Moving Mesh Methods for Singular Problems on a Sphere Using Perturbed Harmonic Mappings

Transcription

1 Moving Mes Metods for Singular Problems on a Spere Using Perturbed Harmonic Mappings Yana Di, Ruo Li, Tao Tang and Pingwen Zang Marc 4, 6 Abstract Tis work is concerned wit developing moving mes strategies for solving problems defined on a spere. To construct mappings between te pysical domain and te logical domain, it as been demonstrated tat armonic mapping approaces are useful for a general class of solution domains. However, it is known tat te curvature of te spere is positive, wic makes te armonic mapping on a spere not unique. To fix te uniqueness issue, we follow Sacks and Ulenbeck [Ann. Mat., 3, -4 (98)] to use a perturbed armonic mapping in mes generation. A detailed moving mes strategy including mes redistribution and solution updating on a spere will be presented. Te moving mes sceme based on te perturbed armonic mapping is ten applied to te moving steep front problem and te Fokker-Plank equations wit ig potential intensities on a spere. Te numerical experiments sow tat wit a moderate number of grid points our proposed moving mes algoritm can accurately resolve detailed features of singular problems on a spere. Introduction In tis work, we will consider efficient moving mes metods for solving singular problems on a spere. Partial differential equations (PDEs) defined on a spere are seen in many LMAM & Scool of Matematical Sciences, Peking University, Beijing 87, P. R. Cina, dyn@mat.pku.edu.cn. LMAM & Scool of Matematical Sciences, Peking University, Beijing 87, P. R. Cina, rli@mat.pku.edu.cn. Matematics Department, Hong Kong Baptist University, Kowloon Tong, Hong Kong, ttang@mat.kbu.edu.k. Te researc of tis autor was supported in part by Hong Kong Researc Grants Council and te International Researc Team on Complex System of Cinese Academy of Sciences. LMAM & Scool of Matematical Sciences, Peking University, Beijing 87, P. R. Cina, pzang@mat.pku.edu.cn. Te researc of tis autor was supported in part by te special funds for Major State Researc Projects and National Science Foundation of Cina for Distinguised Young Scolars.

2 practical cases suc as polymer reological analysis, ocean modeling, and global climate simulation. Te moving mes tecnique is one of te natural coices for solving problems wit localized moving singularities. Examples include moving internal boundaries or interfaces separating different regions wic are present in many problems in nature, suc as interfaces separating two different fluids in a multipase flow, and between solid and liquid regions in a solidification process. Suc interfaces are in general dynamic, i.e. tey evolve in time. Moving mes metods ave been proved powerful for solving time-dependent partial differential equations in two-dimensional domains and simple tree-dimensional domains ([, 3, 4, 5, 4, 3, 3, 34]). In general, a moving mes metod always involves a logical domain and a pysical domain, togeter wit a transformation between tem. In [], Dvinsky suggests tat armonic function teory may provide a general framework for developing useful mes generators. His metod can be viewed as a generalization and extension of Winslow s metod [33]. However, unlike most oter generalizations wic add terms or functionals to te basic Winslow grid generator, is approac uses a single functional to accomplis te adaptive mapping. Te critical points of tis functional are armonic maps. Meses obtained by Dvinsky s metod enjoy desirable properties of armonic maps, particularly regularity, or smootness [5, 7]. Motivated by te work of Dvinsky, a moving mes finite element strategy based on armonic mapping was proposed and studied by te autors in []. Te key idea of tis strategy is to construct te armonic map between te pysical space and a parameter space by an iteration procedure. For two-dimensional domains or tree-dimensional cubic domains as considered in [, ], te metric on te logical domain is Euclidean. Consequently, te construction of te armonic mapping only solves some linear equations. However, tis is not te case for sperical problems. In fact, if te pysical domain is a spere ten it is natural to set te logical domain as a spere rater tan a convex polygon as used in []. In tis case, te fact tat te curvature of a spere is positive makes te armonic mapping on spere not unique. Anoter difficulty is tat te metric on te unit spere is not an identity anymore, wic is different from te cases considered before, see, e.g. []. Our goal is to construct a transformation wic preserves te good properties of te armonic mappings suc as smootness and uniqueness. Moreover, a sceme for solving te nonlinear equation induced by te metric on a spere will be designed, Motivated by te work of Sacks and Ulenbeck [6], we will use te so-called perturbed armonic mapping to construct a unique mapping between te pysical and logical space. It will be seen tat te perturbed armonic mappings are useful for andling sperical domains. Te nonlinear Euler-Lagrange equations from minimizing te mes energy functional will be solved by a local Gauss-Seidel-like iterative metod. Te numerical experiments indicate tat a satisfactory approximation to te perturbed armonic mapping can be obtained using only a few iteration steps. Furtermore, a key step for te moving mes metod is to obtain an improved new mes based on te underlying numerical solu-

