ h p ´ | X | q ›››› L p F v q ď Kh ›››› φ h p ´ | X | q ›››› L p F v q , the CFL condition K ∆ T ă ? h, (17)is sufficient to assure the stability and the convergence of our scheme.
6. Numerical calculations
Our goal is to built a mesh of B F such that it has a fixed size in advance, all the edgesare splitting in the same way, and the meshes of two opposite faces are the image fromeach other by the Clifford translation g i fitting for „ . Of course we also want that allpoints of the mesh of F i,v Ă B F v are on the suitable ellipsoid, and that all points ofthe edges of F i,v are on the right geodesic.First of all we construct the boundary B F v . Each F bi is included in a 2-plane of R andcan be easily meshed with a convenient metric. We choose to built a mesh of F b , with F : “ p S , S , S , S , S q (see Figure 2). We denote F b ,v : “ p p F b q the visualizationof F b . As we want the mesh vertices on each edge of F are equidistant, we choose a priori the desired spherical distance between two consecutive mesh vertices. Thisdetermines the number of mesh vertices on an edge. Then, by a suitable application,we map F b in the 2-plane z “ R , which allows us to use a 2-D mesh generatorable to respect a given metric matrix M . Let us recall the inclusions F b Ă " p x , x, y, z q P R , x “ σ ? , ´ σ x ´ y “ x σ * ,F b ,v Ă " p x, y, z q P R , ´ σ x ´ y “ ? * . ave Computation on the Poincar´e dodecahedral space t of F b ,v by the vector ÝÝÝÝÝÑ OM , “ p , ´ ? , q where M , denotesthe middle of S S , followed by a rotation r in R , with an angle ´ π and axis (cid:126)u “ ÝÝÝÝÝÝÑ S M , “ p ? ´ σ , ´ σ ? ´ σ , q . So, thanks to the quaternionic calculus: r p x i ` y j ` z k q “ „ ? ´ ? ˆ ? ´ σ i ´ ?
22 1 σ ? ´ σ j ˙ r x i ` y j ` z k s „ ? ` ? ˆ ? ´ σ i ´ ?
22 1 σ ? ´ σ j ˙ , and, r p x, y, z q “ ¨˚˝ ´ σ x ´ σ p ´ σ q y ` σ ? ´ σ z ´ σ p ´ σ q x ` σ p ´ σ q y ` ? ´ σ z ´ σ ? ´ σ x ´ ? ´ σ y ˛‹‚ . If p x, y, z q belongs to F b ,v we get p r ˝ t q p x, y, z q “ ˆ x , y , ´ σ ? ´ σ x ´ ? ´ σ ˆ y ` ? ˙˙ “ p x , y , q . Therefore r ˝ t p F b ,v q is in the 2-plane z “ R . Of course we want a metric on R endowed by the metric of S . We denote by f the application that maps r ˝ t p F b ,v q to F , that is f : r ˝ t p F b ,v q Ñ R Ñ F b ,v Ñ F b Ă R p x, y q ÞÑ p x, y, q ÞÑ p x , y , z q : “ p r ˝ t q ´ p x, y, q ÞÑ ´ σ ? , x , y , z ¯ followed by: F b Ă R Ñ F Ă S ´ σ ? , x , y , z ¯ ÞÑ } ´ σ ? ,x ,y ,z ¯ } ´ σ ? , x , y , z ¯ . We deduce that the coefficients of M are given by: m ii : “ ´ ÿ j “ , f j p x, y q B ii f j p x, y q , m ik : “ ´ ÿ j “ , f j p x, y q B ik f j p x, y q . (18)See Appendix B for the values m ij .We used the software FreeFem `` [14] for the generation of the 2-D mesh of r ˝ t p F b ,v q . Then, applying f , we get a mesh of F with the wanted size. Moreover,as we know the equation of the 2-plane containing F b ,v , we are able to control and tocorrect the coordinates of the vertices of the mesh of F b ,v with a great precision. Wecan also control that the mesh vertices on the edges of F are on the convenient geodesic(4), distant from each other with the wanted distance; we correct their coordinates ifnecessary.Then, by the use of rotations in R we get a mesh of the five adjacent faces F b ,v ,....., F b ,v (see Appendix C). After normalizing we have got a mesh of F , ..., F . Atlast, by the use of g , ...., g we deduce a mesh of F , ..., F . Once more we controland correct the mesh vertices on edges. As we also know the equation of the ellipsoidthat contains each face, we also control all the mesh vertices. Meshes of each F i,v areobtained from those of F i by discarding the first coordinate of the mesh vertices. ave Computation on the Poincar´e dodecahedral space ˝ F we used the software Tetgen [29] in several ways. We havecreated a mesh with an inner sphere or an inner dodecahedron, in order to impose twodifferent average sizes of tetrahedra depending on the location. We also used Tetgenwith a .mtr file that gives us the wanted size near each point. In all cases we employedan option so that there is no point introduced into B F v . We can see the accuracy ofour mesh by calculating the sum of the volume of tetrahedra which compose F v . Thissum must be closed to π that is » . Table 1.
