FINITE VOLUME METHODS ON SPHERES AND SPHERICAL...

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FINITE VOLUME METHODS ON SPHERES AND SPHERICAL CENTROIDAL VORONOI MESHES QIANG DU AND LILI JU Abstract. We study in this paper a finite volume approximation of linear convection diffusion equations defined on a sphere using the spherical Voronoi meshes, in particular, the spherical centroidal Voronoi meshes. The high quality of spherical centroidal Voronoi meshes is illustrated through both theoretical analysis and computational experiments. In particular, we show that the error of the approximate solution is of quadratic order when the underlying Voronoi mesh is given by a spherical centroidal Voronoi mesh. We also demonstrates numerically the high accuracy and the superconvergence of the approximate solutions. Key words. Finite volume method, spherical Voronoi tessellations, spherical centroidal Voronoi tessellations, error estimates, convection-diffusion equations. AMS subject classifications. 65N15, 65N50, 65D17 1. Introduction. The numerical solution of partial differential equations defined on spheres is a very active research subject in the scientific community. The subject is related to a number of important applications such as weather forecast and climate modeling. For ex- ample, the numerical solution of linear convection-diffusion equations and nonlinear shallow water equations in the spherical geometry can be used to test numerical algorithms for more complex atmospheric circulation models. Though these models were often solved with spec- tral methods or traditional finite difference methods in spherical coordinates, methods that use quasi-uniform tessellations of the sphere are gradually gaining popularity as the grid-based methods offer great potential when combined with massive parallelism and local adaptivity. To get efficient and accurate numerical solutions of PDEs, it is well known that grid quality plays an important role and high quality grid generation is often a significant part of the overall solution process. Recently, many studies have been made on the development of finite element and finite volume approximations to PDEs defined on spheres using various spherical grids, such as grids based on Bucky balls [15], icosahedral grids [3, 30], skipped grid [21], grids obtained from a gnomonic (cubed sphere) mapping [20], etc. In standard Eu- clidean geometry, the so-called Voronoi-Delaunay grids have always been very popular grids used in both finite element and finite volume methods. Grids generated from the spherical Voronoi tessellations and their variations have also been studied, see for example [14, 31]. In [7] and [8], we proposed a high quality spherical grid based on the Spherical Cen- troidal Voronoi Tessellation (SCVT) which can be used for both data assimilation purposes and for the numerical solution of PDEs on spheres. A finite volume approximation to a sec- ond order linear elliptic equation using the spherical Voronoi meshes was studied in [8], and a first order error estimate for the discrete norm was obtained under some grid regular- ity assumptions. Preliminary numerical experiments was also provided there to demonstrate the good performance when the finite volume scheme was implemented with the Spherical Centroidal Voronoi Meshes (SCVM) that include both the SCVT and its dual (Delaunay) tri- angular grid. A very recent study made in [29] on both the global and the local uniformity of spherical grids indicated that the SCVT grid with a uniform density measures better than many other variations. Moreover, after examining the local truncation errors as well as the This work is supported in part by NSF-DMS 0196522 and NSF-ITR 0205232. Department of Mathematics, Pennsylvania State University, University Park, PA 16802. ([email protected]). Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455. ([email protected]). 1

Transcript of FINITE VOLUME METHODS ON SPHERES AND SPHERICAL...

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FINITE VOLUME METHODS ON SPHERES ANDSPHERICAL CENTROIDAL VORONOI MESHES

QIANG DU�

AND LILI JU �

Abstract. We study in this paper a finite volume approximation of linear convection diffusion equations definedon a sphere using the spherical Voronoi meshes, in particular, the spherical centroidal Voronoi meshes. The highquality of spherical centroidal Voronoi meshes is illustrated through both theoretical analysis and computationalexperiments. In particular, we show that the ��� error of the approximate solution is of quadratic order when theunderlying Voronoi mesh is given by a spherical centroidal Voronoi mesh. We also demonstrates numerically thehigh accuracy and the superconvergence of the approximate solutions.

Key words. Finite volume method, spherical Voronoi tessellations, spherical centroidal Voronoi tessellations,error estimates, convection-diffusion equations.

AMS subject classifications. 65N15, 65N50, 65D17

1. Introduction. The numerical solution of partial differential equations defined onspheres is a very active research subject in the scientific community. The subject is related toa number of important applications such as weather forecast and climate modeling. For ex-ample, the numerical solution of linear convection-diffusion equations and nonlinear shallowwater equations in the spherical geometry can be used to test numerical algorithms for morecomplex atmospheric circulation models. Though these models were often solved with spec-tral methods or traditional finite difference methods in spherical coordinates, methods that usequasi-uniform tessellations of the sphere are gradually gaining popularity as the grid-basedmethods offer great potential when combined with massive parallelism and local adaptivity.

To get efficient and accurate numerical solutions of PDEs, it is well known that gridquality plays an important role and high quality grid generation is often a significant part ofthe overall solution process. Recently, many studies have been made on the development offinite element and finite volume approximations to PDEs defined on spheres using variousspherical grids, such as grids based on Bucky balls [15], icosahedral grids [3, 30], skippedgrid [21], grids obtained from a gnomonic (cubed sphere) mapping [20], etc. In standard Eu-clidean geometry, the so-called Voronoi-Delaunay grids have always been very popular gridsused in both finite element and finite volume methods. Grids generated from the sphericalVoronoi tessellations and their variations have also been studied, see for example [14, 31].

In [7] and [8], we proposed a high quality spherical grid based on the Spherical Cen-troidal Voronoi Tessellation (SCVT) which can be used for both data assimilation purposesand for the numerical solution of PDEs on spheres. A finite volume approximation to a sec-ond order linear elliptic equation using the spherical Voronoi meshes was studied in [8], anda first order error estimate for the discrete ��� norm was obtained under some grid regular-ity assumptions. Preliminary numerical experiments was also provided there to demonstratethe good performance when the finite volume scheme was implemented with the SphericalCentroidal Voronoi Meshes (SCVM) that include both the SCVT and its dual (Delaunay) tri-angular grid. A very recent study made in [29] on both the global and the local uniformityof spherical grids indicated that the SCVT grid with a uniform density measures better thanmany other variations. Moreover, after examining the local truncation errors as well as the

This work is supported in part by NSF-DMS 0196522 and NSF-ITR 0205232.�Department of Mathematics, Pennsylvania State University, University Park, PA 16802.

([email protected]).� Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455.

([email protected]).

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solutions errors for a model Poisson equation on the sphere, it was concluded that the SCVTbased grid tends to produces the smallest errors among all the grids under consideration andsuch a grid would naturally remains as a safe choice in practice.

The SCVT can also be defined with a nonuniform density function and thus makes itsuitable for adaptive computation. In light of the optimization properties they enjoy [7], gridsgenerated by the SCVM, in some sense, may be viewed as optimal grids: they offer bothexcellent local grid regularity and global mesh conformity as well as flexible mesh adaptivity.In this paper, we make further attempts to substantiate the optimality of SCVM both theo-retically and computationally. Our main results include a carefully designed finite volumescheme for a general second order convection-diffusion equation defined on a sphere. Whenimplemented with the SCVM, we present for such a discrete scheme a rigorous quadratic or-der

���error estimate whose proof relies critically on the geometric properties of the SCVT.

We further demonstrate through experiments the superconvergent properties of the numericalsolutions and their gradients solved using our modified finite volume scheme and the SCVTbased grid. All these findings provide compelling reasons for regarding the SCVT’s with theuniform density as arguably the best alternative for near uniform partitions of the sphere andthe SCVT based grids the optimal triangular grids to use for the numerical solution of manypartial differential equations defined on spheres.

We point out that the conclusions given in this paper can be readily adapted to problemsdefined on the two dimensional Euclidean plane. The analysis for the spherical case is some-what more involved than the planar case since we must deal with the differences betweenspherical triangles and planar triangles.

The paper is organized as follows: we first introduce some notation and assumptions usedin the paper, then in section 2, we briefly recall the basic theory of the spherical centroidalVoronoi meshes. The finite volume scheme for linear convection-diffusion equations on thesphere given in [8] is discussed in section 3. With a suitable modification to the finite volumescheme, a rigorous

���error estimate is given in section 4 for spherical centroidal Voronoi

meshes. In section 5, a superconvergent gradient recovery scheme is provided and in section6, we present some numerical experiments. Some concluding remarks are given in section 7.