3 tions and te old meses. Wit te sperical geometry, te mes redistribution as to be designed carefully: efforts ave to be made to ensure tat te mes movement occurs only on te spere, and tat all te grid points stay on te spere during te wole process of mes redistribution. In tis work, appropriate mes redistribution and solution updating strategies will be investigated. Te rest of te paper is arranged as follows. In Section, te teory of te perturbed armonic mapping on a spere will be briefly reviewed. Section 3 describes te general framework of te moving mes metod on a spere, including some details of te mesredistribution and solution-updating strategies. In Section 4, te proposed moving mes sceme will be applied to solve moving socks on a spere and te Fokker-Planck equations derived from liquid crystal simulations. For te latter problem, te solutions of te problem tend to a delta function wen te strengt of te excluded volume potential is large (see []). Wit a uniform mes, an extremely fine mes is required in order to obtain reasonable resolution, but te efficiency can be enanced significantly by using moving mes metods. Some concluding remarks will be given in te final section. Perturbed Harmonic Mapping Let and c be compact Riemannian manifolds of dimension n wit metric tensors d ij and r αβ in some local coordinates x and ξ, respectively. Te energy density of a mapping ξ : x ξ, is defined by e( ξ) = G ij ξ α ξ β g αβ x i x, (.) j were te standard summation convention is assumed. Te corresponding energy functional is ten defined by E( ξ) = e( ξ)dx. (.) Te armonic mapping is te mapping wic minimizes te above energy functional. Existence and uniqueness of te armonic map are guaranteed provided tat te Riemannian curvature of c is non-positive and its boundary is convex (see Hamilton-Scoen- Yau [5, 7]). However, tis result does not apply to te case wen te pysical and logical domains are a spere, i.e., = c = S ; in tis case, te Riemannian curvature of c is positive. In fact, te energy functional (.) as te conformal invariance [6], wic leads to non-uniqueness of te armonic mappings. To recover te uniqueness, we follow te work of Sacks and Ulenbeck [6] to employ a so-called perturbed armonic mapping approac. More precisely, let E α ( ξ) = ( + e( ξ)) α dx, (.3) wit α >. It is noted tat wen α = te functionals (.3) and (.) are equivalent. It 3

4 is known tat te Sobolev space of mappings: L α (M, N) = {ξ Lα (M, Rk ) : ξ(x) N} C (M, N), (.4) is an C separable Banac manifold for α > [5]. In particular, it is sown in [6] tat if α >, ten te critical mappings of E α in L α (M, N) are C and are unique in te sense of isometric transformation. Suc regularity and uniqueness results provide a sound teoretical background for te moving mes metod based on te perturbed armonic mappings. Te Euler-Lagrange equation of te energy functional (.3) is given by L B ξ l (α ) ξ ξ l were L B stands for te Laplace-Beltrami operator: L B f = ( GG ij f G x i x j ξ + [A(ξ)(dξ, dξ)] l =, (.5) and A(ξ) is te second fundamental form of c. For a spere, it is known tat i,j ), A(ξ)(dξ, dξ) = ξ ξ. (.6) Moreover, te derivatives in curvilinear coordinates are given by ( ξ m ) j ij ξm = G x, i ( ) ( ) ξ j = G ij kl ξt ξ t G. x i x k x l t Consequently, te Euler-Lagrange equation (.5) can be written in te following divergence form: (α ξ (α ) ξ l) + α ξ (α ) [A(ξ)(dξ, dξ)] l =. (.7) Using te Stokes law, we can get te weak form of te above equation: { ξ (α ) ξ l λ l } ξ α ξ l λ l dv =, λ L α (M, N). (.8) S l l Te way of obtaining te mapping for te perturbed equation (.7) is similar to tat used for obtaining te conventional armonic mappings described in [, ]. It is sown tat wit te armonic mappings te metric distortion can be minimized [3, 6]. Moreover, it is expected tat te mes associated wit te mapping (.7) as certain smootness and uniqueness. In tis work, te perturbed armonic mapping will be used to generate adaptive meses for unstructured triangles in sperical domains. We point out tat tere are several oter successful ways for adaptive re-mesing also by minimizing mes energy function tat depends on te local pysical scales, see, e.g., Anderson et al. [], Liao et al [3], Sun & Huang [7], and Wang & Wang [3]. 4

