Discrete Solutions of Dirichlet Problems, Finite Volumes, and Circle Packings
Discrete Comput Geom
Discrete Solutions of Dirichlet Problems, Finite Volumes, and Circle Packings¤
T. Dubejko 0
0 Department of Mathematics, Northwestern University , Evanston, IL 60208, USA and Mathematiques Discretes, CP 216 , Universite Libre de Bruxelles , B1050 Brussels , Belgium
Convergence results for discrete solutions of Dirichlet problems for Poisson equations are obtained, where discrete solutions are constructed for triangular grids using finite volumes with sides perpendicular to, but not necessarily bisecting, corresponding edges in underlying triangulations. A method, based on properties of circle packings, is described for generating triangular meshes and associated volumes. Also, the approximation of exit probabilities of the Brownian motion by exit probabilities of random walks on circle packings is discussed.

This paper originated from our studies of discrete harmonic functions given by circle
packings. Such functions were introduced in [Du3] (see also [Du1]) to deal with the
type problem for random walks on infinite planar graphs and the type problem for circle
packings. Here we are interested in properties of these mappings, in particular, in their
connections with classical harmonic functions and approximation issues.
Through most of this paper we actually work with a larger family of maps than the
class of discrete harmonic functions given by circle packings. This family can briefly
be described as consisting of piecewise affine functions, defined for triangulations in
the plane, that are solutions of systems of linear equations derived from classical
Poisson equations using finite volumes and integration. Finite volume techniques in solving
differential equations have been studied in the literature for some time now [BR], [Hn],
[Ha], [Ca], [CMM]. The volumes we introduce here (Section 2) are slightly different
from the ones investigated so far, where it was always assumed (explicitly or implicitly)
that the boundaries of volumes cross edges of underlying triangulations at midpoints.
Instead, we require edges of our volumes to be perpendicular to, but not necessarily to
be bisectors of, the corresponding edges in underlying triangulations. Using this type of
volumes, we show in Section 3 that discrete solutions of Poisson equations satisfy the
maximum principle. Furthermore, we prove convergence of discrete solutions for
Dirichlet problems to the corresponding classical solutions in the H 1norm (Theorem 3.5) and
the L2norm (Theorem 3.6) under rather mild restrictions (i.e., regularity) on
triangulations and volumes involved in the process of generating discrete solutions.
In Section 4 we show that if triangulations used to construct discrete solutions are
close to being uniform, then discrete solutions will approximate the classical counterparts
uniformly on compact subsets. This result is proved for both continuous and
discontinuous boundary conditions, however, in the latter case we require some smoothness on
domains involved in Dirichlet problems.
The question of how to generate “good” triangulations and volumes is addressed in
Section 5. There we show that triangulations and volumes induced by circle packings
have all the desired properties, i.e., regularity, provided some combinatorial (but not
geometric) restrictions on tangency patters in circle packings, which, from practical
point of view, are essentially always satisfied (Corollary 5.1).
We also comment on how a Dirichlet problem for a Poisson equation can be pulled
back to a standard domain (e.g., the unit disk) using the discrete Riemann mapping
theorem for circle packings (Theorem 5.2 and Corollary 5.4).
Finally, we prove that random walks induced by circle packings, which were
introduced in [Du3] and [Du1], have a similar behavior to that of Brownian motion by showing
that exit probabilities of a sufficiently dense circle packing filling a domain in the plain
are close to corresponding exit probabilities of Brownian motion in that domain.
2. Triangulations and Volumes
We begin with a description of triangulations and associated volumetriangulations.
Suppose T is a (finite) triangulation of a simply connected domain in the plane R2. We
denote the set of vertices (nodes), edges, and triangles (faces) of T by T 0, T 1, and T 2,
respectively. We also write IT0 and @ T 0 for the sets of interior and boundary vertices of
T . We use @ T for the (geometric) boundary of the set T . Also, the symbol » is used to
denote adjacent elements in T 0 or in T 2.
Now, T ¤ is said to be a volumetriangulation of T in R2 (i.e., a 2cell dual triangulation)
if the following holds: for every triangle t 2 T 2 there is a unique point zt inside it so that
(
1
) zt can be orthogonaly projected on each side of t , and (
2
) if t and t 0 are two adjacent
triangles, then the segment zt zt0 joining zt and zt0 is perpendicular to and intersects the
common side of t and t 0. For z 2 IT0, Vz denotes the volume associated with z, i.e.,
a polygon bounded by edges zt1 zt2 ; zt2 zt3 ; : : : ; ztn zt1 , where t1; : : : ; tn are consecutive
triangles of T with vertex z. If z 2 @ T 0, then Vz is a polygon bounded by edges zzt01 ,
zt01 zt1 , zt1 zt2 ; zt2 zt3 ; : : : ; ztm¡1 ztm , ztm zt0m , zt0m z, where t1; : : : ; tm are consecutive triangles of
T with vertex z (with t1 and tm being boundary triangles), and zt01 (respectively, zt0m ) is
the image point of the orthogonal projection of zt1 (respectively, ztm ) onto the boundary
edge of t1 (respectively, tm ) originated at z (see Fig. 1).
The regularity constant ¾T of a triangulation T (see [Ci]) is defined by
where in.t / is the radius of the inscribed circle of t and diam.t / is the diameter of t . A
large value of ¾T indicates that T has some rather flat triangles. A family of triangulations
fTn g is said to be regular if there exists ¾ such that ¾Tn < ¾ for all n.
Similarly, we define the regularity constant of a volumetriangulation T ¤ by
¾T :D sup
t2T 2 in.t /
diam.t /
;
¾T ¤ :D sup
t2T 2 dist.zt ; @ t /
;
diam.t /
where dist is the distance function. A family of volumetriangulations fTn¤g is said to be
regular if fTn g is regular and ¾Tn¤ < ¾ ¤ for some ¾ ¤ > 0 and all n.
For every pair t and t 0 of neighboring triangles in T there is the volume Vt\t0 associated
with their common edge t \ t 0: if zi and z j are the endpoints of t \ t 0, then Vt\t0 is built
of two triangles 4zt zt0 zi and 4zt zt0 zi (see Fig. 2(a)). For future references, we remark
that if ½t\t0 :D minfin.4zt zt0 zi /; in.4zt zt0 z j /g and fTn¤g is regular, then there exists ¾
such that
dist.zt ; zt0 /
½t\t0
< ¾;
.}/
for every t ; t 0 2 Tn2, t » t 0, and all n.