Distances beetween two mesh vertices of B F v . Name of Number of mesh vertices between minimal distance maximal distancethe mesh two vertices of an edge of F i F089 89 2 .
06 10 ´ .
11 10 ´ F101 101 1 .
82 10 ´ .
90 10 ´ Table 2.
Characteristics of these two meshes.
Mesh number number number of volume of relative errorof vertices of nodes tetrahedra all tetrahedraF089 182162 730309 4642744 0 . .
22 10 ´ F101 233858 1600118 10198710 0 . .
29 10 ´ We can also look at the tilling of S . In figure 3 we display six images of F v , as wellas F in the first ball of visualization of S . All points of theses images have a positivefirst coordinate in R , except those of g g g p F q . To fully represent this last set, weshould also draw the second ball of visualization of S , in which its representation isthe same as in the first ball. V h space We construct the finite element spaces V h of P type. We take into account theboundary condition (11) in the definition of the finite elements, so that V h Ă W p F v q .We note T h all tetraedra of a mesh, F v,h the set Y K P T h K , and P p K q the set of firstdegree polynomial functions on K . Then we introduce: V h : “ (cid:32) v : F v,h Ñ R , v P C p F v,h q , @ K P T h , v | K P P p K q , M „ M ñ v p M q “ v p M q ( . The equivalent points on B F v are known thanks our construction of meshes. Thenumber of nodes N h is the sum of ˆ n ve ` ˆ n vf ` n vi ` , with n ve thenumber of mesh vertices on an edge of a face that are not a vertex of F v , n vf thenumber of mesh vertices on a face that are not on an edge, n vi the number of meshvertices in ˝ F v .If j is the number of a node and if M i denotes a vertex of the mesh, we constructa basis ` e hj ˘ ď j ď N h of V h by: ave Computation on the Poincar´e dodecahedral space F v , g p F v q , g g p F v q , g p F v q , g g p F v q , g g p F v q , g g g p F v q Figure 3.
The Fundamental Domain (dark blue) and six images of it. (i) If j is associated to a node that does not belong to B F v : e hj p M i q “ δ ij .There are n vi functions of this kind.(ii) If j is associated to a node that is a vertex of F v : e hj p M i q “ " if M i „ M j , otherwise. There are five functions of this kind.(iii) If j is associated to a node that belongs to a face of B F v and not to an edge: e hj p M i q “ " if M i “ P i , otherwise. There are 6 ˆ n vf functions of this kind.(iv) If j is associated to a node that belongs to an edge of a face of B F v and is not avertex of F v : e hj p M i q “ " if M i “ P i , otherwise. There are 10 ˆ n ve functions of this last kind. K p i, j q , D p i, j q and M p i, j q are found with a numerical integration using CUBPACK[10]. These matrices are sparse and symetric. So we choose a Morse storage oftheir lower part, and all of the calculations will be performed with this storage. Tosolve the linear problem we use a preconditioned conjugate gradient method. Thepreconditioner is an incomplete Choleski factorisation, and the starting point is thesolution obtained with a diagonal preconditioner.In order to control K ` D we verify that ∆ F v X “ X with all its coordinatesequal to 1. Consequently, we compute the sums of the elements of each line of K ` D ;all these sums must be equal to 0. For F089 mesh we obtain sums less than 5 10 ´ .This maximal sum is reached for very few mesh vertices of the interior of F v . Thisresult is slightly lower for meshes with more mesh vertices on the edges of F i . Higher ave Computation on the Poincar´e dodecahedral space We choose different initial data, all with B t ψ p , . q “ ψ p . q “ ψ p X q “ e d p X,X q d p X,X q´ r , for d p X, X q ă r , and ψ p X q “ , for d p X, X q ě r . with X “ p , , q and r “ .