We now introduce some basic notation used in the paper. Let � � denote the (surface ofthe) sphere having radius ���� , i.e.,� �� ��������

��� � � � ��������� � �! � �#" �where

%$& denotes the Euclidean norm. Throughout the paper, the term sphere denotes the

surface of a ball in� �

. Let ')( denote the tangential gradient operator [10, 13] on � � definedby ')(+* ���,�-�� ')(/. � � ')(/. � � ')(/. � � * ���,�- '�* �0�,�21� '�* �0�,� $�346587 . 9 � 34,5�7 . 9 �where ' :�<;

��� ; � � ;)�=� denotes the general gradient operator in� �

and34,5 7 . 9 is the unit

outer normal vector to � � at�

. We consider the second order elliptic equation on the spheregiven by

' ( $?> 1A@B���,� ' ( * �0�,�,C 3D ���,� * �0�,�FEGCIH����,� * �0�,��KJ2�0�,� for�L� � �BM (1.1)

Note that, since � � has no boundary, there is no boundary condition imposed.We use the standard notation for Sobolev spaces on � � (viewed as a compact, two-

dimensional Riemannian manifold) [13]:��N � � �O��QP * ���,� �SR 5 7 � * �0�,� � NUTWV ���,�YXKZ\[ �2

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��� . N � � � �- � * � � � � � � � � '��( * � � N � � � � for ��� � �Y� ��� " �where

� � �� � � � � � � � , ' �( ' �(/. � ' � 7(+. � ' ���(/. � , and

� �Y� ��C � � C � � . Also,

* � ��� ��� 5 7�� �� � >������! � � � ' �( * N " � � 5�7 � E �$# N for %&�(' XKZ)+*�, �-�� � � � ' �( * "/. � 5�7 � for ' Z MFor the case ' 10

, we let � � � � � �- � � . � � � � � .Let the data in (1.1) satisfy the following set of assumptions.ASSUMPTION 1.

J�� ��� � � � � , @ �32 � � � � � , H � �54 � � � � , and3D �62 � � � � � � � � such that@B���,�87 �

� �� , H����,�97 � , and ')( $ 3D ���,�6C H��0�,�87 � � �� , a.e..For any * �;: � � � � � � � , define the bilinear functional < such that

< � * �;: �- R 5 7 @B���,�>= ' ( * ���,� $ ' ( : �0�,�;?YC * ���,�>= 3D ���,� $ ' ( : �0�,�;? TWV ���,�C R 5 7 H����,� * �0�,� : ���,� TWV ���,� (1.2)

then we have (for some constant2 � � )< � * �;: � � 2 * -@ �BA 7;� -@ �BA 7;� M (1.3)

Since � � is compact, if Assumption 1 holds the problem (1.1) has a unique weak solution* � � � � � � � such that < � * �$: �-��<J �$: � �DCE: � � � � � � � (1.4)

where � J �;: �� R 5 7 J2�0�,� : ���,� TWV ���,�is the standard

� �inner product, and consequently * satisfies (for some constant

2 �K� ) theestimate * F@ 7 � 5�7 � � 2 J " 7 � 5�7 � M (1.5)

2. Spherical centroidal Voronoi meshes. LetT ��� �$G � denote the geodesic distance be-

tween�

and G on � � , i.e., for� �;G � � � ,T �0� �;G �� �+H *JILK-K-M�N > �

$ G� � E/O MGiven a set of distinct points P �RQTS�UQWV �YX � � , the corresponding spherical Voronoi regionsP�Z Q;S�UQBV � are defined by

Z Q6\[ G � � � � T ���]Q �;G �YX T ���_^ �;G � for ` % � $=$ $ �$a and `cb1d-e � %&� d � a MP�Z Q;S UQBV � forms a Voronoi tessellation or Voronoi diagrams of � � associated with the set ofgenerators P �fQgS�UQWV � . Each Voronoi cell Z Q is an open convex spherical polygon on � � withgeodesic arcs making up its boundary. It is also well-known that the dual tessellation (in agraph-theoretical sense) to a Voronoi tessellation of � � consists of spherical triangles whichform the Delaunay triangulation.

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Given a density function � defined on � � , for any spherical region Z X � � , the con-strained mass centroid

���of Z on � � is given by the solution of the following problem:)����9�� � �0�,� � where

� ���,�� R � � G � G 1 � � TWV � G � M (2.1)

As in [7, 8], a Voronoi tessellation of � � is called a constrained centroidal Voronoi tessella-tion (CCVT) of � � or specifically Spherical Centroidal Voronoi Tessellation (SCVT) if andonly if the points P �fQgS �QWV � which serve as the generators of the associated spherical Voronoitessellation P�Z QTS �QWV � are also the constrained mass centroids of those Voronoi regions.

Given any set of points P���]Q;S�UQBV � on � � and any spherical tessellation P �Z QTS�UQWV � of � � , wedefine the corresponding energy by

� = P���]Q � �Z Q;S UQBV � ? U� QWV�

R���� �

���,� ��1 ��]Q � TWV ���,� MIt can be shown [7] that

� � $ �is minimized only if P ��]Q � �Z Q;S UQBV � are a spherical centroidal

Voronoi tessellation. Consequently, spherical centroidal Voronoi meshes have many goodgeometric properties, see [7, 8]. If the density function � �0�,� is a constant on � � , then theSCVT generators will be uniformly distributed. Alternatively, a non-constant density functionprovides systematically a non-uniform distribution of points while the accumulation of SCVTgenerators still remains locally regular.

Constructing a constrained mass centroid from (2.1) may be cumbersome. In [7], it hasbeen shown that one can compute first the centroid in

� �by

� �Q R �� � ���,� TWV ���,�R

�� ����,� TWV ���,� � for

d� % � M M=M �;a �then compute

���Qusing the fact that it is the projection of

� �Qonto � � along the normal di-

rection at���Q

. Both deterministic and probabilistic algorithms have been provided in [7, 8] toconstruct a SCVT. The deterministic version requires an explicit geometric construction ofgeneral spherical Voronoi tessellations which was done by the algorithms given in [28]. Onemay also extend the work in [18] to parallelize some versions of the probabilistic algorithms.

Fig.2.1 shows some examples of SCVT’s associated with a constant density. More ex-amples, including SCVT’s with nonuniform densities, can be found in [8].

3. A finite volume method on spherical Voronoi tessellations. Given a spherical Voronoimesh � P � Q � Z Q S UQBV � , following the discussion in [8], we refer to a pair of generators

� Qand

� ^as neighbors if and only if � Q . ^ Z Q�� Z ^ b��

. Note that, for Voronoi meshes, � Q . ^can only be a point or a geodesic arc on the sphere. Let

� ��� �� P the area of�

if it is a nonempty subdomain of � �the length of

�if it is a geodesic arc on � �BM

Then, for each�fQ

, we denote by � Q the set of the indices of its neighbors� ^

’s such that� � � Q . ^8� � � , i.e., of those neighbors for which � Q . ^ is a geodesic arc. We also denote by�]Q<�_^the vector from

�]Qto�_^

, by �]Q<�_^ the geodesic arc joining�fQ

and�_^

, and by�fQ ^

themidpoint of the geodesic arc �]Q0�_^ . From the construction of spherical Voronoi tessellations,

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FIG. 2.1. Spherical centroidal Voronoi tessellations of a sphere for a constant density function with 162 (left-top), 642 (right-top), 2562 (left-bottom) and 10242 (right-bottom) generators.

it is known that � Q � ^ is perpendicular to � Q ^ and the plane determined by � Q . ^ and the originbisects �]Q<�_^ at its midpoint

�]Q ^[28], see Fig. 3.1. Thus, �]Q?1L� �_^%1 �

and34 9W. �

�]Q<�_^for

� � � Q ^ � � 1d � ` � (3.1)

where the outer unit normal vector to the boundary of Z Q , denoted by34 9W. � , is taken to lie in

the tangent plane of � � at�

.

x i

x j

x k

Vi

ij

FIG. 3.1. A spherical Voronoi region and its dual triangles.

Meshes of the type � P �fQ � Z QTS�UQWV � are used as control (or finite) volumes for thediscretization method discussed below.