It is convenient in what follows to use the following notation: for t 2 T 2, jt j denotes the
area of t ; if Vz and Vt\t0 are volumes, then jVzj and jVt\t0 j denote their areas, respectively.
Furthermore, if t; t 0 2 T 2 are adjacent and their common side has the endpoints zi and
zj , then we write zi j :D t \ t 0, zi¤j :D zt zt0 , ½i j :D ½t\t0 , Vi j :D Vt\t0 , jzi j j D jzi ¡ zj j :D
dist.zi ; zj /, and jzi¤j j :D jzt ¡ zt0 j (Fig. 2(b)).
3. Finite Volume Method
In this section we describe a finite volume method. Let Ä be a domain in R2. By
H k .Ä/, 0 · k, we denote the standard kth Sobolev space, i.e., the set of functions
in Ä with finite k ¢ kHk.Ä/ norm, kukHk.Ä/ :D .Pj®j·k RÄ jD®uj2 d x /1=2, where D®u
is a weak derivative of u, and ® is a multiindex (for details, see [GT]). We write
jujHk.Ä/ :D .Pj®jDk RÄ jD®uj2d x /1=2 for the seminorm in H k .Ä/. Furthermore, we
define H 1.Ä/ :D fu : kukL1.Ä/ < 1g, where kukL1.Ä/ denotes the essential supremum
of u in Ä.
We now introduce discrete versions of the above (semi)norms. Let T be a triangulation
of a domain in R2. Denote the set of realvalued functions defined on vertices of T by
60.T /, and the set of continuous functions w: T ! R that are linear on each t 2 T 2
by 61.T /. If w 2 60.T /, then its linear extension is denoted by wb 2 61.T /.
We first introduce the following inner product, and discrete H 0 and supnorms in
60.T /: for u; w 2 60.T /,
.u; w/T :D
X u.z/w.z/jVzj;
z2T 0
kuk02;T :D .u; u/T ;
kuk1;T :D sup ju.z/j:
z2T 0
The discrete H 1 seminorm and norm in 60.T / are defined by
juj12;T :D
X .Di j u/2jVi j j;
zij 2T 1
kuk12;T :D kuk02;T C juj12;T :
where
Di j u :D
u.zj / ¡ u.zi / ;
jzi j j
For future reference we make the following observation:
Remark 3.1. juj12;T D 12 Pzij 2T 1 .jzi¤j j=jzi j j/ju.zi / ¡ u.zj /j2:
Finally, we extend the definitions of the above discrete (semi)norms to functions
defined in T as follows: if w: T ! R, then kuk1;T :D kujT 0 k1;T and juj1;T :D
jujT 0 j1;T , where ujT 0 is the restriction of u to the set T 0. If ! is a subset of T , then
kuk1;! :D supz2!\T 0 ju.z/j.
For functions in 61.T /, the classical and discrete definitions of H 1seminorms are
closely related in the following way (for a proof see, e.g., [BR]):
Proposition 3.2. There exists a constant C D C .¾T / depending only on the regularity
constant ¾T of T such that, for u 2 61.T /,
1
C jujH1.T / · juj1;T · C jujH1.T /:
We now introduce an operator whose domain is H 2.T / [ 61.T / [ 60.T / and the
range is the space of realvalued functions defined over I T 0. If w 2 H 2.T / [ 61.T /,
then
1 Z
AT w.z/ :D ¡ jVzj @Vz 5w ¢ ¡!´ ds
for z 2 IT0;
where ¡!´ denotes the outward unit normal vector on the boundary @ Vz of Vz. If w 2
60.T /, then AT w :D AT wb. We extend the operator AT to AN T : H 2.T / [ 61.T / [
60.T / ! 60.T / by AN T w.z/ :D AT w.z/ if z 2 IT0 and AN T w.z/ :D 0 for z 2 @ T 0.
Suppose that f 2 L2.T /, ' 2 C .@ T /, and u is the solution to the Dirichlet problem
¡4u D f in T and u D ' on @ T . Then the corresponding discrete problem is defined
as follows:
find
w: T 0 ! R
such that
½ AT w.z/ D fT .z/
w.z/ D '.z/
where fT .z/ :D .1=jVzj/ RVz f d x .
Remark 3.3. 1. Notice that the discrete problem defined above is modeled on a classical
approach where a solution of the equation ¡4u D f is found by replacing the differential
equation by the integral condition: ¡.1=jV j/ R@V 5u ¢¡!´ ds D .1=jV j/ RV f d x for every
subset V ½ Ä with Lipschitz boundary.
.?/
2. The discrete problem is a linear problem. In general, if .F; 8/ 2 RjI T 0jCj@T 0j, then
the discrete Dirichlet problem, AT w D F in IT0 and w D 8 on @ T 0, has the following
explicit formulation:
8> 1
<
X jzi¤j j .w.zi / ¡ w.zj // D F .zi /
jVzi j zj »zi jzi j j
>:w.z/ D 8.z/
for zi 2 IT0;
Solutions of the above linear problem have the following important property.
Maximum Principle. If F ¸ 0, then a solution w of .?/ attains its minimum on @ T 0.
In particular, if 8 ¸ 0, then w ¸ 0.
Proof. From the equations in .?/ involving interior vertices and the assumption that
F ¸ 0, it follows that if w attains its global minimum at an interior vertex then w must
be constant, in particular, attaining minimum on @ T 0.
Remark 3.4. 1. From the Maximum Principle one obtains that the linear system of
equations .?/ is always uniquely solvable.
2. The above Maximum Principle can also be derived from a probabilistic
interpretation of equations .?/ as discussed in the last section of this paper.
The next result gives some estimates on an error between the classical solution of a
Dirichlet problem and its discrete counterpart; the result is essentially due to Cai and
coworkers [Ca], [CMM]. Differences are in the assumptions on families of triangulations
and boundary conditions; we do not require sides of volumes in Tn¤ to be bisectors of
sides of triangles in Tn nor do we impose any conditions on angles of triangles of Tn,
and the boundary of Tn does not need to coincide with the boundary of the domain
considered.