3. In the following we refer to this initial data as thename
Init c . And for the one depicted in figure 7, we have taken a similar function witha smaller support, and especially a support not centered at the origin. In the followingwe refer to this another initial data as the name Init exc . Both these initial data havea kink at X : they are not in C p F v q , and by computing their derivatives in the senseof the distributions, we can check that they belong only to W p F v qz W p F v q . Hencethis singularity is rather weak and we may approximate these functions in V h . Themain interest of the kink is its ability of exciting a large amount of eigenmodes, and itprovides an efficient tool to compute a lot of eigenvalues with an excellent agreement.We also have considered a third initial data in C p F v q , denoted Init exc, in thesequel, and of the form: ψ p X q “ e d p X,X q d p X,X q´ r , for d p X, X q ă r , and ψ p X q “ , for d p X, X q ě r . Since our scheme is based on the approximation by finite element of order one, wecannot expect a best accuracy but this function will be very convenient if we want usea higher finite element method to improve the accuracy of our computation.We note that the support of all these initial data is far from B F v , therefore theyrespect obviously the constraint of the equivalent points. It is interesting to considerinitial data involving several equivalent points. On S , any function Φ defines aninitial data Ψ that is invariant under the action of I ˚ by the formulaΨ p x , x, y, z q “ ÿ g P I ˚ Φ p g p x , x, y, z qq , hence by (9), we can introduce on the unit ball of R (that is how we visualize S ) : ψ p X q : “ ÿ g P I ˚ Φ p g p f p X qqq , ϕ p X q : “ Φ p f p X qq . After transcribing the property (3) in R , and denoting g v the application R Ñ R induced by g P I ˚ on F v , we can see that when the support of Φ is small enough,such an initial data satisfies: @ X P F v , D ! g v g v p X q P supp ϕ Therefore we can define ψ on F v by ψ p X q “ ϕ p g v p X qq . For example we have made calculations with an initial data denoted by
Init sum ,and defined by: ψ p X q “ $&% ϕ p X q , if X P supp ϕ X F v ϕ p g ,v p X qq if g ,v p X q P supp ϕ X g ,v p F v q ϕ p g ,v p X qq if g ,v p X q P supp ϕ X g ,v p F v q ave Computation on the Poincar´e dodecahedral space ϕ p X q “ e d p X,X q d p X,X q´ r , for d p X, X q ă r , and ϕ p X q “ , for d p X, X q ě r . where X “ p´ . , ´ . , . q and r “ . X belongs to F v and is closed to themiddle of the edge Ŕ S S of F v (see Appendix A). There are three regions of F v where ψ is non equal to 0. On a cutting plane passing through the zone which contains X and one of the two others, we can see that the second one is the complement of thefirst one (Figure 4, Figure 8). Figure 4.
Init sum on a cutting plane.
In order to see the stability of our method, we compute E d p t q , the discrete energyassociated to our numerical scheme at the time t : E d p t q : “ (cid:28) M X n ´ X n ´ ∆ t , X n ´ X n ´ ∆ t (cid:29) ` (cid:10) p K ` D q X n ´ , X n (cid:11) . It is well known that our scheme is conservative, hence E d must be invariant as afonction of time: this is the case in our calculations. For example, with the previousinitial data and the use of the mesh F089 (see Table 2), we obtain:Initial data E d p q E d p q Init c . . Init exc . . Init sum . . Figure 5.
The solution ψ p t, X q from t “ t “ ave Computation on the Poincar´e dodecahedral space Figure 6.
The solution ψ p t, X q with the initial data Init c at t “ t “ . t “ . t “ . t “ . t “ . Figure 7.
The solution ψ p t, X q with the initial data Init exc at t “ t “ . t “ . t “ . t “ . t “ . Figure 6, Figure 7 and Figure 8 contain some characteristic pictures for a giveninitial data. At each time there are a view of the solution on B F v , and, in a bounding ave Computation on the Poincar´e dodecahedral space Figure 8.