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3.1. A finite volume discretization scheme. Denote by���

the space of all piecewiseconstant grid functions associated with a spherical Voronoi mesh � P � Q � Z Q S�UQBV � of � � ,i.e.,

��� P=* � * �0�,� is constant on each cell Z Q S (3.2)

Set P8*��Q *�� ��� Q �LS�UQWV � and denote by � Q ^ the approximate diffusion flux defined by

� Q ^ � * � �- 1 � � � Q ^8�F@Q ^ *��^ 1 *��Q � ^ 1 � Q � R ��� �O1 @B���,� ' ( * �0�,�O� $ 34 9W. � T� ���,� � (3.3)

where

� � � Q ^8�O@ Q ^ R ��� @B���,� T� ���,� MBased on the Green’s formula, a finite volume method for seeking the approximate solution*�� � � � of (1.1) was proposed in [8]. It used an up-wind approximate convection flux

� Q . ^[17] defined by

� Q . ^ ����Q ^ * �Q C����Q ^ * �^�� R ��� � 3D �0�,� * ���,�O� $�34 9W. �� T� �0�,� � (3.4)

where

���Q ^ %0 ���/Q ^�C � �/Q ^ � � � ���Q ^ %0 ���_Q ^ 1 � �_Q ^ � � and�_Q ^ R

���3D ���,� $ 34 9W. � T� ���,� M

For all Z Q , letJ Q

andH Q

denote respectively the mean value ofJ

andH

on Z Q , i.e.,

J Q %� � Z Q � R �� J2���,� TWV �0�,� andH Q %� � Z Q � R �� H��0�,� T V �0�,� M (3.5)

Then, the finite volume scheme given by [8] is defined by the following system:

��� � * � � Q %� � Z Q � �^ ��� � = � Q ^ C � Q ^ �,CIH Q * �Q KJ Q ford2 % � M=M M �$a M (3.6)

REMARK 1. An approximate convection flux which leads to a central difference schemeis defined by

� Q . ^ �/Q ^0 � * �Q C * �^ � M (3.7)

Then, a stability condition such as

� Q )+*�,^ ��� �� �/Q ^ ��$W �]Q?1 �_^ 0 � � � Q ^8�F@Q ^ � % for

d % � $ $=$ �;a)�is needed, where

� Qis called the local Peclet number [17, 25].

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3.2. Previous results. For given grid functions * �;: � ���, we define, similar to [25],

the following discrete inner products and norms associated with a spherical Voronoi mesh� P �]Q � Z QTS�UQWV � :�������� �������� * �;: � � U� QBV

�� � Z Q � * �0� Q � : �0� Q � � * �� . � � * � * � � �� * � � � . � %0 �� QBV�

�^ ��� � � � � Q ^�� T �0�]Q � �_^��� * ���]Q � 1 * �0�_^�� � Q 1 � ^ �� � � * � � . � * �� . � C � * � � � . � M

Norms for general function spaces can also be defined.We define the mesh quality norm � by

� )+*�,QWV� .������ . U � Q � where � Q6 )+*J,� � � T �0�]Q �;G � M

Thus, � Q gives the maximum geodesic distance between a particular generator� Q

and thepoints in its associated cell Z Q and � gives the maximum of all � Q ’s. The above mesh norm �has been used in [7] in the context of polynomial interpolation on the sphere.

Given a Voronoi mesh � P �fQ � Z Q;S �QBV � , we define the mesh regularity norm � by

� )����QWV� .������ . U � Q � where � Q ) � �^ ��� � � Q ^ and � Q ^ T �0� Q � � ^ �0 � Q M

(3.8)

� can be used as a measure of the uniformity of a mesh; the larger the value of � , the moreuniform is the mesh. In addition, the value of � provides us a measure of the mesh regularity,i.e., the local uniformity of a mesh. Again, if a mesh is locally uniform in the sense thatthe cells in a neighborhood of any cell are nearly congruent to that cell, then the value of �will again be large. We will refer to � as regular if � is not too small. To get a reasonabletheoretical estimate, we will need to require a regularity condition on the mesh. The followingresult has been proved in [8].

THEOREM 1. Suppose that Assumption 1 is satisfied. Let � Q ^ , � Q ^ , andHFQ

be defined by(3.3)–(3.5). Then the discrete system (3.6) has a unique solution * � � � �

. Furthermore,assume that the unique variational solution * of (1.1) belongs to �

� � � � � , then there exists aconstant

2 �� only depending on3D ,H, and � such that

� � . � � 2 � * -@ 7 � 5 7 � (3.9)

where� �0�,��

�Q * ���]Q �21 *��Q for� � Z Q .

Note that Theorem 1 holds for general regular spherical Voronoi meshes. For moreexisting studies on the finite volume methods, especially when applied to solve second orderelliptic on the two dimensional plane, we refer to [2, 4, 5, 9, 11, 12, 16, 23, 22, 24, 26, 32, 33].

4.� �

error estimate on spherical centroidal Voronoi meshes. An improved errorestimate in the

���norm is generally expected for finite element and finite volume approx-

imations of second order elliptic equations. However, in this section, it is shown that thequadratic order error estimate for the scheme under consideration here can only be provedwhen the grid satisfies certain geometric constraints. In fact, a part of the estimate dependscritically on the property that if � P �RQ � Z Q;S�UQBV � is a spherical centroidal Voronoi tessella-tion of � � corresponding to a density function � , then,R

� ����,� ��� �Q 1L�,� TWV ���,�� � �DC d6 % � 0 � M=M M �$a (4.1)

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where� �Q

is standard mass centroid of Z Q , whose projection (through the standard map�

)onto the sphere coincides with

� Q.

Without loss of generality, we set � % , that is, we consider the case of the unit sphere.For the rest of the section, only those schemes based SCVM are analyzed.

If�]Q � �_^ and

� � are neighbors for each other in � , we denote by �� Q ^ � the sphericaltriangle determined by

� Q � � ^ and� � , and by

� Q ^ � the corresponding planar triangle (see Fig.3.1). We use the notation

� P d ` �� d � ` � �

are neighbors for each other in � S ��� Q ^ � � � � G �� G � G � G � � Q ^ � " � � � Q ^ � � �

�� P��� Q ^ � � d ` � � � S � and� P � Q ^ � � d ` � � � S M

It is easy to see that �� is the corresponding spherical Delaunay triangulation of � � associatedwith the generators P �fQgS�UQWV � . We can also view

�as a one-to-one smooth function that maps� � �� � � � � � � Q ^ � to � � ���� � � � � �� �� Q ^ � .

Clearly,� � � � � P � Q S�UQBV � . For any

� � � � and�� � � � � �� Q ^ � , we have��� ��

� 1 ����0�,� � � � �J0 �� % 1 0 � � � T �0� � � � � � � ��

������21 ��

��0� � � � � % C 0 � � � T ��� � � � � � �� � � Q ^ � � ��� � �� Q ^ � � � � % C 0 � � � � � � � Q ^ � � (4.2)

where��� � Q ^ � � denotes the area of the triangle

� Q ^ � . Set

� P � � % 1 � �UX � X % C � �JS �we see that

� Q ^ � X �and �� Q ^ � X �

. For any * �� �� � � � , define the function � * , the

extension of * in�

, by

� * � G �� * > G G E � G � � MThe following results have been established in [8]:

PROPOSITION 1. For any G � � and�� G � G � � � , and

d � ` % � 0 ��� ,����� ����' ( * �0�,�� '�� * ���,� �' �0; Q � * � ���,�� ')( � ')(/. Q * � ���,�21 � ')(/. Q * ���,�O� 34,5�7 . 9 � G '�� * � G �� '�� * ���,� � G � ; Q<; ^ � * � G �� ; Q ; ^ � * �0�,� �

(4.3)

From Proposition 1, we naturally haveCOROLLARY 1. There exist a constant

2 � � such that for any G � � ,�� G � G ,��Q . ^$V �

� ' (/. Q ' (/. ^ * ���,� � � � G ����Q . ^$V �� ; Q ; ^ � * � G � � � � 2 ����! � � �

� ' �( * ���,� � � M (4.4)

We call * " a piecewise linear function on� �

if and only if

* " � G ���/Q * " ���]Q �,C��^ * " ���_^8�,C�� � * " �0� � � �DC G � � Q ^ � M8

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where� Q � � ^ � � � are the barycentric coordinates of G in the planar triangle