Theorem 3.5. Suppose u 2 H 2.Ä/ is a solution of ¡4u D f in Ä, f 2 L2.Ä/.
Let fTng be a regular family of triangulations with Tn µ Ä and ¹n ! 0 as n ! 1,
where ¹n D supt2Tn diam.t /. Assume that fTn¤g is the corresponding family of
volumetriangulations of fTng. Denote by un the discrete solution, in Tn, of
Then
½ ATn w.z/ D fTn .z/;
w.z/ D u.z/;
ju ¡ unj1;Tn · C ¹njujH2.Ä/;
where C is a constant that depends only on the regularity of families fTng and fTn¤g.
Proof. Since the proof requires only minor modifications to the one in [CMM], we
outline here the major steps=differences, and for details the reader is refered to [CMM].
First, we notice that, for w 2 60.Tn /, one has . AN w; w/Tn D jwj21;Tn , that follows from
direct calculations:
. AN w; w/Tn D
D
D
X w.zi /
zi 2I Tn0
X w.zi /
zi 2I Tn0
X w.zi /
zi j 2Tn1 jzi j j
X
zj »zi
¡
Z
z¤
i j
5wb ¢ ¡!´ds
!
X jzi¤j j .w.zi / ¡ w.z j //
zj »zi jzi j j
X jzi¤j j .w.zi / ¡ w.z j //
zj »zi jzi j j
!
!
D 12 X jzi¤j j .w.zi / ¡ w.z j //2 D jwj21;Tn :
Second, because u 2 H 2.Ä/, the Sobolev embedding theorem implies that u 2 C .Ä/
(and u 2 C . ÄN/ if Ä has the exterior cone property (see [Ad] and [GT])). Next, define
en :D u ¡ un and enI :D u ¡ u nI , where u nI :D ujOTn0 , i.e., u nI is the linear interpolant of
un over Tn . Then, from the definition of discrete solutions and the fact that un D u nI on
@ Tn0, we obtain
jen j12;Tn D . AN en ; en /Tn D
X .en .z j / ¡ en .zi //
zi j 2Tn1
0
X jzi¤j j Ã Z
zi j 2Tn1 jzi j j
¡
z¤
i j
0
X jzi¤j j Ã Z
zi j 2Tn1 jzi j j
¡
z¤
i j
5enI ¢ ¡!´ds
5enI ¢ ¡!´ds
!211=2
A
:
!211=2
A
;
0
X
zi j 2Tn1
Now, from the regularity of families fTn g and fTn¤g, and the property .}/, we obtain
(exactly the same way as in Lemma 3 of [CMM])
¯
¯Z
¯
¯¯ zi¤j
¯
5enI ¢ ¡!´ds¯¯¯ · C jzi j j5=2jzi¤j j1=2½i¡j 2jujH 2.Vi j /;
¯
where C is a constant depending only on the regularity of families fTn g and fTn¤g. Hence
C 2¾ 4jzi j j2juj2H 2.Vi j /A
· C ¾ 2¹n jujH 2.Ä/;
11=2
where ¾ is a constant as in .}/.
We now investigate the L2convergence of discrete solutions. Suppose Ä is a
Jordan domain and ' is a continuous function on @Ä. Suppose, further, that there is a
neighborhood Ä" ½ Ä of @Ä and a method for construction of a continuous function
'N: Ä" [ @Ä 7! R such that 'N D ' on @Ä. (From Tietze’s theorem, we know that
such an extension always exists, however, it may be hard to construct it in a manageable
way.) For example, when @Ä is C 2 (i.e., a twice continuously differentiable curve) then
there exists " such that when dist.z; @Ä/ < " then there is a unique point z@ 2 @Ä with
dist.z; z@ / D dist.z; @Ä/, and 'N can be defined by a projection, i.e., 'N.z/ :D '.z@ /.
Let f 2 L2.Ä/ and let u be the solution of the Dirichlet problem
½¡4u D f
u D '
If T is a triangulation with the boundary as a Jordan curve, and T µ ÄN, @ T ½ Ä" [ @Ä,
and there is an associated volumetriangulation T ¤, then we define the corresponding
approximate solution uT of the above continuous problem by
.¤/
.¤¤/
½ AN T uT .z/ D fT .z/
uT .z/ D 'N.z/
i.e., the system of equations .¤¤/ is just a generalization of the earlier definition to a case
where the boundary of T does not coincide with that of Ä.
The following result addresses a question of the L2convergence of discrete solutions
to the classical one.
Theorem 3.6. Let Ä be a Jordan domain with C 2boundary, ' 2 C .@Ä/, and f 2
L2.Ä/. Suppose 'N is a continuous extension of ' to an inside neighborhood of @Ä.
Denote by u the solution to the Dirichlet problem .¤/. Assume that fTng is a family of
triangulations such that, for each n, Tn is simply connected, Tn µ ÄN, Tn ! Ä as n ! 1
(i.e., sets Tn exhaust Ä), and ¹n ! 0, where ¹n is the mesh size of Tn. Suppose, further,
that fTn¤g is an associated family of volumetriangulations, which is regular. For each n,
let un be the discrete solution of .¤¤/ in Tn. Then limn!1 ku ¡ unkL2.Tn/ D 0.
Before we give a proof, we make a few remarks.
Remark 3.7. 1. The above result is true for any Jordan domain, not necessarily with
a C 2boundary. By adopting techniques used in the next section together with the proof
below one can give a proof of the general case. However, it should be noted that as @Ä
gets more bizarre, it is much harder to get a good construction for an extension map 'N.
2. The above result can also be extended to Dirichlet problems for nonsimply
connected domains.
3. The sequence of maps un is bounded by the Maximum Principle. We conjecture
that it is in fact a locally equicontinuous family of mappings. If this is true then fung
forms a normal family, and hence the un’s converge uniformly on compacta of Ä to
some continuous function uQ. Then it would follow from the above theorem that uQ must
be equal to u, i.e., un ! u uniformly on compact subsets of Ä.
Proof of Theorem 3.6. Let " > 0 and let Ã 2 C 1.@ Ä/ such that jÃ ¡ 'j1;Ä < ".