5 3 Moving Mes Metods for Sperical Domain Consider te time-dependent PDEs on a spere = S, Ψ t = L(Ψ) in (, T ], (3.) were L is some spatial partial differential operator, wit initial value Ψ t= = Ψ, in. (3.) Let us consider a moving mes metod based on te perturbed armonic mappings to solve tis time-dependent problem. We triangulate te domain into a mes T wit sperical triangular elements. Te equation (3.) is discretized wit finite element approximation on te mes T. Assume a PDE solver is given so tat te numerical solution at time level t n+ can be obtained by using te numerical approximation at t = t n. Since our mes varies wit respect to time, we denote te mes at time step t n as T (n). Te procedure to combine te mes motion and te PDE solutions is illustrated below: Step : Initialization. Set n = and t =. Project te initial function on te initial mes T (), wic gives Ψ() ; Step : Solution evolution. Obtain te numerical solutions at t = t n+ on te mes T (n) by an appropriate PDE solver, to obtain Ψ (n+) ; Step : Mes quality evaluation. Ceck te mes quality by certain criteria; if satisfactory, set T (n+) = T (n), n = n +, and go to Step ; Step 3: Mes quality enancement. Obtain a more appropriate mes by some mes moving tecniques, as described later in tis section; Step 4: Solution updating. Update te numerical solution Ψ (n+) obtained in Step 3 and go to Step. on te new mes We point out tat te mes quality (mentioned in Step above) takes into account te sape, alignment and adaptation features of te elements. For example, te minimum angle, te maximum angle, and te aspect ratio ave been used widely to caracterize te mes quality. More details can be found in Huang [8]. 3. Finite element space on spere Te spere is triangulated into a mes wit sperical triangular elements. Te Voronoi sperical geodesic grid (see, e.g. []) is used in our work. Here we give a finite element space for te discretization of te problem (3.)-(3.). We will project te linear basis functions of te triangles on polyedron onto te sperical geodesic triangles. Te vertices 5

6 C P W A B O Figure : Te sape function defined on a sperical triangle. (anti-clockwise) of a sperical geodesic triangle K are denoted by A, B and C. Te plane triangle K p as te same vertices. For a point P K, te intersection point of line OP and K p is denoted by W (see Figure (3.) ). Te tree basis functions on sperical geodesic triangle K are defined by λ ( x) = D ( w a a 3 ), λ ( x) = D ( w a 3 a ), (3.3) λ 3 ( x) = D ( w a a ), were OA = a, OB = a, OC = a 3, OW = w, D = ( a a a 3 ), d = a a + a a 3 + a 3 a. OP = x, w( x) = D x x d, Wit tese notations, a finite element space on spere is presented: H { Ψ C () : Ψ K span {λ, λ, λ 3 }, K }. (3.4) 6