� Q ^ � . Denote by� �the space of all functions * � on � � such that* � �0�,�� * " � � � � ���,� � � � � � � M

where * " is a piecewise linear function on� �

with P8* " �0�]Q �- * � ���]Q � S UQBV � , i.e,

� * � � G �� * " � G � � C G � � � MLet ' � be the tangential surface gradient operator on

� �, i.e., for any function * defined on� �

, ' � * � G �� � ' � . ��� ' � . � � ' � . �=� * � G �� '�* � G �21 � '�* � G � $ 34 � � �� 34 ��� � �DC_G � � Q ^ �

where34 ��� � is the unit outer normal vector to

� Q ^ � . Similarly, we could define the standardSobolev space on

� �. It is easy to get * � � � � � � � � for any * � � � � using (4.3) and the

fact that � * � * " � � � � � � � . In addition, as in Proposition 1 and Corollary 1, after sometechnical derivation, we have, for any G � � Q ^ � ,�������� �������

� ' � � * � G � � � � � ' � * � G � � � �� �Q . ^$V � � ' � . Q ' � . ^ � * � G � � � � � �Q . ^$V � � ; Q<; ^ � * � G � � � �' � . Q ' � . ^ � * � � G �� � � d � ` % � 0 ��� � G ' �0; ^ � * � � � G �� 1 � 34 ��� �

$B34,5�7 . � � � � � �<;�Q � * � � G �O� 34 ��� � �34 � � �

$ 3465 7 . � � � � 7 % 1 0 � ��M(4.5)

From Proposition 1 and (4.5), we obtain,COROLLARY 2. There exist a constant

2 � � such that for any�A� �� Q ^ � ,��Q . ^$V �

� ')(/. Q ')(/. ^ * � ���,� � � � 2 � � ��-�� � � �� ' �( * � ���,� � � M (4.6)

Let �* � * � ��� , using (4.5) with Proposition 1 and Corollary 1, we also get that, thereexists some constant

2 � � , �* @ � � ��� �

� � 2 * @ � � � � � �� � � � � % � 0 M (4.7)

for anyd ` � � �

. We now give estimates on the interpolants.PROPOSITION 2. For any * � � � � � � � , let

��� � * � and��� � * � be the interpolants of * on� �

and� �

respectively, then there exist some constants2��� 2 � � 2�� �� such that��� ��

* 1 ��� � * � " 7 � 5 7 � � 2� � � * F@ 7 � 5 7 � * 1 � � � * � F@ � 587 � � 2 � � * F@ 7 � 5�7 � * 1 ��� � * � " 7 � 5 7 � � 2�� � * -@ 7 � 5 7 � (4.8)

Proof. The proof requires only standard techniques. Define* " � G ���� Q * ��� Q �,C�� ^ * ��� ^ � C�� � * ��� � � �DC_G � � Q ^ � Mthen

��� � * � ���,� * " � �� � �0�,�O� . Furthermore, using the estimate for the linear interpolationon planar triangles and the relation* ���,�21 ��� � * � �0�,�� � * � � � � �0�,�O�21 * " � � � � ���,� � �

9

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we obtain by (4.2) that * 1 � � � * � " 7 � 5�7 � > �� ��� � � ��

R� � � �

� * ���,��1 � � � * � ���,� � � TWV ���,�FE �;# � > �

� ��� � � ��R� � � �

�� * �0�,��1 * " � � � � �0�,�O� � � TWV ���,� E �;# �

� 0 > � ��� � � �

R � � �

��* � G �21 * " � G � � � TWV � G �OE �;# � 0 > �

��� � � � �* � G ��1 * " � G � �" 7 � � � �

� E �$# �� �2 � � � > �

��� � � � �* �@ 7 � ��� �

� E �;# �� 2

� � � > �� ��� � � ��

* �@ 7 � � ��� �� E �;# � 2

� � � * F@ 7 � 5�7 �

(4.9)

The other estimates can be proved in similar manners. We omit the details.Next, we provide some equivalence between various norms. For convenience, we make

the assumption that that all three angles of� Q ^ � are less than

� ��� in this section, such astatement generally holds for the triangles in the spherical centroidal Voronoi meshes withsufficient large (no smaller than 42, for example for the constant density) number of vertices(generators), or equivalently, sufficient small � .

LEMMA 1. For any * � � � � , there exists some constants2� � 2 � ��2

* � � . � � * � " 7 � 5 7 � � 2 � * � � . �2 � * � � . � � * � -@ � 5 7 � � 2 � * � � . � (4.10)

Proof. First, from (4.2), we have%0 � � � � � �

R ��� �

� � * � � � � G � TWV � G � � * � �" 7 � 5 7 � �� ��� � � ��

R� ��� �

* � � ���,� TWV ���,�� 0 �

� � � � �R ��� �

� � * � � � � G � TWV � G � (4.11)

Since * � � � � , � * � is linear on� Q ^ � . ThenR

��� �

� � * � �F� � G � TWV � G �� %% 0�� � * � � ��� Q � C * � � ��� ^ � C * � � �0� � � �C � * � ���]Q �,C * � ���_^�� C * � ��� � �O� ��� � � � Q ^ � � �and consequently,

��� � * � � �0� Q �6C * � � �0� ^ �,C * � � ��� � � � ��� � Q ^ � � � R ��� �

� � * � � � � G � TWV � G �� �� � � * � � ���]Q � C * � � ���_^�� C * � � ��� � � � � � � Q ^ � � (4.12)

Let� Q ^ � ���� Q � * � � �0�]Q �6C ����/^8� * � � ���_^�� C � �� � � * � � ��� � � �

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clearly,� �� Q �,C � �� ^ �,C � �� � �� ��� � Q ^ � � (see Fig. 4.1), we have

� ��� � � �

� Q ^ � � * � �� . � U� QWV�� � Z Q � * � � ���]Q � � �

��� � � �0 � Q ^ � (4.13)

Since��� � Q ^ � ��� � X � �� � � X � � � Q ^ � ���J0 for � �d � ` � �

, with (4.11), (4.12) and (4.13), wehave 2

* � � . � � * � " 7 � 587 � � 2 � * � � . � (4.14)

for some constants2� � 2 � � � .

xi

xk

xj

S i

S jS k

z

zz

z 0

i

j

k

FIG. 4.1. ��� ���

We now consider� * � � � � . � and

� * � � �@ � 5 7 � . By Proposition 1 and (4.2), we have� * � � �@ � 5 7 � �� ��� � � ��

R� ��� �

� ')(+* � �0�,� � � TWV �0�,� �� ��� � � ��

R� ��� �

� '�� * � �0�,� � � TWV ���,�� 0 �

��� � � �R ��� �

� '�� * � � G � � � TWV � G �(4.15)

and similarly, � * � � �@ � 5�7 � 7 %0 � ��� � � �

R ��� �

� '�� * � � G � � � TWV � G � (4.16)

Let �* � � * � � � � , clearly, for any G � � Q ^ � , ' � �* � � G � 3� Q ^ � is a constant vector and

� Q ^ � $ 34 � � � . � � . Since3465 7 . � � � $ 34 ��� �

7 % 1Y0 � � , we have� ' � �* � � G � � � � '�� * � � G � � � � % C 0 � � � � ' � �* � � G � � �which leads to

��� � Q ^ � � � 3� Q ^ � � � � R � � �

� '�� * � � G � � � T V � G � � � % C 0 � � � � � � � Q ^ � � � 3� Q ^ � � � M (4.17)

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Without loss of generality, suppose that the triangle� Q ^ � is parallel to the two dimensional��� � � � -plane. Again since � * is linear on

� Q ^ � , then we have

' � �* � ��0;�� � * � � ; � � * � � � � ��� Q ^ � � � ��0; � � * � � � CK�<; � � * � � � � ���B1L� � � ������?1L� � � � C � � �B1 � � � � � � � � d � ` � � �

where� � ���� � � � � � � � � for � 1d � ` � �

. Since * � � � � , we have on� Q ^ � ,� ;�� � * � � � ��� �

� � * � ���]Q � � � ^%1 � � �,C * � �0�_^�� � � � 1 � Q<�,C * � ��� � � � � Q?1 � ^8� � �;� � * � � � ��� �

� � * � ���]Q � ��� � 1L� ^�� C * � ���_^�� ���_Q 1 � � �,C * � ��� � � �0� ^ 1L�_Q � � M (4.18)