Write uQ for the solution of .¤/ with boundary condition Ã . Denote by uQ nI the linear
interpolant of uQ in Tn , i.e., uQ nI :D uQ djTn0 . Also, denote by uQ n the solution of .¤¤/ in Tn with
boundary condition ujTn0 . Then
n n
ku ¡ un kL2.Tn/ · ku ¡ uQ kL2.Tn/ C kuQ ¡ uQ I kL2.Tn/ C kuQ I ¡ uQ n kL2.Tn/ C kuQ n ¡ un kL2.Tn/: .1/
We are going to give estimates on terms on the righthand side of the above inequality.
The smoothness assumptions on @ Ä and Ã imply [GT, Theorem. 8.12] that uQ 2 H 2.Ä/.
Recall that if a function w 2 H 1.Ä/ is such that wj@Ä D 0, then the following Poincare´
inequality [GT] holds:
kwkL2.Ä/ ·
µ jÄj ¶1=2
¼
jwjH 1.Ä/:
n
Because uQ I ¡ uQ n D 0 on @ Tn , from the Poincare´ ineqality, Theorem 3.5, and
Proposition 3.2, it follows that
n
kuQ I ¡ uQ n kL2.Tn/ ·
un
j Q I ¡ uQ n jH 1.Tn/ · C
un
j Q I ¡ uQ n jH 1.Tn/
µ jÄj ¶1=2
¼
µ jTn j ¶1=2
¼
· C
µ jÄj ¶1=2
¼
¹n juQ jH 2.Ä/:
for large n. Finally, from uQ 2 H 2.Ä/ it follows [Ca, Theorems 3.1.6 and 3.2.1] that
kuQ n ¡ un kL2.Tn/ · 2"jÄj;
n
lim kuQ ¡ uQ I kL2.Tn/ D 0:
n!1
Since @ Ä is C 2, we get [GT, Theorem 9.30] that u; uQ 2 C .ÄN /. As u ¡ uQ is a harmonic
function, we obtain that ju ¡ uQ j1;Ä · ju ¡ uQ j1;@Ä · ". Thus
ku ¡ uQ kL2.Tn/ · ku ¡ uQ kL2.Ä/ · jÄj":
The definitions of uQ n and un together with the fact that ju ¡ uQ j1;Ä · " imply that
juQ n ¡ un j1;@Tn · 2" for all large n. By applying the Maximum Principle to discrete
solutions uQ n and un , we obtain that juQ n ¡ un j1;Tn · " for all large n. Hence
Thus, by combining (
1
)–(5) we obtain the assertion of the theorem.
L1Convergence
In this section we show that, under some additional conditions on families of
triangulations involved in the construction of discrete solutions, we obtain convergence in
supnorm on compact subsets.
.2/
.3/
.4/
.5/
Let fTng and fTn¤g be regular families of triangulations and volumetriangulations.
Throughout this section we assume in addition that fTng is quasiuniform, i.e., there
exists a constant ¾ such that
supt2Tn diam.t /
inft2Tn diam.t / · ¾;
for every n (see [Ci]). As in the previous section, suppose Ä is a Jordan domain, f 2
L2.Ä/, and ' 2 C .Ä/. We assume that we also have a continuous map 'N defined inside
Ä in some neighborhood of @Ä, which is an extension of '. Let u be the solution of .¤/.
We denote by un the discrete solution of .¤¤/ for T :D Tn. Then the main result is the
following theorem.
Theorem 4.1. If fTng and fTn¤g are regular families of triangulations and associated
volumetriangulations, fTng is quasiuniform, the sets Tn exhaust Ä from inside, and
¹n D supt2Tn diam.t / ! 0, then the sequence of maps un converges uniformly on
compact subsets of Ä to u.
Remark 4.2. 1. As we have pointed out in Remark 3.7, we believe that the conclusion
of the theorem is true without quasiuniform condition on fTng. We hope to resolve this
issue in a sequel.
2. There are related results that address the convergence in supnorm in Chapter 3.3
of [Ci] and in [Hn]. The main differences are that the boundaries of Tn’s are not that
rigorously associated with @Ä here as they are in [Ci] and [Hn], and the volumes here
are different from the ones in [Hn]. This allows for consideration of a broader class of
domains but yields loss in estimates for the rate of convergence.
The proof of Theorem 4.1 is given in a sequence of lemmas, where the assertion of
the theorem is first proved for C 2domains and then the general case is split into two
parts: the case of harmonic solutions and the case with zero boundary condition.
Lemma 4.3. Suppose Ä is a C 2domain, f 2 L2.Ä/, and ' 2 C .Ä/. Then under the
assumptions of Theorem 4.1, ku ¡ unk1;Tn ! 0.
Proof. Let " > 0, and let '" 2 C .R2/ be such that k' ¡ '"kL1.@Ä/ < ". Define u" to
be the solution of .¤/ with boundary condition '". Then, since '" 2 C .R2/, it follows
[GT, Theorem 8.12] that u" 2 H 2.Ä/ \ C .ÄN /. Let u"n be the discrete solution of .¤¤/ in
Tn for boundary condition u"n.z/ D u".z/, z 2 @ Tn0. By applying the Maximum Principle
to the discrete solutions un and u"n, and to the classical solutions u and u", we obtain the
following inequalities:
ku ¡ unk1;Tn · ku ¡ u"k1;Tn C ku" ¡ u"nk1;Tn C ku"n ¡ unk1;Tn
· ku ¡ u"kL1.Ä/ C ku" ¡ u"nk1;Tn C ku"n ¡ unk1;@Tn
D k' ¡ '"kL1.@Ä/ C ku" ¡ u"nk1;Tn C ku"n ¡ unk1;@Tn
· " C ku" ¡ u"nk1;Tn C k'" ¡ 'Nk1;@Tn :
Since 'N is a continuous extension of ', and @ Tn ! @Ä as n ! 1, we get that
limn!1 k'N ¡ '"k1;@Tn < 2". To give an estimate on the term ku" ¡ u"nk1;Tn we need
the following result which is due to Oganesyan nad Rukhovets [OR, pp. 74–77]: there
exists a constant C D C .¾ /, depending only on ¾ , such that, for every triangulation
T that is regular and quasiuniform with corresponding constants no bigger that ¾ and
every w 2 61.T / with wj@T D 0, we have kwkL1.T / · C j log ¹T j1=2kwkH1.T /, where
¹T :D supt2T diam.t /. Now, from Theorem 3.5, the Poincare´ inequality, Proposition 3.2,
and the above result, we obtain
ku" ¡ u"nk1;Tn D kudj"Tn0 ¡ ub"nkL1.Tn/ · C j log ¹nj1=2kudj"Tn0 ¡ ub"nkH1.Tn/
· C 0j log ¹nj1=2judjTn0 ¡ ub"njH1.Tn/ · CQ j log ¹nj1=2ju" ¡ u"nj1;Tn
"
· CQ j log ¹nj1=2¹nju"jH2.Ä/;
where C , C 0, CQ , and CQ 0 are just constants independent of u" or the mesh size of Tn. Thus
limn!1 ku" ¡ u"nk1;Tn D 0. Hence limn!1 ku ¡ unk1;Tn · 3", and as " is arbitrary,
this completes the proof.