The solution ψ p t, X q with the initial data Init sum at t “ . t “ . t “ . t “ . t “ . t “ . box, a view on a cutting plane. On Figure 7 we can observe that the equivalencerelation is satisfied. Progressively, equivalent faces are in the support of the solutionand we have ψ p t, x q “ ψ p t, y q ‰ x and y of B F v . OnFigure 8 the cutting plane is the same as in Figure 4. At t “ . „ mbachelo/PDS.html). We test our scheme in the time domain, by looking for the eigenvalues that areexplicitly known (see [1], [15], [17], [19]). Since the Laplace-Beltrami operator ∆ K on PDS is a non positive, self-adjoint elliptic operator on a compact manifold, itsspectrum is a discrete set of eigenvalues ´ q ď
0, and by the Hilbert-Schmidt theorem(see e.g
Theorem 6.16 in [23]), there exists an orthonormal basis in L p K q , formed ofeigenfunctions p ψ q q q Ă H p K q associated to q , i.e. ´ ∆ F v ψ q “ q ψ q , ψ q P W p F v q . One has: q “ β ´ , with β P t , , , , , , , , , , , , , , u Y t n ` , n ě u , and wetake ψ “ ? π . Therefore any finite energy solution ψ p t, X q of B t ψ ´ ∆ K ψ “ ave Computation on the Poincar´e dodecahedral space at ` b ` ř q ` c q e iqt ` c q e ´ iqt ˘ ψ q p X q . More precisely, if wedenote ă , ą the scalar product in L p K q , we write ψ p t, X q “ π pă B t ψ p , . q , ą t ` ă ψ p , . q , ąq` ÿ q ‰ ă B t ψ p , . q , ψ q ą sin qtq ψ q p x, y q` ă ψ p , . q , ψ q ą cos qt ψ q p X q . To compute the eigenvalues q we investigate the Fourier transform in time of thesignal ψ p t, X q in the case where B t ψ p , X q “
0, for different X . Practically, duringthe time resolution of the equation, we store the values of the solution at some vertices M p X q for the discrete time k ∆ t , N i ď k ď N f . We choose the initial step N i in orderto the transient wave is stabilized, that to say N i ∆ t is at least greater than thediameter of F v , i.e. N i ∆ t ě ˚ d p , S q “ d p S , S q “ arccos ` σ ´ ˘ » . p ψ h p k ∆ t, X qq N i ď k ď N f with the free licensed (GNU GPL)FFT library fftw3. After we search the values j max for which the previous result p Ψ j p X qq ,N f ´ N i ` has a local maximum. The eigenvalues found by the algorithm areexpressed as: q “ π p N f ´ N i ` q ∆ t j max . We have made tests by varying parameters such as: mesh, initial data, N i and N f .With F089 mesh, in all cases we have obtained the 36 first consecutive eigenvalueswith an relative error on q of about 10 ´ for the first nine values, and 7 10 ´ for thefollowing. The values are the same with all our initial data: Init c , Init exc , Init exc, and Init sum . We have also performed some calculations with a mesh having 135mesh vertices on edges of F v . In this case the error is smaller. It was expectedfor two reasons: the approximation of F v is best, and also, in order to respect theCFL condition (17), we must take a smaller time step. Therefore ψ p t, X q is betterapproximated. In the next table we present the results for the nine first eigenvaluesfor this last mesh. β q result relative error β q result relative error13 168 167 . .
306 10 ´
37 1368 1373 .
049 3 .
691 10 ´
21 440 439 .
107 2 .
029 10 ´
41 1680 1687 .
761 4 .
619 10 ´
25 624 623 .
184 1 .
308 10 ´
43 1848 1857 .
283 5 .
023 10 ´
31 960 959 .
561 4 .
571 10 ´
45 2024 2034 .
916 5 .
393 10 ´
33 1088 1087 .
757 2 .