So, we finally obtain (see [23]),� * � � � � . � %0 � �QWV�

� ^ ��� � � � � Q ^ � T ��� Q � � ^ � >� � � 9 � � � � � 9 � �� 9 � � 9 � � E �� � % C � � � � � � � � � � � 0 ����� � Q � � � ^ ��� Q ^ C 0 � ��� � ^ � � � � ��� ^ �C&0 � ��� � � � � � Q ��� � Q �� 0 � � � Q ^ � � � �0;�� � * � � � � �O� � CK�<; � � * � � � � � � � � 0 � � � Q ^ � � � 3� Q ^ � � �

(4.19)

where

����� � ��� � � �� %0 � � 1 � � � � 1 � � � � � � � * � ����� �21 * � ��� � � � � 1L� � � � �� ��� �]Q � � �_^�� C � ��� �_^ � � � � �,C � ��� � � � � �]Q �� � � � Q ^ � � M

Similarly, � * � � � � . � 7 %0 ��� � Q ^ � � � 3� Q ^ � � � (4.20)

Combining (4.15) - (4.20), we have2�� � * � � � � . � � � * � � �@ � 5 7 � � 2�� � * � � � � . � (4.21)

for some constants2 � � 2 � �� . From (4.14) and (4.21), we get that there exist some constants2�� � 2 � �� 2�� * � � . � � * � F@ � 587 � � 2 � * � � . � M

REMARK 2. The results of the lemma can actually be proved for more general Voronoi-Delaunay meshes that satisfies the local mesh regular properties. In fact, we need, for eachspherical triangle �� Q ^ � , that the number of spherical Voronoi regions Z � having non-emptyZ � � �� Q ^ � is bounded by a constant integer, independent of �� Q ^ � . Furthermore, any sphericalVoronoi region Z � can be covered by the union of the finite set of �� Q ^ � . The maximum number

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of spherical triangles needed in such a union should also be bounded by a constant integerindependent of

�. Under those conditions and the mesh regularity conditions, the above equiv-

alence of norms still holds. The assumption that any triangle only has acute angles makesthe above derivation much simpler.

Since� �

is the space of grid functions on � � with respect to � P �]Q � Z QTS�UQWV � , the setof functions P � QTS�UQBV

� where

� Q ���,��QP % � � � Z Q� � � � � � 1 Z Qgives a set of basis functions for

� �. For any * � � � � � � � and : � � � � , define the bilinear

functionals < � and < � such that

< � � * �;: � �- U� QBV�: � �0�]Q � < � � * � � Q<� � (4.22)

< � � * �;: � �- U� QBV�: � ���]Q � < � � * � � Q<�

(4.23)

where

< � � * � � Q �- R� ��F1 ')(/* ���,�,C 3D �0�,� * ���,�O� $ 34 9W. �� T� �0�,�,C R � H����,� * �0�,� T V �0�,� �

< � � * � � Q �- �^ ��� � � Q ^ � * �,C R � � � �� * � � 3D ���,� $B34 9W. � � T� �0�,�6C R � H � � � * � T V �0�,� M

Comparing < � with the finite volume scheme (3.6), we have in fact replaced Z Q ^ by � � �� ��� � * � � 3D �0�,� $34 9W. � � T� ���,� and � � Z Q � H Q * ���]Q � by � � H � � � * � T V �0�,� . Note that no change is made for apure diffusion problem containing only second order terms 'G( $ �<@ �0�,� ')( * ���,� � . Our discreteproblem here is then: find * � � � � such that< � � * � �$: � �- � J �;: � � �DCE: � � ��� M

(4.24)

Formulations like the above for the finite volume methods have been used, for instance, in[23]. The original scheme (3.6) may be viewed as an approximation of (4.24) through numer-ical integration. It is not difficult to show that the error estimate of Theorem 1 still holds forthe above * � using similar analysis in [8]. Combining with Lemma 1 and Proposition 2, wehave the following result:

THEOREM 2. Suppose that Assumption 1 is satisfied. Let � Q ^ and < � be defined by(3.3) and (4.23) respectively. Then the discrete system (4.24) has a unique solution * � � � � .Furthermore, assume that the unique variational solution * of (1.1) belongs to �

� � � � � , thenthere exists a constant

2 �\� only depending on3D ,H, and � such that for

��0�,� * ���,��1* � �0�,� , we have

� -@ � 5�7 � � 2 � * F@ 7 � 5�7 � M (4.25)

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Proof. Notice that � � � * �21 * � � . �

� � . � , then we have

� F@ � 587 � * 1 * � F@ � 587 �� * 1 � � � * � F@ � 5 7 � C ��� � * �21 * � F@ � 5 7 �� 2

* F@ 7 � 5 7 � C 2 � � � � * � 1 * � � . � 2�

* F@ 7 � 587 � C 2 � � � . �� 2 � * -@ 7 � 5 7 � �where the conclusion of the theorem 1 has been used.

LEMMA 2. Suppose that � P �fQ � Z QTS�UQWV � is a spherical centroidal Voronoi tessellationof � � with the density function � satisfying � �32 � � � � � and � �0�,� � � for any

�A� � � . Then,for any �

��� � � � � , there exits a constant

2 � � such that� R �� � 1 � � � � �O� TWV ���,� � � 2 � � � � � Z Q � � �$# � � F@ 7 � � � � d2 % � M=M M �$a (4.26)

Proof. Let us assume that ��62 � � � � � , then it is easy to see that � � �62 � � � �

. Considerthe spherical Voronoi region Z Q associated with

� Q, for any

�L� Z Q , we have

� ��� Q � 1 � ���,�� � � ��� Q �21 � � �0�,�� '�� � ���,� $ ��� Q 1L�,�C R ���� � � � ��� � C � % 1��O� � Q � �0� Q 1 �,� $ �0� Q 1L�,��� T �

where �� � � � ���,� denotes the Hessian matrix of � � at

�. So� R

�� �1 � � � � � TWV ���,� � ��� � C � � C � �

where

� � � R

�� '�� � ���,�$ �0� �Q 1 �,� T V �0�,� � � R

� ')( � �0�,�$ ��� �Q 1 �,� TWV ���,� � �

� � � R�� '�� � ���,�

$ �0� Q 1 � �Q � TWV ���,� � � R�� ')( � ���,�

$ �0� Q 1L� �Q � TWV ���,� � �� � R �

R �� ��� � � � ��� � CK� % 1��O�F�]Q � ���]Q,1L�,� $ ���]Q,1L�,� � � T � T V ���,� M

where� �Q

is the standard mass centroid of Z Q associated with the density function � .Using the properties (4.1) of the SCVT, we have

� � � R

� ' ( � ���,�$ ��� �Q 1L�,�21 � �0�,�� ��� Q ��� � � ' ( � � $ ��� �Q 1L�,� T V ���,� �� �/R

��� �0�,�21 � ��� Q �� ���]Q � ')( � ���,� $ ��� �Q 1L�,� TWV ���,� �C � R �

� ���,�� ��� Q � � ' ( � �0�,��1 � � � ' ( � �O� $ ��� �Q 1L�,� TWV ���,� � � � C � �where � � denotes the

���projection on

� �.