We now look into the harmonic part of the solution u.
Lemma 4.4. Suppose Ä is a Jordan domain, f ´ 0, and ' 2 C .Ä/. Then under the
assumptions of Theorem 4.1, ku ¡ unk1;Tn ! 0.
estimates:
Proof. Denote by ¿ : Ä ! D a Riemann mapping, where D is the unit disk. Let fÄ"g be
a sequence of C 2 Jordan domains such that ÄN ½ Ä", ÄN"0 ½ Ä" for "0 < ", T Ä" D Ä,
and the boundary of Ä" converges to @Ä in the sense of Fre´chet (see p. 27 of [LV]
and [Wa]) as " ! 0. We define ¿": Ä" ! D to be the Riemann mapping such that
¿".¿ ¡1.0// D 0 and ¿".¿ ¡1. 12 // > 0. Then ¿" ! ¿ uniformly in ÄN (see [Wa] or [Du2]),
and hence ¿ ¡1 ± ¿" ! i d uniformly in ÄN .
Let u" :D u ± ¿ ¡1 ± ¿" : Ä" ! R. Then 4u" D 0 in Ä", u" 2 H 2.Ä/, and
ku ¡ u"kL1.ÄN / ! 0 as " ! 0.
u"n.Lze/t Du"n ub"e.zt/hefodriszcr2ete@sTon0lu.tUiosninogf t.h¤e¤/MianxTimnuwmithPrfiTnnci´ple0 wanedobbotauinndtahrey fcoolnlodwitiinogn
" " " "
ku ¡ unk1;Tn · ku ¡ u k1;Tn C ku ¡ unk1;Tn C kun ¡ unk1;Tn
" " " "
· ku ¡ u k1;Tn C ku ¡ unk1;Tn C kun ¡ unk1;@Tn
· ku ¡ u"kL1.@Ä/ C ku" ¡ u"nk1;Tn C ku"n ¡ u"k1;@Tn
C ku" ¡ uk1;@Tn C ku ¡ unk1;@Tn
· 2ku ¡ u"kL1.@Ä/ C 2ku" ¡ u"nk1;Tn C ku ¡ unk1;@Tn :
Let ± > 0. Recall that 'N is a continuous extension of ' near @Ä. Hence 'N ¡ u is a
continuous function in some neighborhood of @Ä, inside of Ä, and 'N ¡ u D 0 on @Ä.
Hence, there is some neighborhood of @Ä such that, for any point z in this neighborhood,
j'N.z/ ¡ u.z/j < ±. Therefore, from the boundary condition on un’s it follows that, for
all sufficiently large n, ku ¡ unk1;@Tn < 2±. Next, we choose "± to be small enough so
that ku"± ¡ ukL1.@Ä/ < ±; this is guaranted by the fact that ku ¡ u"kL1.ÄN / ! 0. Then
Lemma 4.2 and the fact that @Ä"± is C 2 imply that limn!1 ku"± ¡ u"n± k1;Tn D 0. Hence
we obtain that limn!1 ku ¡ unk1;Tn · 5±.
In the next result we deal with the case of homogeneous boundary data.
Lemma 4.5. Suppose Ä is a Jordan domain, f 2 L2.Ä/, and ' ´ 0. Then under the
assumptions of Theorem 4.1, un ! u uniformly on compacta of Ä.
Proof. Let fÄ"Cg and fÄ"¡g be sequences of C 2 Jordan domains such that ÄN ½ Ä"C,
ÄN"C0 ½ Ä"C for "0 < ", T Ä"C D Ä, and ÄN ¾ Ä"¡, ÄN "¡0 ¾ Ä"¡ for "0 < ", S Ä¡ D Ä.
"
Denote by fN the extension of f which is 0 in R2nÄ. We write u"C for the solution
¡4u"C D fN in Ä"C and u"C D 0 on @Ä"C. Similarly, we denote by u"¡ the solution
¡4u"¡ D f in Ä"¡ and u"¡ D 0 on @Ä"¡.
We also introduce the corresponding discrete solutions as follows. Let u"C be the
n
solution of .¤¤/ in Tn with boundary condition u"nC.z/ D u"C.z/ for z 2 @ Tn0. Denote by
Tn;" the “intersection” of Tn with Ä"¡, i.e., the largest part of Tn contained in Ä"¡ which is
still a triangulation of a simply connected domain. Then let u"n¡ be the solution of .¤¤/
in Tn;" with boundary condition u"n¡.z/ D 0 for z 2 @ Tn0;".
Suppose first that 0 · f . By applying the Maximum Principle to classical solutions
we get
u · u"C in Ä;
u"¡ · u in Ä¡:
"
and
and
u"n¡.z/ · un.z/ for z 2 Tn0;"I
.\/
Similarly, in the discrete setting we have
un.z/ · u"nC.z/ for z 2 Tn0;
to obtain the first inequality above, we have used that u · u"C in Ä and u"nC D u"C on
@ Tn0.
Since fu"¡g is an increasing sequence of functions as " & 0, and u"¡ ¡ u are
harmonic, from Harnack’s theorem we get that u"¡ ! u uniformly on compact subsets
of Ä. Similarly, as fu"Cg is a decreasing sequence of functions as " & 0, and u ¡ u"C
are harmonic, we obtain that u"C ! u uniformly on compacta of Ä.