7 3. Mes redistribution on spere A key step for te moving mes metod is to obtain an improved new mes based on te underlying numerical approximations. In our computations, we simply coose te given pysical domain, i.e., te spere, as te logical domain, and we use te Voronoi sperical geodesic grid (see, e.g., []) to generate a quasi-uniform mes wit triangular elements on te logical domain. Obviously, tis also gives te initial partition for te pysical domain. Te nodes in te pysical mes are denoted by X i, i N, and te nodes in te logical mes by Ξ i, i N. Initially, since we coose te logical mes te same as te pysical mes, we ave Ξ i = X i. Te pysical mes will be redistributed at eac time step, wile te logical mes will be kept uncanged during te entire process of te numerical computations. For eac time step, by numerically solving te Euler-Lagrange equation (.7), we can obtain a discrete transformation between te pysical domain x and te logical domain ξ. Te logical mes is ten given suc tat Ξ i = ξ (X i ). Te metric on te pysical domain is given according to te given PDEs and teir numerical solution, wic is called monitor function. On te spere, tere are no local coordinates to cover te wole domain S. Terefore, we ave to coose suitable local coordinates. Te metric for te logical domain is te natural metric of spere as a sub-manifold embedded in R 3. Since te Euler-Lagrange equation (.7) is nonlinear, it will be solved numerically by a point by point iterative approac. Assume tat we are considering a node in te pysical mes, X k, wose corresponding node in te logical mes is denoted by Ξ k. Denote te set of triangles in te pysical domain wit X k as a vertex by S k and te corresponding set in te pysical domain by S c,k. We will set te local coordinate around X k in te pysical domain and te local coordinate around Ξ k in te logical domain. It is natural to eliminate one component wose absolute value is te largest among te tree components. Tis will give two-dimensional local coordinates. For example, for X k = (x k, x k, x3 k ) set, if x k x k and x k x3 k ten we will let (x k, x k ) (x k, x3 k ) to be te local coordinate around X k. Te same strategy is employed to coose te local coordinate around Ξ k. We ten ave te Euler-Lagrange equation (.7) around te node Ξ k : ( ξ (α ) ξ l) + ξ α ξ l = for l =,, 3, ξ Sk = ξ b. (3.5) In te pysical domain, around X k te patc of surface on te unit spere can be expressed as ( x 3 ) = ( x ) ( x ), (3.6) and te metric of te pysical domain is taken as a sub-manifold embedded in R 3 : G ij i,j=, = ( (x ) x x x x (x ) 7 ). (3.7)

8 Te metric G ij on te pysical domain around X k is ten set as G ij = m(u ) G ij, (3.8) were m(u ) is te monitor function given for te problem under consideration. Te coice of te monitor function m is very subtle, wic is an indicator of te solution singularity and is problem dependent. A general discussion of te monitor function can be found in [6, 9]. Now on te patc of triangles S k, we will solve a Diriclet boundary-value problem for (3.5), wit linear boundary mapping segment-wise on S k determined by te vertex mapping Ξ n = ξ(x n ), X n S k, n k. Te mapping in S k is approximated by a piecewise linear mapping on every triangle in S k. Tus te mapping can be given only if ξ(x k ) is given. Using te standard Galerkin approximation for (3.5) in S k, we ave only one test function wic is te pyramid-like basis function located at X k. Tis gives S k { ξ (α ) ξ l λ k ξ α ξ l λ k} dv = for l =,, 3, (3.9) were λ k is te test function for X k, wic is an algebraic system wit order α + for tree unknowns. We will solve (3.9) approximately. On S k, we denote ξ by ξ = n Ξ n λ n + Ξ kλ k, (3.) were te summation for n is taken over all vertices in S k except X k. By taking ξ in (3.9) explicitly, i.e., ξ in ξ takes te currently available Ξ k as its value at X k, we ten get an explicit approximation for (3.9): Ξ l k S k n ξ α Ξ l n λl λ k dv ξ (α ) G ij Ξ l λ n λ k n n S k x i x dv j ). (3.) ( ξ α λ k λ k + ξ (α ) ij λk λ k G dv x i x j S k Below we will describe te algoritm for mes redistribution, wic is similar to te one proposed in []. In te first step, we iterate for k from to N, using (3.) to update all logical nodes Ξ k until convergence (i.e., te difference between te obtained Ξ k and initial (fixed) Ξ () k is smaller tan a given tolerance). Te tolerance for te convergence criteria is of te order O( l ) were l is te typical element size in te logical mes. Te converged result is denoted by Ξ k, and we ten get te difference between Ξ k and te initial logical mes as δξ k = Ξ () k Ξ k. (3.) In te second step, we use te above difference to obtain te mes redistribution information in te pysical mes difference. Again we will coose te same local coordinate as 8