14

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Since � � is compact, we have

� � � � R �� �0�,�21 � �0�]Q �� �0� Q � ' ( � ���,� $ ��� �Q 1 �,� T V ���,� �

� � R �� )+*�, 9�� �� '�� � ���,� �� � �0�]Q � � � ')( � �0�,� � � �Q 1 � T V �0�,�

� 0 � � R � ) *J, 9�� �� ' ( � ���,� �� � ��� Q � � � ' ( � ���,� � TWV ���,�

� 2 � � R � � ')( � ���,� � TWV ���,�� 2 � � >6R � � ' ( � ���,� � � T V �0�,� E �$# � > R � TWV ���,�;? �$# �� 2 � � � � � Z Q � � �$# � � F@ 7 � � �(4.27)

and

� � � R �)+*�, 9�� � � � �0�,� �� � ���]Q � � � ' ( � ���,��1 � � � ' ( � � � � �Q 1L� TWV ���,� �

� 2 ' ( � 1 � � � ' ( � � " 7 � � � > R � � �Q 1L� � T V ���,�OE �$# �� 2 � � � � � Z Q �O� �;# � � -@ 7 � �� �(4.28)

Combining (4.27) and (4.28), we get

� � � 2 � � � � � Z Q �O� �$# � � F@ 7 � � � M (4.29)

Consider � � , using the fact that�fQ6 � �Q � � �Q

and (4.2), we know that �_Q?1L� �Q X � � �J0Then, we get

� � � � � >2R �� ')( � �0�,� � � TWV ���,�FE �;# � > R � T V ���,�OE �$# �� � � � � � Z Q �O� �;# � � -@ � �� � M (4.30)

On the other hand, for � � , by doing a change of variables G � �UC � % 1 �O�F�RQ and settingZ �Q P G � �GC � % 1��O� �]Q � �A� Z Q;S , we have thatTWV ���,� � 0 TWV � G ��� � � . Then

� � � 0 � � R �� R������ � � � �� TWV � G � T � M

It is easy to see that � � Z �Q � � � � � � Z Q � . By a proof similar to that given in [8], we get

� � � 0 � � R �� > R������ � � � � � TWV � G �OE �;# � � � � Z �Q � � �$# � % � T �� 2 � � � � � Z Q � � �$# � � F@ 7 � � � R �� %� � T �� 2 � � � � � Z Q � � �$# � � F@ 7 � � �

(4.31)

Finally, we obtain (4.26) for * � � � � � � � by combining (4.29) and (4.30) with (4.31)and invoking a density argument.

15

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THEOREM 3. Let Assumption 1 be satisfied. Suppose that � P � Q � Z Q S�UQBV � is a spher-ical centroidal Voronoi tessellation of � � with the density function � satisfying � � 2 � � � � �and � ���,� � � for any

� � � � . Let � Q ^ and < � be defined by (3.3) and (4.23) respectively.Then the discrete system (4.24) has a unique solution * � � � � . Furthermore, assume thatthe unique variational solution * of (1.1) belongs to �

� � � � � , then there exists a constant2 �� only depending on � ,3D ,H, and � such that � " 7 � 5�7 � � 2 � � * -@ � � 5�7 � (4.32)

where��0�,�� * �0�,�21 * � �0�,� .

Proof. For simplicity, we assume@B���,�� % . Since * 1 * � � � � � � � � , according to (1.4),

we know that there exists a weak solution ���� � � � � satisfying

< � � �;: ���� * 1 * � �$: � �DC : � � � � � � �Put : * 1 * � in the above equality, then we get� � * 1 * � � � �" 7 � 5 7 � �� * 1 * � � * 1 * � �� < � � � * 1 * � � (4.33)

Furthermore, from (1.5), we know,� ��� � @ 7 � 5�7 � � 2 � � * 1 * � � � " 7 � 587 � (4.34)

for some constant2 � � .

Denote by� � � � � and

� � � � � the interpolants of � on� �

and���

respectively, then wehave

< � � * � � � � � � �� �<J � � � � � �O� � < � � * � � � � � � � �� �<J � � � � � �O� MConsequently, we get * 1 * � �" 7 � 5�7 � � � < � * 1 * � � � 1 ��� � � �O� �C � < � * 1 * � � � � � � � ��1 < � � * 1 * � � � � � � � � �C � < � � * � � � � � � �O�-1 < � � * � � � � � � �O� � (4.35)

According to Theorem 2, Proposition 2 and (4.13), we get� < � * 1 * � � � 1 � � � � � � � � 2 * 1 * � -@ � 5 7 � � 1 � � � � � -@ � 5 7 �� 2 � � * -@ 7 � 5�7 � � � � @ 7 � 587 �� 2 � � * -@ 7 � 5�7 � * 1 * � " 7 � 587 � (4.36)

16

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On the other hand, by Green’s formula, we have

< � * 1 * � � � � � � �O�� R 5 7 ')( � * 1 * � � $ ')( � � � � � TWV ���,�C R 587 � * 1 * � � � 3D $ ' ( � � � � � � TWV �0�,�6C R 5�7 H�� * 1 * � � ��� � � � TWV ���,� �� ��� � � ��

> R� ��� �

' ( � * 1 * � � $ ' ( ��� � � �,C � * 1 * � � � 3D $ ' ( ��� � � �O� TWV ���,� EC R 587 H�� * 1 * � � ��� � � � T V �0�,� �

� ��� � � ��> R� ��� �

1 � ( � * 1 * � � � � � � �6C � ')( $ � * 1 * � � 3D � � � � � � TWV � G �C R

� � � � �

� ')( � * 1 * � � $ 34 9W. � ��� �� � � � � �-1� * 1 * � � � 3D $�34 9W. � � � �

� � � � � � T� � G �OEC R 587 H�� * 1 * � � ��� � � � T V �0�,� (4.37)

where34 9W. � � � � is one of the tangent vector of � � which is the unit outer normal vector toward

the boundary on �� Q ^ � . Also, we have

< � � * 1 * � � ��� � � � �� U� QBV�

��� � � � �0�]QF� < � � * 1 * � � � Q � U� QBV

> R� �

1 � ' ( � * 1 * � � $ 34 9W. � � ��� � � �,C � * 1 * � � � 3D $ 34 9W. � � ��� � � � T� ���,�C R �H�� * 1 * � � � � � � � TWV �0�,�FE �

� ��� � � ���9 � � ��� �

R� ��� � � � �

1 � ')( � * 1 * � � $ 34 9W. � � � � � � �C � * 1 * � � � 3D $ 34 9W. � � � � � � � T� ���,�6C R 5 7 H�� * 1 * � � � � � � � TWV ���,� �

� ��� � � ��> R� � � �

1 � ( � * 1 * � � ��� � � �6C � ' ( $ � * 1 * � � 3D � � � � � � TWV ���,�C R

� � � � �

� ' ( � * 1 * � � $ 34 9W. � ��� �� ��� � � ��1 � * 1 * � � � 3D $ 34 9W. � ��� �

� � � � � � T� ���,� EC R 5 7 H � * 1 * � � � � � � � TWV �0�,� (4.38)

So, we obtain

< � * 1 * � � ��� � � �O��1 < � � * 1 * � � � � � � �O�� � � C � � C � ��C � � C � � C � �17

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where

� �\1 �

� ��� � � ��R� � � �

� ( * � ��� � � �-1 ��� � � �O� TWV ���,� �� � �

� � � � � ��R� ��� �

� ( * � � � � � � ��1 ��� � � �O� TWV ���,� �� � �

� � � � � ��R

� � � � �

� ' ( � * 1 * � � $B34 9W. � ��� �� � � � � � ��1 ��� � � �O� T� �0�,� �

� � �� � � � � ��

R� ��� �

� ' ( $ � * 1 * � � 3D � � ��� � � ��1 � � � � �O� T V ���,� �� � �

� � � � � ��R

� � � � �

� * 1 * � � � 3D $ 34 9W. � � � �� � ��� � � �-1 ��� � � �O� T� ���,� �

� � R 5 7 H�� * 1 * � � � � � � � �21 � � � � � � TWV �0�,� MConsider � � , we have�

� �� � �

� � � � � ��R� ��� �

� ( * � ��� � � �21 � � � � �O� T V �0�,� � � �

� � � � � ��R� ��� �

� ( * � ��� � � �21 � C � 1 ��� � � � � TWV �0�,� �� � �

� � � � � ��R� ��� �

� ( * � ��� � � �21 � � TWV �0�,� �C � �� ��� � � ��

R� � � �

� ( * � � � � � ��1 � � T V ���,� �� � �

� � � � � ��R� ��� �

� ( * � � � � � �21 � � TWV �0�,� �C � U� QBV

R � � �

��� ( * � � � � � � ��1 � � TWV �0�,� �C � U� QWV

R����� ( * �0�,�21 � � ��� (/* �O� � � � � � ��1 � � TWV �0�,� � (4.39)

where � � denotes the���

projection on���

. Using Lemma 1 and Cauchy-Schwartz inequal-ity, we get� �

� ��� � � ��R� ��� �

� ( * � � � � � �21 � � T V �0�,� � � 2 � � * -@ 7 � 5 7�� � -@ 7 � 5 7��� 2 � � * -@ 7 � 5�7 � * 1 * � " 7 � 5�7 � M

18

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Using Lemma 2, we have� U� QBV�

R � � �

��� ( * � � � � � � ��1 � � TWV ���,� �� 2 � � U� QWV

�� � ��� ( * � � � � � � � Z Q � � �$# � � F@ 7 � � � 2 � � � � ��� ( * � " 7 � 5 7 � � �@ 7 � 5�7 �� 2 � � * F@ 7 � 5�7 � * 1 * � " 7 � 587 �

and � U� QBV�

R ���� (/* ���,��1 � � ��� ( * �O� � � � � � �-1 � � T V ���,� � � 2 � � * F@ � � 5 7�� � F@ 7 � 5 7��� 2 � � * F@ � � 5 7 � * 1 * � " 7 � 5 7��

So, we obtain �� �� � 2 � � * F@ � � 5�7 � * 1 * � " 7 � 5�7 � (4.40)

As for � � , using Corollary 2 and Lemma 1, we have�� � � � �

� ��� � � ��R� � � �

� � (/* � � � � � � ��1 � � � � �O� � TWV ���,�� 2 � �

� � � � � ��R� ��� �

� ')(+* � �#� � � � � � �21 � � � � � � � TWV ���,�� 2 � � * � F@ � 5�7 � � F@ 7 � 5�7 �� 2 � � � * F@ � 5 7�� C 2 * -@ 7 � 5 7 � � * 1 * � " 7 � 5 7 �� 2 � � * -@ 7 � 587 � * 1 * � " 7 � 5�7 �(4.41)

According to the continuity of ' ( * on each� �� Q ^ � , we have

�� ��� � � ��

R� � ��� �

� ' ( * $ 34 9W. � � � �� � � � � � �-1 ��� � � �O� T� ���,�� �

so we get

� � �� ��� � � ��

R� � � � �

� ' ( * � $ 34 9W. � ��� �� � � � � � ��1 ��� � � � � T� �0�,� M

(4.42)

Additionally, on each edge �� of �� Q ^ � , by symmetry (see Fig. 4.2), we have

' ( * � � � � � $B34�� . � ��� � ')(/* � � � � � $ 34�� 7 . � � � � �� ��� � � ��1 ��� � � �O� � �

��- 1 � � � � � ��1 � � � � � � � � � �

which also implies R�" � ' ( * � $ 34 9W. � ��� �

� � ��� � � ��1 � � � � �O� T� ���,�- � M19

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T ijk Tijk

M

M

L

L

zz

12

~

~

~

FIG. 4.2.������

are middle points of � and�� respectively, ����� �� � ����� �

�� � .

So we have

� � � M (4.43)

About � � , we have�� �� � �

� � � � � ��R� ��� �

� ')( $ � * 1 * � � 3D � � � � � � �-1 � � � � � � T V ���,�� N����9�� 5 7 �

� 3D � C � ')( 3D � � � �� � � � � ��

R� ��� �

� ')( � * 1 * � � �#� � � � � �-1 � � � � � � TWV ���,�� 2 � * 1 * � F@ � 587 � � -@ 7 � 5�7 � (4.44)� 2 � � * -@ 7 � 587 � * 1 * � " 7 � 5�7 � (4.45)

About � � , using Trace theorem [13], we have�� � � � �

� � � � � ��R

� � � � �

� � * 1 * � � � 3D $ 34 9W. � ��� �� � � � � � ��1 � � � � � � � T� ���,�

� N����9�� 5�7 ��<3D � � > �

� � � � � ��R

� � � � �

� * 1 * � � � T� �0�,�FE �$# �$ > �� � � � � ��

R� � ��� �

� � � � � �21 � � � � � � � T� �0�,� E �$# �� 2 > �

� ��� � � �� * 1 * � �@ � � ��� �

� E �$# � > �� � � � � �� � �

� �@ 7 � � ��� �

� E �$# �� 2 � * 1 * � -@ � 5�7 � � -@ 7 � 5�7 � (4.46)� 2 � � * -@ 7 � 5 7�� * 1 * � " 7 � 5 7�� (4.47)

About � � , we have �� � � � 2 * 1 * � F@ � 5 7�� � � � � �21 � � � � � " 7 � 5 7��� 2 � � * -@ 7 � 5�7 � � F@ � 5�7 �� 2 � � * -@ 7 � 5�7 � * 1 * � " 7 � 5�7 � (4.48)

By (4.40), (4.41), (4.43), (4.44) (4.46) and (4.48), we get� < � * 1 * � � � � � � �O�-1 < � � * 1 * � � ��� � � � � � 2 � � * -@ � � 5 7 � * 1 * � " 7 � 5 7 � (4.49)

20

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On the other hand, we have

< � � * � � ��� � � �O�-1 < � � * � � ��� � � �O�� U� QBV�

> R� ���O1 ' ( * � ���,� $ 34 9W. � � � � � � � T� �0�,�C R

� � * � �3D $ a 9W. � � � � � � � T� ���,� C R � H��0�,� * � �0�,� ��� � � � TWV �0�,�FE1 U� QBV

> �^ ��� � � Q ^ � * � � ��� � � �6C R � � ���� * � � � 3D $ a 9W. � � � � � � � T� �0�,�C R

��H ��� � * � � � � � � � TWV ���,� E � � C ���

C ��� (4.50)

where

� � U� QWV�

�^ ��� � � � � Q ^ ��� Q ^ � �0� Q ����

U� QWV�

R� �� * � 1 � � � * � �O� � 3D $ a 9W. �� � � � � � � T� �0�,�

� � U� QWV�

R �H�� * � 1 ��� � * � �O� ��� � � � TWV ���,�

with

�-Q ^ 1 %� � � Q ^8� R � ��� ' ( * � ���,�$ 34 9W. � T� �0�,�6C * � ��� Q � 1 * � ��� ^ � �]Q?1L�_^ M

Notice that

* � �0�]Q ��1 * � �0�_^8� �]Q 1L�_^ '�� * � � G � $ 34 9W. �� �DC �L� � Q ^ �;G � � � ���,� MSince

��1 G � � � �J0 , we know by Proposition 1 that� '�� * � � G �21 ' ( * � ���,� � � 0 � � � '�� * � � G � � MThen we get

�-Q ^ � 0 � � � * � ���]Q �21 * � ���_^�� � � Q 1 � ^ M(4.51)

It is also easy to find that

� � U� QBV�

�^ ��� � � � � Q ^ ��� Q ^ � ��� Q �- 1 %0 U� QBV�

�^ ��� � � � � Q ^ ��� Q ^ � Q 1 � ^ ��0� Q ��1 � �0� ^ � �]Q 1L�_^ M

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By the Cauchy-Schwartz inequality and Theorem 2, we have that�� �� � U� QBV

�^ ��� � � � � Q ^����-Q ^ T �0�]Q � �_^���� �0�]Q ��1 � �0�_^�� � � Q 1L� ^

� 0 > %0 U� QWV�

�^ ��� � � � � Q ^ � T ��� Q � � ^ ��� �Q ^ E �;# �$F> %0 U� QBV�

�^ ��� � � � � Q ^�� T �0�]Q � �_^8� > ����]Q � 1 � ���_^�� � Q 1L� ^ E � E �$# �

� � � � � * � � � . � � � � � . �� 2 � � * � F@ � 5 7 � � � � � � � � � . �� 2 � � * � F@ � 5�7 � � � � � � � � @ � 5�7 �� 2 � � � * F@ � 5�7 � C * 1 * � F@ � 5�7 � � � � F@ � 587 � C � 1 � � � � � F@ � 587 � �� 2 � � � * F@ � 5�7 � C 2 � * F@ 7 � 587 � � � � � @ � 5�7 � C 2 � � F@ 7 � 5�7 � �� 2 � � * -@ 7 � 5�7 � � -@ 7 � 5�7 �� 2 � � * -@ 7 � 5 7�� * 1 * � " 7 � 5 7 � (4.52)

As for � � and � � , since��� � * � �� * � , we have

� � � � � (4.53)

So we get� < � � * � � ��� � � �O�-1 < ��� * � � ��� � � � � � � 2 � � * -@ 7 � 5 7 � * 1 * � " 7 � 5 7 � M (4.54)

By (4.36), (4.49) and (4.54), we get * 1 * � �" 7 � 5 7�� � 2 � � * -@ � � 5�7 � * 1 * � " 7 � 5�7 �which means

� " 7 � 587 � * 1 * � " 7 � 5�7 � � 2 � � * -@ � � 587 � M

This completes the proof of the theorem.REMARK 3. Extra regularity on the exact solution is required to get the quadratic or-

der error estimates, though in the standard finite element literature, such a requirement isnot needed in general. This is seen as a consequence of the quadrature approximations tothe standard weak forms of the equations. The regularity in �

� � � � � can in fact be furtherweakened to, for instance,

� � . N � � � � for ' � % .REMARK 4. The quadratic order error estimates depend on critically the properties

of the spherical centroidal Voronoi meshes. The proof is not valid for a general sphericalVoronoi mesh. Such an

� �error estimate has not been given in the literature even for the

planar finite volume methods based on the general Voronoi-Delaunay meshes. With otherchoices of the co-volumes which are not of the Voronoi-Delaunay type, a quadratic orderestimate has been proved in [23] for two dimensional diffusion equations. A first order

� �error estimate has been given in [16].