From Lemma 4.3, we have that, for a fixed ", ku"¡ ¡ u"n¡k1;Ä"¡ ! 0 as n ! 1.
Also from Lemma 4.3, the fact that, for a fixed ", u"C 2 H 2.Ä/, and that u"nC D u"C on
@ Tn0, we obtain that limn!1 ku"C ¡ u"C
n k1;Tn D 0.
Now, let ! be a compact subset of Ä and let ± > 0. If n is large enough so that ! ½ Tn,
then from .\/ we have
un.z/ ¸ u"n¡.z/ D .u"n¡.z/ ¡ u"¡.z// C .u"¡.z/ ¡ u.z// C u.z/
and
for every z 2 ! \ Tn0. By choosing first "± so that ku"±¡ ¡ ukL1.!/ < ± and ku"±¡ ¡
ukL1.!/ < ±, and then N D N .±; "±/ large enough so that, for all n ¸ N , ku"±¡.z/ ¡
u"±¡.z/k1;! < ± and ku"±C.z/ ¡ u"n±C.z/k1;! < ±, we get
n
Since ! and ± are arbitrary, this shows that un ! u uniformly on compact subsets of Ä
in the case when f ¸ 0. By symmetry, the same is true for f · 0, and the general case
follows.
We can now prove Theorem 4.1.
Proof of Theorem 4.1. Let uh be the solution of .¤/ for f ´ 0, and let uo be the solution
of .¤/ for ' ´ 0. From the uniqueness of solutions it follows that u D uh C uo.
The same is true for discrete solutions. If unh denotes the solution of .¤¤/ for f ´ 0,
and uon denotes the solution of .¤¤/ for ' ´ 0, then un D unh C uno. Now, the convergence
un ! u is an immediate consequence of Lemmas 4.3 and 4.4.
Theorem 4.1 can, of course, be extended to include discontinuous boundary
conditions. However, as more general cases are considered, it is getting much harder to define
in a “practical” way boundary conditions for discrete solutions. We finish this section
with a result related to discontinuous boundary conditions, which is applied in the next
section.
Example 4.6. Suppose Ä is a C 2 Jordan domain, f 2 L2.Ä/, and ' D Â° , where ° is
an arc in @Ä and Â° : @Ä ! f0; 1g is the characteristic function of ° (i.e., Â° .z/ is equal
to 1 if z 2 ° and 0 otherwise). Denote by u the solution of .¤/ with the above data. Let
fTng and fTn¤g be as in Theorem 4.1. Write un for the discrete solution of .¤¤/ in Tn with
the boundary condition un.z/ D Â° .z@ /, z 2 @ Tn0, where z 7! z@ is the projection of
@ Tn0 to @Ä defined earlier for C 2domains. Then un ! u uniformly on compact subsets
of Ä.
Proof. The proof is similar to that of Lemma 4.4. Let '"C; '"¡ 2 C .@Ä/ be such that
'"¡ · Â° · '"C, and the linear measure of sets fz 2 @Ä : j'"C.z/ ¡ Â° .z/j C j'"¡.z/ ¡
Â° .z/j > 0g goes to 0 as " ! 0. In other words, '"C and '"¡ are two continuous “step”
functions on @Ä that approximate Â° from above and below, respectively.
We define u"C to be the solution of ¡4u"C D f in Ä and u"C D '"C on @Ä. Similarly,
we write u"¡ for the solution of ¡4u"¡ D f in Ä and u"¡ D '"¡ on @Ä. Then, from
Harnak’s theorem, it follows that u"C ! u and u"¡ ! u uniformly on compact subsets
of Ä as " ! 0.
Now let u"nC and u"n¡ be corresponding discrete solutions of .¤¤/ in Tn with boundary
conditions u"nC D '"C on @ Tn0 and u"n¡ D '"¡ on @ Tn0, respectively. Then, from the
Maximum Principle for discrete solutions, we obtain
un.z/ ¸ u"n¡.z/ D .u"n¡.z/ ¡ u"¡.z// C .u"¡.z/ ¡ u.z// C u.z/
and
un.z/ · u"nC.z/ D .u"nC.z/ ¡ u"C.z// C .u"C.z/ ¡ u.z// C u.z/;
for z 2 Tn0. As in the proof of Lemma 4.5, it now follows from the above inequalities and
convergence of their terms in brackets to 0 on compact subsets, that, for every compact
subset ! of Ä, ku ¡ unk1;! ! 0 as n ! 1.
5. Circle Packings and Random Walks
As we mentioned in the Introduction, this paper was motivated by the results in [Du1]
and [Du3], where discrete harmonic functions for circle packings were introduced. In
this section we discuss connections among circle packings, volumetriangulations, and
random walks. We show how to generate triangulations and associate with them volumes
for domain approximation by means of circle packings. We also describe how a Dirichlet
problem from a reasonable domain can be pulled back to a standard domain, such as the
unit disk, using the discrete Riemann mapping theorem.
We begin with a definition of circle packings (see also [BeS1], [BoS], [Du1], and
[RS]). Let K be a simplicial 2complex that is simplicially isomorphic to a triangulation
of a closed disk in R2. We assume that K carries an orientation (induced, for example,
from R2). Denote by K0, I K0, @K0, K1, and K2 the sets of vertices, interior vertices,
boundary vertices, edges, and faces of K, respectively. A collection P D fCP .³ /g³ 2K0
of circles in R2 is said to be a circle packing for K if for every face h³1; ³2; ³3i in K with
the vertices ³1, ³2, and ³3, listed in positive order, hCP .³1/; CP .³2/; CP .³3/i is a triple of
mutually and externally tangent circles in R2 listed in positive order (in R2) (see Fig. 3).
We remark here that for any K there is a continuum family of associated circle packings,
and any of such packings is uniquely determined by values of radii of boundary circles
up to isometries (see [Du1], [BeS2], and [CdV]).
Assumption. Unless stated otherwise, we assume from now on that if P is a circle
packing, then all circles in P have disjoint interiors.