9 described above. For a node X k in te pysical mes, we first compute te mes motion vector: E xl δxk l = K S k ξ δξ k K, for l =,, (3.3) K S k E were te Jacobian matrix x/ ξ K is computed by using te fact tat te linear element is used (see also []): ( Ξ E Ξ E Ξ E Ξ E Ξ E Ξ E Ξ E Ξ E ) x x ξ ξ x x ξ ξ = ( X E X E X E X E X E X E X E X E ) (3.4) and E, E and E are te subscripts of te tree vertices of te element K in te pysical mes. Te coordinate in te tird dimension is excluded, as explained earlier (i.e., wit te way of coosing te local coordinate). On te oter and, to ensure tat te mes movement occurs only on te spere, te tird component can be recovered by using δx 3 k sgn(x3 k ) (X k + δx k ) (X k + δx k ) X 3 k. (3.5) In te tird step, we will control te magnitude of eac redistribution to avoid mes tangling. For te element K T wit vertices E, E and E, we solve te following cubic equation (in terms of µ), det ( X E + µ δx E X E + µ δx E X E + µ δx E ) =, (3.6) were δx Ek is determined by (3.3). Denote te smallest positive root of (3.6) by µ K. Te mes motion parameter is ten set as µ = min K T {; µ K /}. (3.7) It is obvious tat te mes tangling can be avoided if we move te mes using te formula X k X k + µ δx k. In te last step, we update te mes moving direction by δx k X k + µ δx k X k + µ δx k X k. (3.8) Tis is to make sure tat all te grid points are moving along te spere. Te new mes nodes are ten given by X k + δx k. It can be verified tat te new nodes are all on te spere. 9

10 3.3 Solution interpolation on te new mes After redistributing te mes in te pysical domain, we need to transfer te solution information on te old mes to te new mes. Let δ x := N i= δx iλ i, were λ i is te basis function of H. Assume tat a finite element solution Ψ (n) = N i= Ψ(n) i λ i is given at t = t n. Let τ be a virtual time variable. We will regard te mes motion from te old mes to te new mes as a linear omotopy for τ goes from to. Consequently, te mes flow wit te virtual time τ is x(τ) x old + τδ x. For an arbitrary point x T, te basis function λ i ( x) associated wit x will be varying linearly wit τ. Since λ i ( x) is defined on te pysical mes, it is denoted by λ i ( x; τ). Direct calculation sows tat dλ i ( x; τ) dτ Assume tat te numerical solution on mes x(τ) is of te form = δ x x λ i ( x; τ). (3.9) Ψ ( x; τ) = N Ψ i (τ)λ i ( x; τ), (3.) i= wic takes te starting value Ψ ( ; ) = Ψ (n). Te main idea of te solution updating is to let te variation of Ψ ( ; τ) wit respect to τ vanis in te test function space: dψ ( ; τ), Φ =, Φ H. (3.) dτ Using (3.9) and (3.) gives N i= { } dψi dτ λi ( ; τ)λ j ( ; τ) + Ψ i δ x x λ i ( ; τ)λ j ( ; τ) dx =, j N. (3.) Te above ODE system can be solved by some appropriate ODE solver, e.g., a tree-step Runge-Kutta metod ([, ]). 4 Numerical experiments We first discuss te perturbation parameter α in (3.5): in te numerical computations reported in tis section it is cosen as.5. It is known [5] tat α must be larger tan in order to ensure te uniqueness of te mapping. If α is close to, te critical mapping of E α is close to a armonic mapping. Tis implies tat α sould be cosen approximately. On te oter and, it is found tat larger values of α increase te solution efficiency for (.8). In fact, larger values of α can speed up te convergence of te mes movement: note tat te nonlinear Euler-Lagrange equations are solved by a local Gauss-Seidel-like iterative metod. Wit a few numerical tests, it is found tat one of te good coices for te parameter α sould be approximately.5.