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5. Superconvergent gradient recovery. In this section, we discuss how to post-processthe finite volume solutions to obtain their tangential gradients in the longitude and latitudedirections. Let us rewrite the equation (1.1) in the spherical coordinate system

� � � � � definedby ���� � N ��� � K-M�N � � � N ��� � N ��� � � � K-M�N � � for

� � � � ��� � and� � � � � 0 � � M

Ignoring the radial component, we may denote3D � � � � � � : � � � � � � �;: � � � � � �O� , where : �

and : � are the (orthogonal) components of3D in the

�and

�directions, respectively, on the

tangential surface of � � at�

. We also have

' ( * � � � � �� � %� � *� � � %� N ��� � � *� � � (5.1)

' ( $ 3D � � � � �� %� N � � � � �� � � : � N � � � �,C � : �� � � M

(5.2)

For any generator�fQ

in � , defineZ 9 � � 9 � � ��� �� Q ^ �

We first project Z 9 � onto the tangential plane� 9 � of � � at

� Q, i.e.,

� 9 � is perpendicular to346587 . 9 � at�]Q

. Denote by34�� . 9 � the unit vector on

� 9 � along the�

direction. Then, move� 9 �

to the��� � � � -plane by an affine map satisfying that

3465�7 . 9 � is mapped to the � -axis and34�� . 9 �

to the�

-axis. See Fig. 5.1. We now define a map � 9 ���fZ Q� � �by the above procedure

such as Z�Q � 9 � � Z Q � .n S^2,x_i

n N,x_ix i

x j

x k

n

n

S^2 ,x_i

N,x_i

k

j

i

x’

x’

x’ x

y

z

Vx_i x_iV’

FIG. 5.1. The mapping �� � .On each planar triangle

� � Q � ^ � � � 9 � � � Q ^ � � , we uniquely determine a linear function* � � � by setting* ��� ���� Q �- * � �0� Q � � * � � �

��� ^ �- * � �0� ^ � � * ��� ���� � �- * � ��� � � M

Now we define

' ( * � ���]Q �� %�

� � � ��� �� �

> � * � � �� � � � * � � �

� � E(5.3)

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where � Card� P � Q ^ � � � Q ^ � X Z 9 � S�� . We also let;�� * � * � �- > � Q � � � ' ( * �0�]Q ��1 ' ( * � �0�]Q � � � � � Z Q � E �$# � M (5.4)

The index set�

may be taken to be the set of all Voronoi generators or a large portion of thegenerator set. In light of the recent studies on the finite element gradient recovery [34] at meshsymmetric points, the close relationship between finite element and finite volume schemes[4, 32], and the nice properties of SCVM, we expect that for the finite volume solution withSCVM, there exists the estimate: ;�� * � * � �-��)� � � � M (5.5)

Such results are to be numerically investigated in the next section.

6. Numerical experiments. Let � � be the unit sphere. We now present numerical re-sults that are summarized in the following two examples with each example containing twoseparate experiments (corresponding to two different exact solutions) but with one identi-cal exact solution. In our experiments, the finite volume meshes are taken to be the sphericalcentroidal Voronoi meshes corresponding to a constant density function with various differentnumbers of generators.

For our first example, we choose the exact solution to be* � � � � �� N ��� � � KFM N � � (6.1)

and study two different model problems whose data are given in Table 6.1.

TABLE 6.1Data for two model problems.@B� � � � � : � � � � � � : � � � � � � H � � � � �

I no convection 1 0 0 1II convection dominated 0.05 % C N ��� � % C N � � � �

M � C N � � � �Approximate solutions were obtained using the finite volume scheme (3.6) with the cen-

tral difference scheme and the uniformly distributed spherical centroidal Voronoi meshes inFig. 2.1 which were based on the constant density function � ���

� � � � � ���8� % . In Table 6.2,errors in the approximate solution are plotted against the number of generators.

For the second example, the exact solution of (1.1) is chosen to be* � � � � � �� N ��� ��� 0 � � KFM N � � � � M (6.2)

In Table 6.3, errors in the approximate solution against the number of generators are given.Since the exact solution (6.2) is more complex than (6.1), the largest 2% of the pointwisegradient errors

� ' (+* ��� Q ��1 ' (/* � �0� Q � � was removed from the estimate when computing the;�� * � * � � in Table 6.3. These relatively larger errors concentrates near the 12 defect pointsof the SCVM (i.e., those Voronoi cells with only 5 neighbors) where the mesh lacks perfectsymmetry.

From the numerical values given in the tables we see that, for both the�%�

errors and thegradient recovery errors

;�� * � * � � , the trend of quadratic order convergence is very evidentas we refine the mesh.

In Fig. 6.1, the errors of the finite volume approximation were plotted (with only halfof the sphere shown). The numerical solutions corresponds to those simulations with a %=� 0 � 0 . It can be observed that the errors distribute very evenly on the sphere, thus, theSCVM with a constant density is already adequate.

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TABLE 6.2Finite volume errors for (6.1) vs. the number of generators � .

a * 1 * � " 7 � 587 � ; � * � * � �162 I 4.038E-02 1.331E-01

II 6.406E-02 1.655E-01642 I 1.021E-02 3.444E-02

II 1.612E-02 4.214E-022562 I 2.556E-03 8.788E-03

II 3.994E-03 1.161E-0210242 I 6.362E-04 2.221E-03

II 1.004E-03 3.072E-0340962 I 1.631E-04 5.269E-04

II 2.375E-04 8.132E-04

TABLE 6.3Finite volume errors for (6.2) vs. the number of generators � .

a * 1 * � " 7 � 587 � ; � * � * � �162 I 4.032E-01 4.313E-00

II 5.962E-01 4.314E-00642 I 1.370E-01 1.178E-00

II 1.241E-01 1.115E-002562 I 2.687E-02 3.115E-01

II 3.033E-02 2.908E-0110242 I 7.445E-03 7.902E-02

II 7.577E-03 7.346E-0240962 I 2.080E-04 1.745E-02

II 2.072E-04 1.797E-02

7. Conclusion. High quality spherical grids have many applications. Many strategieshave already been studied in atmospherical and geophysical simulations [29] for producinggood spherical grids such as those based on the concept of discrete geodesic grids and thosebased on spherical Voronoi-Delaunay tessellations. For global data analysis purposes, ideassuch as the EASE-grid [19] or grids based on Platonic solids have also been explored. Thoughmany of these choices lead to good quality grids, it can be argued that the recently proposedconcept of SCVT [7, 8] in general gives grids which are superior to most of existing ones.By providing both theoretical and computational evidences, our study here on a finite vol-ume approximation of linear convection diffusion equations defined on a sphere based on thespherical centroidal Voronoi meshes demonstrated further the optimality of the SCVT grids.

In the future, we will make further studies on the properties on the SCVT grids such asthe local energy equipartition and hierarchical SCVT grids for multiresolution analysis. Thevalidity of superconvergent gradient recovery estimate 5.5 will be explored through analyticalmeans. We will also study the application of the SCVT grids to more complex physicalproblems.

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FIG. 6.1. Plots of� � ����� on half a sphere. Top row: the example with exact solution (6.1); second row: with

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