If P is a circle packing for K, then the carrier carr.P/ of P is the collection
fhsP .³1/; sP .³2/; sP .³3/i : h³1; ³2; ³3i 2 K2g of triangles in R2, where sP .³ / denotes
the center of the circle in P associated with vertex ³ 2 K0. It follows from our
assumption about disjointness of interiors of circles that carr.P/ is in fact a (piecewise linear)
triangulation of a simply connected domain in R2, and it is simplicially isomorphic to
the complex K.
We now describe the volumetriangulation carr.P/¤ that corresponds to the
triangulation carr.P/. To do this, it is sufficient to define a point zt for every triangle t in carr.P/.
If t D hsP .³1/; sP .³2/; sP .³3/i, then we define zt to be the radical center of circles
CP .³1/, CP .³2/, and CP .³3/. (For more information, the reader is refered to [Du3], [Co],
and [Ya].) Equivalently, zt can be described as the center of the inscribed circle of t (see
Fig. 4(a)). Then the volume Vz, z D sP .³ /, is a polygon circumscribed on CP .³ /, as in
Fig. 4(b).
We now address the regularity issues concerning triangulations and volumes generated
by circle packings. First, we define the degree deg.T / of a triangulation T as the least
upper bound on the number of edges coming out of any vertex in T . The degree deg.P/ of
a packing is then defined by deg.P/ :D deg.carr.P//. The key result regarding regularity
is the Ring Lemma [RS], which implies the following circle packing regularity.
CPRegularity. There exists a constant · D ·.d/, depending only on d, such that for
any circle packing P with deg.P/ · d,
radius.CP /
radius.CP0 / · ·
for every pair of adjacent circles CP and CP0 that are also interior.
In the above, a circle of P is called interior if its center is an interior vertex of carr.P/,
otherwise it is called a boundary circle.
Since the length of an edge in carr.P/ is the sum of radii of two circles centered at the
two endpoints of the edge, and since the radius of the inscribed circle of a triangle can
be explicitly computed (see [Du3]) from lengths of its edges, the following conclusion
is immediately given by CPregularity.
Corollary 5.1. For a circle packing P, let carr.P±/ denote the triangulation obtained
from the triangulation carr.P/ by removing all triangles having at least one boundary
vertex. Then carr.P±/ and the associated volume triangulation carr.P±/¤ have their
regularity constants depending only on the degree of P.
As circle packings can be quite easily generated (see [St3]) once a tangency pattern
is given (i.e., a simplicial complex K), the above result shows that triangulations and
volumes that are regular can also be delivered. In particular, approximation results from
earlier sections can be applied.
We define the quasiuniformity constant of a circle packing P as the least upper bound
on the ratio radius.CP /=radius.CP0 / for any two circles CP and CP
0 of P. We say that
a collection of circle packings fPng is regular (respectively, quasiuniform) if degrees
(quasiuniformity constants) of packings Pn’s are all uniformly bounded. If this is the
case, then it follows that the corresponding families of triangulations fcarr.Pn±/g and
fcarr.Pn±/¤g are regular (quasiuniform). From the results of Section 3 and 4 we obtain
the following:
Theorem 5.2. Let Ä be a Jordan domain. Suppose that fPng is a collection of circle
packings contained in Ä such that
(
1
) radii of circles in Pn go to 0 as n ! 1,
(
2
) there is a constant d > 0 such that deg.Pn/ · d for all n, and
(
3
) carriers carr.Pn/ exhaust Ä.
Denote by u the solution of the Dirichlet problem ¡4u D f in Ä and u D ' on @Ä,
where f 2 L2.Ä/ and ' 2 C .@Ä/. Suppose 'N is a continuous extension of ' to some
neighborhood of @ Ä. Write un for the corresponding discrete solutions for triangulations
carr.Pn±/ and volumes carr.Pn±/¤. Then ku ¡un kL2.carr.Pn±// ! 0 as n ! 1. Furthermore,
if fPn g is a quasiuniform family, then un ! u uniformly on compacta of Ä as n ! 1.
Remark 5.3. Notice that “traditional” conditions, such as bounds on angles of triangles
in grids [BR], [Ca], [Ci] to ensure the regularity of grids and additional restrictions on
these angles [CMM, (3.5)], [Hn, Section 2.3] to ensure the regularity of volumes, are
replaced in the above theorem by a single combinatorial condition, i.e., every vertex has
no more than d neighbors. (This combinatorial condition is also closely linked with the
assumption that circles in packings have disjoint interiors).
We now recall a result about the convergence of discrete Riemann mappings given by
circle packings. Suppose that Ä is a Jordan domain. Let a; b 2 Ä be two points. Suppose
fPn g is a collection of circle packings satisfying conditions (
1
)–(
3
) of Theorem 5.2.
Denote by D the unit disk in R2. From the Andreev–Koebe–Thurston theorem [An],
[Th1] it follows that for each n there exists a circle packing PQn contained in D, with
all boundary circles internally tangent to @ D, whose carrier is simplicially isomorphic
to that of Pn . Moreover, PQn is normalized so that if a circle in Pn contains the point a,
then the corresponding circle in PQn is centered at 0, and if a circle in Pn contains the
point b, then the corresponding circle in PQn is centered in the .0; 1/ interval. Let ¿n be
a piecewise linear map ¿n : carr.Pn / ! D that maps the center of a circle in Pn to the
center of the corresponding circle in PQn (see Fig. 5). Also, let ¿n# be a piecewise linear
map ¿n#: carr.Pn / ! .0; 1/ whose value at the center of a circle in Pn is the ratio of the
radius of the corresponding circle in PQn to the radius of the circle in Pn . Then we have
the following theorem (see [HR], [HS], [RS], [St1], [St2], and [Th2]), where ¿ 0 denotes
the complexvariable derivative of ¿ .
Discrete Riemann Mapping Theorem. The sequence of maps ¿n converges uniformly
on compact subsets of Ä to the Riemann mapping ¿ : Ä ! D with ¿ .a/ D 0 and ¿ .b/ > 0.
Moreover, ¿n# converge uniformly on compacta of Ä to j¿ 0j.
The next observation is a straightforward consequence of the above approximation
result and Theorem 5.2.