11 Example 4. (Moving steep front on a spere) Our first example is to compute a moving steep front along te direction b := (,, ) T governed by te following equation: Ψ t + Ψ b Ψ = ɛ L B Ψ, (4.) wic is defined on te unit spere, were is te surface gradient. Te initial condition is given by ( Ψ(x, y, z; ) = + exp( x + y + z + )). (4.) ɛ We first briefly discuss te PDE algoritm to be used. For tis example, a simple finite element discretization as described in Section 3. is employed. Te weak formulation of te equation (4.) reads: Find Ψ H () suc tat Ψ t vdx + Ψ b Ψvdx = ɛ Ψ vdx, v H (). (4.3) In te finite element space H, te above weak form can be approximated as: Find Ψ H, suc tat Ψ t v dx + Ψ b Ψ v dx = ɛ Ψ v dx, v H (). (4.4) For te temporal approximation, an explicit tree-step Runge-Kutta sceme is applied, see, e.g., [7]. In our computations, te viscosity coefficient ɛ is cosen as.. For te convection dominant problem, more mes points are required around te wave front, wic makes te use of te moving mes metods meaningful. Te monitor function m defined in (3.8) is taken as te standard gradient-based one, i.e., m(ψ) = + β Ψ. (4.5) For problems wit moving steep front, te parameter β as to be set as a large number, say between to 3 ; see, e.g., [6]. A large value of β can usually cluster more grid points around te steep front. On te oter and, if Ψ in (4.5) is replaced by Ψ / max Ψ ten te parameter β can be cosen as an O() number, see, e.g., [7]. In our computations, te parameter β in (4.5) is cosen as 3. In Fig., te moving meses obtained by using our moving mes sceme wit 8 nodes (about 6 triangular elements) are plotted. Togeter in tis figure are te numerical solutions for Ψ at various time levels. Fig. (a) sows te initial state: te initial front is at x + y + z + =, were a stage can be seen. Te solution is on te left-top side of te front and on te rigt-bottom side. It can be seen tat te front moves along te direction (,, ) T on te spere. In Figs. (b)-(f), te area wit solution value becomes bigger, and te area of value becomes smaller as time increases. At t =.5, te solution is mostly on te spere and is only on a small concave area. In order to display

12 te numerical solution more clearly, te figures are plotted from various visual angles at different time levels. Moreover, in order to ave a better idea of te mes distribution te meses are projected to a planispere and are plotted in Fig. 3. It is seen tat our moving mes sceme adapts te mes extremely well in te regions wit large solution gradients. Fig. 4 sows te meses and solutions obtained by te proposed moving mes algoritm wit 8 nodes (left) and by using a uniform mes approac wit 3 nodes (rigt), at t =.6. It is observed from te figure tat te moving mes metod wit fewer nodes resolves te sarp front muc more accurately tan te corresponding uniform mes approac wit more nodes. Furtermore, te numerical oscillation in te uniform mes solution is not seen in te moving mes solution. In fact, it is found in our numerical computations tat it requires more tan 7 nodes in a uniform mes in order to obtain te same resolution of te moving mes solution wit 8 nodes. Example 4. (Solution of te Fokker-Planck equation) Te second example is concerned wit te Doi model [9, 6] wic is commonly used in studying rod-like polymer fluids. In tis model, a omogeneous population of equal rods is described by an orientational distribution function Ψ( x, t). Te evolution of Ψ( x, t) is modeled wit te Smolucowski equation or te Fokker-Planck equation []: Ψ t = De R (RΨ + ΨRV ev), on = S, (4.6) were De is te Debora number, R is te rotational operator R x / x, and V ev is an effective excluded-volume potential wic takes te Maier-Saupe form: V ev ( x) = 3 U x x Ψ( x )dx. (4.7) In (4.7), U is te non-dimensional potential intensity wic is proportional to te concentration of te polymers. Te initial distribution function is taken as Ψ( x, ) = 4π + 5 ( ) 3( x x), 6 π were x is a constant unit vector. As te potential intensity increases, te orientational distribution function becomes sarply peaked and beaves like a δ-function. To resolve te solutions in tis case, it is necessary to use some adaptive grid metods. Here we will use te framework introduced in te last section, wit a few additional details described below. Weak formulation. Te weak formulation of te equation (4.6) reads: Find Ψ H (), suc tat Ψ t vdx + RΨ Rvdx = ΨRV (Ψ) Rvdx, v H (). (4.8) De De