231 10 ´
7. Conclusion
We note a good agreement of the eigenvalues obtained by the spectral analysis of thetransient waves that we have computed. We emphasize that this result is a strongevidence of the accuracy of our numerical approximation since the waves are rapidlyoscillating due to the large value of the eigenvalues. Therefore we conclude that wehave validated this computational method of the waves on the Poincar´e dodecahedralspace, and we plan to extend it, in the future, to the case of the non-linear waves onthis manifold. The finite element method is much more appropriate to the nonlinearproblems than the spectral method that has a very high complexity due to thenonlinearities. Among the nonlinear dynamics that could be treated, we can mention,in field theory on curved space-times, the Higgs field that has been studied for the ave Computation on the Poincar´e dodecahedral space R t ˆ S by Y. Choquet-Bruhat and D. Christodoulou [8], and obeysthe semilinear Klein-Gordon equation l g Ψ “ F p Ψ q , F P C p R q , and also the more complicated Yang-Mills system ([8], [9]) that has the form l g F µν “ N µν p A, F, B A, B F q , where the nonlinearities N µν are cubic polynomials. Acknowledgments
The author wishes to thank J. Fresnel for his stimulating discussions on the Coxeter’sbook [12], R. Cools for having given me acces to the software CUBPACK.
Appendix A:
Description of F and F v .(i) Coordinates of the vertices S i of F : S “ ? ` σ , ´ σ , σ , ´ σ ˘ , S “ ? ` σ , , σ , ˘ ,S “ ? ` σ , ´ σ , ´ σ , σ ˘ , S “ ? ` σ , σ , ´ σ , ´ σ ˘ ,S “ ? ` σ , , ´ , ´ σ ˘ , S “ ? ` σ , σ , σ , σ ˘ ,S “ ? ` σ , ´ σ , , ˘ , S “ ? ` σ , , , σ ˘ ,S “ ? ` σ , ´ σ , σ , σ ˘ , S “ ? ` σ , σ , , ˘ ,S “ ? ` σ , , , ´ σ ˘ , S “ ? ` σ , ´ , σ , ˘ ,S “ ? ` σ , ´ σ , , ´ ˘ , S “ ? ` σ , σ , ´ σ , σ ˘ ,S “ ? ` σ , σ , , ´ ˘ , S “ ? ` σ , ´ σ , ´ σ , ´ σ ˘ ,S “ ? ` σ , σ , σ , ´ σ ˘ , S “ ? ` σ , ´ , ´ σ , ˘ ,S “ ? ` σ , , ´ σ , ˘ , S “ ? ` σ , , ´ , σ ˘ . (ii) Images by the Clifford translation g i of the face F i of F and of its edges. g maps F to F , and we have for the vertices and the edges: g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p Ŕ S S q “ Ő S S , g p Ŕ S S q “ Ŕ S S , g p Ŕ S S q “ Ŕ S S , g p Ŕ S S q “ Ŕ S S ,g p Ŕ S S q “ Ő S S . g maps F to F , and we have for the vertices and the edges: g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p Ŕ S S q “ Ŕ S S , g p Ŕ S S q “ Ŕ S S , g p Ő S S q “ Ŕ S S , g p Ő S S q “ Ŕ S S ,g p Ŕ S S q “ Ŕ S S . g maps F to F , and we have for the vertices and the edges: g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p Ő S S q “ Ŕ S S , g p Ŕ S S q “ Ŕ S S , g p Ŕ S S q “ Ŕ S S ,g p Ŕ S S q “ Ŕ S S , g p Ŕ S S q “ Ŕ S S . ave Computation on the Poincar´e dodecahedral space g maps F to F , and