13 Z Z Y X XY Z Z X Y X Y Z Z X Y X Y Figure : Example 4.. Te adaptive meses on a spere wit 8 nodes from t = to t =.5. 3

14 Figure 3: Example 4.. t =.5. Te adaptive meses on planispere using 8 nodes from t = to In te finite element space H, te above weak form can be approximated as: Find Ψ H, suc tat Ψ t v dx + RΨ Rv dx = Ψ RV (Ψ ) Rv dx, v H (). (4.9) De De For te temporal approximation, similar to [7] we use a semi-implicit tree-step Runge- Kutta sceme to solve (4.9): Find Ψ n+ H, suc tat Step : Ψ () Ψn v dx + RΨ () t/3 De Rv dx = Ψ n De RV (Ψ n ) Rv dx, v H. (4.) 4

15 Z Z X Y X Y Figure 4: Example 4.. Te adaptive mes on spere wit 8 nodes (left) and uniform mes wit 3 nodes (rigt) at t =.6. Step : Step 3: Ψ () = De Ψn t/ Ψ n+ = De v dx + De Ψ () RV (Ψ() Ψ n v dx + t De Ψ () RV (Ψ() RΨ () Rv dx ) Rv dx, v H, RΨ n+ Rv dx ) Rv dx, v H, (4.) (4.) Note tat in eac step above te diffusion term is treated implicitly wile te nonlinear term is treated explicitly. Wit tis treatment, a standard stability criterion of CFL-type applies; and in our computations a uniform time step is used. By doing tis, it is well known tat te moving mes metod does not gain muc in time-stepping, since te time steps are controlled by te smallest size of te finite elements. One possible way of furter speed-up in temporal discretization is to use te local time stepping tecnique, see, e.g., [8]. Monitor functions. For a piecewise linear approximation φ to a function φ, te following formula is adopted to approximate te computational error: [ ] φ φ, Rφ t l dl, (4.3) l : interior edge were t l is te unit tangential vector of te edge l and [ ] is te jump along te edge l; see, e.g., [7, 9]. Wit tis a posteriori interpolant error estimator, a monitor function is 5 l

16 ..9.8 S Time Figure 5: Example 4.. Evolution of te scalar order parameter S wit U =. proposed as m(ψ ) K = K l K [ ] l RΨ t l dl, K T. (4.4) l Te instantaneous average molecular orientation is quantified by te order parameter tensor S = xx I. (4.5) 3 Te scalar order parameter S = 3 λ represents te degree of molecular alignment, were λ is te eigenvalue wit te largest absolute value for S. It lies in te range wit S = for an isotropic pase, and S = for rods tat are all aligned in te same direction. Fig. 5 sows te evolution of te scalar order parameter S wit U =. In tis case, te stationary scalar order parameter S is very close to, indicating tat all te molecular directions converge to one direction and te distribution function is very singular. Fig. 6 sows te meses and te corresponding orientational distribution functions wit U = and. To better demonstrate te singular beaviors of te solutions, te spere is projected into a planispere. Te peak value of te distribution function wit U = reaces Ψ. Since te integral of te distribution function equals to, te area of nonzero Ψ is about, were te solution as large gradients. Moreover, since te area of te unit spere is equal to 4π, te compact support occupies only /(4π) portion of te spere. In order to put 5 (= 5 5) mes points in suc a small area, a uniform mes requires a total of 5 mes points wile our moving mes approac needs 6

17 4 3 Ψ Ψ Figure 6: Example 4.. Te orientational distribution function obtained by using 6 nodes and 396 elements, for U = (top) and U = (bottom). Te mes on te spere is projected to a disk (left) and te corresponding solution is sown on te rigt. 7

18 Figure 7: Example 4.. Te close-up of te adaptive mes for U = (bottom figure of Fig. 6). 6 points only. By a close-up look of te mes structure in Fig. 7, it is seen tat wit te moving mes approac more grid points are pused into te (small) blow-up region. More comparisons are made for Example 4. wit U =. In Fig. 8, we compare te numerical solutions obtained by using te uniform mes (nodes 568, 554 and 566, respectively) and by using te adaptive moving mes (6 nodes). It is seen tat te moving mes solution wit 6 nodes is more accurate tan te uniform mes solution wit 554 nodes, and is comparable to te uniform mes solution wit 566 nodes. Tis is consistent wit our estimate above quite well. Finally, it is necessary to compare te total CPU time usage for tese computations. Table gives te relevant CPU time usages wit various numbers of nodes. Te time step used is t = 6 and te total time is T = 3. For tis particular computation, te memory savings wit te moving mes 8