we have for the vertices and the edges: g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p Ŕ S S q “ Ő S S , g p Ŕ S S q “ Ŕ S S , g p Ŕ S S q “ Ŕ S S , g p Ő S S q “ Ŕ S S ,g p Ŕ S S q “ Ŕ S S . g maps F to F , and we have for the vertices and the edges: g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p Ő S S q “ Ŕ S S , g p Ŕ S S q “ Ő S S , g p Ŕ S S q “ Ő S S , g p Ŕ S S q “ Ő S S ,g p Ŕ S S q “ Ŕ S S g maps F to F , and we have for the vertices and the edges: g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p S q “ S , g p Ŕ S S q “ Ŕ S S , g p Ŕ S S q “ Ő S S , g p Ŕ S S q “ Ŕ S S ,g p Ŕ S S q “ Ŕ S S , g p Ŕ S S q “ Ŕ S S .(iii) F bi , the set of all barycenters in R of the vertices of F i , is included in a 2-planeof R . The equations of these 2-planes are: F b Ă ! p x , x, y, z q P R , x “ σ ? , ´ σ x ´ y “ x σ ) ,F b Ă ! p x , x, y, z q P R , x “ σ ? , ` σ x ` y “ x σ ) ,F b Ă ! p x , x, y, z q P R , x “ σ ? , ´ x ` σ z “ x σ ) ,F b Ă ! p x , x, y, z q P R , x “ σ ? , ` x ´ σ z “ x σ ) ,F b Ă ! p x , x, y, z q P R , x “ σ ? , ´ σ y ` z “ x σ ) ,F b Ă ! p x , x, y, z q P R , x “ σ ? , ` σ y ´ z “ x σ ) ,F b Ă ! p x , x, y, z q P R , x “ σ ? , ` σ x ´ y “ x σ ) ,F b Ă ! p x , x, y, z q P R , x “ σ ? , ´ σ x ` y “ x σ ) ,F b Ă ! p x , x, y, z q P R , x “ σ ? , ´ σ y ´ z “ x σ ) ,F b Ă ! p x , x, y, z q P R , x “ σ ? , ` σ y ` z “ x σ ) ,F b Ă ! p x , x, y, z q P R , x “ σ ? , ´ x ´ σ z “ x σ ) ,F b Ă ! p x , x, y, z q P R , x “ σ ? , ` x ` σ z “ x σ ) . (iv) So, after having normalized the points of F bi , we get that F i is included in an ave Computation on the Poincar´e dodecahedral space R . The equations of these 3-planes are: F Ă (cid:32) p x , x, y, z q P S , ´ σ x ´ y “ x σ ( ,F Ă (cid:32) p x , x, y, z q P S , ` σ x ` y “ x σ ( ,F Ă (cid:32) p x , x, y, z q P S , ´ x ` σ z “ x σ ( ,F Ă (cid:32) p x , x, y, z q P S , ` x ´ σ z “ x σ ( ,F Ă (cid:32) p x , x, y, z q P S , ´ σ y ` z “ x σ ( ,F Ă (cid:32) p x , x, y, z q P S , ` σ y ´ z “ x σ ( ,F Ă (cid:32) p x , x, y, z q P S , ` σ x ´ y “ x σ ( ,F Ă (cid:32) p x , x, y, z q P S , ´ σ x ` y “ x σ ( ,F Ă (cid:32) p x , x, y, z q P S , ´ σ y ´ z “ x σ ( ,F Ă (cid:32) p x , x, y, z q P S , ` σ y ` z “ x σ ( ,F Ă (cid:32) p x , x, y, z q P S , ´ x ´ σ z “ x σ ( ,F Ă (cid:32) p x , x, y, z q P S , ` x ` σ z “ x σ ( . (v) We deduce from the previous items that F bi,v , the set of all barycenters in R ofthe vertices of F i,v , is included in a plane of R . The equations of these planesare: F b ,v Ă ! p x, y, z q P R , ´ σ x ´ y “ ? ) ,F b ,v Ă ! p x, y, z q P R , ` σ x ` y “ ? ) ,F b ,v Ă ! p x, y, z q P R , ´ x ` σ z “ ? ) ,F b ,v Ă ! p x, y, z q P R , ` x ´ σ z “ ? ) ,F b ,v Ă ! p x, y, z q P R , ´ σ y ` z “ ? ) ,F b ,v Ă ! p x, y, z q P R , ` σ y ´ z “ ? ) ,F