19 a. 3 b c d. Figure 8: Example 4.. Te orientational distribution function obtained by using a uniform mes wit (a) 568 nodes, (b) 554 nodes, (c) 566 nodes; and (d) by using a moving mes 6 nodes. U =. 9

20 metod is about :3 and te speed-up is over 5. Table : Example 4.: CPU time(seconds) used for uniform and moving mes computations. # of Nodes CPU time used uniform mes s uniform mes s uniform mes s moving mes 6 344s moving mes 3 598s 5 Conclusions In tis paper, an effective moving mes metod is proposed to resolve steep solutions generated by PDEs on a spere. Te metod exploits armonic maps and invokes a regularization procedure to overcome te lack of convexity in te set-up. Some special tecniques for andling local coordinates on a spere are also designed. Efforts ave been made to ensure tat te mes movement occurs only on te spere and all te grid points remain to stay on te spere in te entire process of mes redistribution. Two examples are given, one for a convection-dominated problem and te oter for te Fokker-Planck equation on te spere. It is demonstrated tat te moving mes metods cannot only enance te solution speed but also reduce substantially te computational storage. Te importance of increasing efficiency is obvious, but te latter part (of reducing storage) is also important for many practical problems, in particular iger-dimensional problems. For example, a large class of problems in complex fluid simulations are posed on speres, for wic Example 4. is a simple case. For te microscopic model of te polymer fluid, a well-known model is te rigid dumbbell model of Doi [9, 6]. In te Doi model, te orientation of a rod is determined by a pseudo-vector u on te unit spere. Te existing numerical scemes include a spectral type metod wit sperical armonic functions as basis functions (sperical armonic expansion metod), see, e.g., [4]; and standard finite element approac, see, e.g., []. Since te underlying equations for te complex fluid models (suc as te Fokker-Planck type) involve several dimensions in bot pysical and pase spaces, te required storage for te classical metods may be unrealistically large. It is seen tat te moving mes approac can use muc less storage to obtain te required resolution. Te numerical tecnique developed in tis work is currently being used to study te reological beavior of rod-like polymers governed by te Doi model; see, e.g., [8]. Finally, we point out tat many important practical problems ave been posed in sperical geometries, suc as climate modeling and ocean modeling. One of our future

21 projects is to extend te numerical tecniques developed in tis work to solve some practical problems defined on speres wit solution singularity. Acknowledgments Te researc of Di and Li was supported in part by te Joint Applied Matematics Researc Institute between Peking University and Hong Kong Baptist University. Te researc of Tang was supported by CERG Grants of Hong Kong Researc Grant Council, FRG grants of Hong Kong Baptist University, and NSAF Grant #4763 of National Science Foundation of Cina. Te researc of Zang is partially supported by te special funds for Major State Researc Projects 5CB74 and National Science Foundation of Cina for Distinguised Young Scolars 53. References [] A. Anderson, X. Zeng, and V. Cristini, Adaptive unstructured volume remesing - i: Te metod, J. Comput. Pys. 8 (5), [] B.N. Azarenok and T. Tang, Second-order Godunov-type sceme for reactive flow calculations on moving meses, J. Comput. Pys. 6 (5), [3] M.J. Baines, M.E. Hubbard, and P.K. Jimack, A moving mes finite element algoritm for te adaptive solution of time-dependent partial differential equations wit moving boundaries, Appl. Numer. Mat. 54 (5), [4] G. Beckett, J. A. Mackenzie, and M. L. Robertson, A moving mes finite element metod for te solution of two-dimensional Stepan problems, J. Comput. Pys. 68 (), [5] W.M. Cao, W.Z. Huang, and R.D. Russell, An r-adaptive finite element metod based upon moving mes PDEs, J. Comput. Pys. 49 (999), 44. [6], A study of monitor functions for two-dimensional adaptive mes generation, SIAM J. Sci. Comput. (999), [7] Y. Di, R. Li, T. Tang, and P.W. Zang, Moving mes finite element metods for te incompressible Navier-Stokes equations, SIAM J. Sci. Comput. 6 (5), [8] Y. Di and P.W. Zang, Te sear kinetic analysis for nematic polymers wit ig potential intensities, Commun. Comput. Pys. (6).

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