b ,v Ă ! p x, y, z q P R , ` σ x ´ y “ ? ) ,F b ,v Ă ! p x, y, z q P R , ´ σ x ` y “ ? ) ,F b ,v Ă ! p x, y, z q P R , ´ σ y ´ z “ ? ) ,F b ,v Ă ! p x, y, z q P R , ` σ y ` z “ ? ) ,F b ,v Ă ! p x, y, z q P R , ´ x ´ σ z “ ? ) ,F b ,v Ă ! p x, y, z q P R , ` x ` σ z “ ? ) . (vi) As x “ ´ x ´ y ´ z , it follows from item (iv) that the faces F i,v of F v areincluded in an ellipsoid. F ,v and F ,v are included in the same ellipsoid: (cid:32) p x, y, z q P R , p σ ` q x ` σ y ` z ` σ xy “ ( .F ,v and F ,v are included in the same ellipsoid: (cid:32) p x, y, z q P R , σ x ` y ` p σ ` q z ´ σ xz “ ( .F ,v and F ,v are included in the same ellipsoid: (cid:32) p x, y, z q P R , x ` p σ ` q y ` σ z ´ σ yz “ ( .F ,v and F ,v are included in the same ellipsoid: (cid:32) p x, y, z q P R , p σ ` q x ` σ y ` z ´ σ xy “ ( . ave Computation on the Poincar´e dodecahedral space F ,v and F ,v are included in the same ellipsoid: (cid:32) p x, y, z q P R , x ` p σ ` q y ` σ z ` σ yz “ ( .F ,v and F ,v are included in the same ellipsoid: (cid:32) p x, y, z q P R , σ x ` y ` p σ ` q z ` σ xz “ ( . Appendix B
We give the expression of the metric matrix on R endowed by the metric of S whichis written in 18. We denote by f the application that transfoms the 2-D mesh to amesh of F , that is f : R Ñ R Ñ F b ,v Ă R Ñ F b Ă R p x, y q ÞÑ p x, y, q ÞÑ p x , y , z q : “ p r ˝ t q ´ p x, y, q ÞÑ ´ σ ? , x , y , z ¯ followed by F b Ă R Ñ F Ă S ´ σ ? , x , y , z ¯ ÞÑ ›››´ σ ? ,x ,y ,z ¯››› ´ σ ? , x , y , z ¯ We simplify: ››››ˆ σ ? , x , y , z ˙›››› “ ” x ` y ` ? ? x ` ? p ` ? q y ` ` ? ı , and we define g p x, y q by g p x, y q : “ ››››ˆ σ ? , x , y , z ˙›››› . So: f : “ p x, y q ÞÑ ? p `? q? g p x,y q ,f : “ p x, y q ÞÑ p ` ? q x ´ ? y ´? ? g p x,y q ,f : “ p x, y q ÞÑ ´ ? x ` p ´ ? q y ´ p `? q ? ? g p x,y q ,f : “ p x, y q ÞÑ ? `? pp ´? q ? x ` ? y `? q? g p x,y q . Thanks to Maple we obtain: m : “ “ x ` y ` ? ? x ` ? y ` ? ? y ` ` ? ‰ ´ ˆ “ x y ` y ` ? ? x y ` ? x y ` ? ? xy ` ? y ` ? ? y ` x ` ? x ` xy ` ? xy ` ? y ` y ` ? ? x ` ? x ` ? ? y ` ? y ` ` ? ‰ ,m : “ “ x ` y ` ? ? x ` ? y ` ? ? y ` ` ? ‰ ´ ˆ “ x ` x y ` ? ? x ` ? x y ` ? ? x y ` ? ? xy ` x ` ? x ` ? xy ` xy ` y ` ? y ` ? x ` ? ? x ` ? ? y ` ? y ` ? ` ‰ , ave Computation on the Poincar´e dodecahedral space m : “ “ x ` y ` ? ? x ` ? y ` ? ? y ` ` ? ‰ ´ ˆ “ x y ` x y ` x y ` x y ` xy ` ? ? x ` ? x ` ? ? x y ` ? ? x y ` ? x y ` ? ? x y ` ? ? x y ` ? x y ` ? ? x y ` ? ? x y ` ? x y ` ? ? x y ` ? xy ` ? ? xy ` ? ? y ` x ` ? x ` ? x y ` x y ` x y ` ? x y ` x y ` ? x y ` ? x y ` x y ` x y ` ? x y ` x y ` ? x y ` xy ` ? xy ` ? y ` y ` ? ? x ` ? x ` ? ? x y ` ? x y ` ? ? x y ` ? x y ` ? ? x y ` ? x y ` ? ? x y ` ? x y ` ? ? x y ` ? x y ` ? xy ` ? ? xy ` ? ? y ` ? y ` x ` ? x ` x y ` ? x y ` x y ` ? x y ` x y ` ? x y ` ? x y ` x y ` xy ` ? xy ` y ` ? y ` ? ? x ` ? x ` ? x y ` ? ? x y ` ? ? x y ` ? x y ` ? x y ` ? ? x y ` ? ? xy ` ? xy ` ? y ` ? ? y ` x ` ? x ` x y ` ? x y ` ? x y ` x y ` xy ` ? xy ` y ` ? y ` ? ? x ` ? x ` ? x y ` ? ? x y ` ? ? xy ` ? xy ` ? y ` ? ? y ` x ` ? x ` ? xy ` xy ` ? y ` y ` ? ? x ` ? x ` ? ? y ` ? y ` ` ? ‰ . Appendix C
We present the rotations that map F b ,v to each F bi,v for i belonging to t , u . Usingthem, we can built a mesh of the five adjacent faces F b ,v , ....., F b ,v of F b ,v . • F b ,v is sent to F b ,v and F b ,v by rotations in R with an angle ˘ π and anaxis (cid:126)u : “ } ÝÝÑ OG } ÝÝÑ OG , where G denotes the center of F b ,v . We have: (cid:126)u “ ? ? ` σ ` ´p σ ` q i ` ? k ˘ . It is easier to calculate a rotation with an angle equal to π than with an angle equal to π . Hence we begin to calculate ρ a rotation in R ave Computation on the Poincar´e dodecahedral space π and an axis (cid:126)u : ρ p x i ` y j ` z k q“ “ cos ` π ˘ ` sin ` π ˘ (cid:126)u ‰ r x i ` y j ` z k s “ cos ` π ˘ ´ sin ` π ˘ (cid:126)u ‰ “ ” σ ` ? ` σ ? ? ` σ ` ´p ` σ q i ` ? k ˘ı r x i ` y j ` z k s ” σ ´ ? ` σ ? ? ` σ ` ´p ` σ q i ` ? k ˘ı . So ρ p x, y, z q “ ¨˝ ´ σ ´ σ σ ´ σ ´ σ ´ ´ σ ˛‚¨˝ xyz ˛‚ . Hence ρ : F b ,v Ñ F b ,v , and ρ : F b ,v Ñ F b ,v . We have: ρ p x, y, z q “ ¨˝ σ ´ σ ´ σ ´ σ ´ σ σ ˛‚¨˝ xyz ˛‚ , ρ p x, y, z q “ ¨˝ σ ´ ´ σ σ σ ´ σ ´ σ ˛‚¨˝ xyz ˛‚ . • F b ,v is sent to F b ,v and F b ,v by rotations in R with an angle ˘ π and anaxis (cid:126)u : “ } ÝÝÑ OG } ÝÝÑ OG , where G denotes the center of F b ,v . We have: (cid:126)u “ ? ? ` σ ` ´? j ` p σ ` q k ˘ . Once more we begin to calculate ρ a rotation in R with an angle π and an axis (cid:126)u : ρ p x i ` y j ` z k q “ ” σ ` ? ` σ ? ? ` σ ` ´? j ` p ` σ q k ˘ı r x i ` y j ` z k s ” σ ´ ? ` σ ? ? ` σ ` ´? j ` p ` σ q k ˘ı . So ρ p x, y, z q “ ¨˝ ´ σ ´ ´ σ ´ σ ´ σ σ ´ σ ˛‚¨˝ xyz ˛‚ . Hence ρ : F b ,v Ñ F b ,v , and ρ : F b ,v Ñ F b ,v . We have: ρ p x, y, z q “ ¨˝ σ σ ´ σ ´ σ ´ ´ σ σ ˛‚¨˝ xyz ˛‚ , ρ p x, y, z q “ ¨˝ σ ´ σ ´ σ ´ σ ´ σ σ ˛‚¨˝ xyz ˛‚ . • F b ,v is sent to F b ,v by a rotation in R with an angle ˘ π and an axis (cid:126)u : “ } ÝÝÑ OG } ÝÝÑ OG ,where G denotes the center of F b ,v . We have: (cid:126)u “ ? ? ` σ ` ´p σ ` q i ´ ? k ˘ . Oncemore we begin to calculate ρ a rotation in R with an angle π and an axis (cid:126)u : ρ p x i ` y j ` z k q “ ” σ ` ? ` σ ? ? ` σ ` ´p ` σ q i ´ ? k ˘ı r x i ` y j ` z k s ” σ ´ ? ` σ ? ? ` σ ` ´p ` σ q i ´ ? k ˘ı . So ρ p x, y, z q “ ¨˝ σ σ ´ σ ´ σ σ ´ ´ σ ˛‚¨˝ xyz ˛‚ . ave Computation on the Poincar´e dodecahedral space ρ : F b ,v Ñ F b ,v . We have: ρ p x, y, z q “ ¨˝ σ ´ σ σ ´ σ σ σ ˛‚¨˝ xyz ˛‚ . References [1] Aurich R and Lustig S and Steiner F 2005
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