Corollary 5.4. Let fQ 2 L2.D/ and 'Q 2 C .@D/. Denote by uQ the solution of the
Dirichlet problem
½¡4uQ D fQ
uQ D 'Q
.¤/
f
Let Ä be a Jordan domain and let ¿ : Ä ! D be a Riemann mapping. Write u for the
solution of the Dirichlet problem ¡4u D f in Ä and u D ' on @Ä, where f .z/ :D
j¿ 0.z/j2 fQ.¿ .z// and '.z/ :D 'Q.¿ .z//. Suppose that fPng is a collection of circle packings
contained in Ä and satisfying (
1
)–(
3
) of Theorem 5.2. Let fPQng be an associated family
of circle packings in D such that the corresponding maps ¿n: carr.Pn/ ! carr.PQn/
and ¿n# converge uniformly on compacta of Ä to ¿ and j¿ 0j, respectively. Write uQn for
the discrete solution of .f¤/ for triangulations carr.PQn±/ and volumes carr.PQn±/¤. Then
ku ¡ uQn ± ¿nkL2.carr.Pn±// ! 0 as n ! 1. Furthermore, if fPQng is a quasiuniform family,
then uQn ± ¿n ! u uniformly on compact subsets of Ä as n ! 1.
We now move to random walks on circle packings. For details on the subject of
random walks in general, the reader should consult, for example, [So] or [Wo]. The
notion of random walks on circle packings was introduced in [Du3], and we recall it
briefly here. We first define a conductance along an edge. If P is a circle packing for K
and ³; ³ 0 2 K0, ³ » ³ 0, then sP .³ /sP .³ 0/ is an edge in carr.P/, and the conductance
induced by P along this edge is defined by
EP .³; ³ 0/ :D
jzt ¡ zt0 j
jsP .³ / ¡ sP .³ 0/j
;
where t and t 0 are two triangles in carr.P/ with the common edge sP .³ /sP .³ 0/ and, as
before, zt and zt0 denote the centers of inscribed circles in triangles t and t 0, respectively.
Then the transition probability from a vertex sP .³ / to another vertex is defined by
QP .³; ³ 0/ :D < P³ 00»³ EP .³; ³ 00/
:0
8
EP .³; ³ 0/
for ³ » ³ 0;
for ³ ¿ ³ 0:
Because P³ 0»³ QP .³; ³ 0/ D 1 for every interior vertex ³ , the matrix QP is a probability
matrix. We refer to the random walk given by the matrix QP as the random walk induced
by the packing P.
Next, it is standard to introduce the Laplace operator for a random walk by
LP u :D .I ¡ QP /u;
where I is the identity matrix and u is a realvalued function defined on the set of vertices
of carr.P /. A function u is said to be harmonic (with respect to the random walk) if
L P u D 0 at every interior vertex. In other words, a function is harmonic if its value
at any interior vertex is equal to the weighted average of its values at the neighboring
vertices.
Recall that, in Section 3, for a triangulation T we have defined an operator AT . For
T D carr.P /, we write AP for AT . By comparing the definitions of operators L P and
AP we easily get
Proposition 5.5. Let u: carr.P /0 ! R. Then L P u.z/ D 0 for every interior vertex
'z:if@acnadrro.nPly/0if!APRu, .tzh/e DDir0icfholreetvperroyblienmterfioorrthveerrtaexndzo. mFuwrathlkerimndourec,edfobryaPny function
and the discrete Dirchelet problem
½L P u D 0
u D '
½ AP u D 0
u D '
in I carr.P /0;
on @ carr.P /0;
have the same solution.
We apply the approximation results from Section 3 to obtain some information on
exit (hitting) probabilities for random walks induced by circle packings. Recall that if X
is a subset of the boundary vertices @ carr.P /0 and z 2 I carr.P /0, then the probability
MP .z; X / that the random walk (given by P ) starting at z will reach a boundary vertex
for the first time and such a vertex will be in X , is called the exit probability from z
through X . It follows that, for a fixed z 2 I carr.P /0, MP .z; ¢/ is a probability measure
on @ carr.P /0.
A similar notion is available in the continuous case. If Ä is a domain, z 2 Ä, and
X ½ @ Ä, then the probability M .z; X / that a Brownian particle starting at z will leave the
set Ä for the first time through the set X is called the exit probability from z through X .
The next result shows that random walks induced by circle packings mimic the Brownian
motion, and that they can be used to estimate exit probabilities of the Brownian motion.
Theorem 5.6. Let Ä be a C 2 Jordan domain. Let ° be an arc in @ Ä. Suppose fPn g is a
quasiuniform family of circle packings that exhaust Ä (i.e., (
1
)–(
3
) of Theorem 5.2 are
satisfied). Denote by °n the set fz 2 @ carr.Pn /0 : z@ 2 ° g, where, as before, z@ denotes
the nearest point on @ Ä to the point z. Then, for any compact subset ! of Ä,
lim sup jMPn .z; °n / ¡ M .z; ° /j D 0:
n!1 z2!
Proof. Let un be the solution of the Dirichlet problem: L Pn un .z/ D 0 for z 2 I carr.Pn /0
and un .z/ D 1 if z 2 °n and un .z/ D 0 if z 2 @ carr.Pn /0n°n . Then un .z/ D MPn .z; °n /
for every z 2 I carr.Pn /0 [DS], [KSK]. Similarly, if u is the solution of the classical
Dirichlet problem 4u D 0 in Ä and u.z/ D 1 if z 2 ° and u.z/ D 0 if @ Än° , then
u.z/ D M .z; ° / [KS]. Since un is also the solution of the corresponding Dirichlet
problem for the operator APn by Proposition 5.5, the assertion of the theorem now follows
from Example 4.6.
We conclude this paper with some final remarks.
Remark 5.7. 1. Once again, it should be observed that if we had that the discrete
solutions to a Dirichlet problem converge uniformly on compact subsets, regardless of the
quasiuniform condition, then such a condition could be removed from the assumptions
in the above theorem.
2. The results of this section can easily be extended to circle packings with
overlaps (see [Du3]). Volumes for such circle packings are defined exactly the same as for
circle packings without overlaps, that is corners of volumes (i.e., vertices of the dual
triangulation) are going to be radical centers of triples of circles. However, volumes
will no longer be circumscribed on circles of underlying packings. Nevertheless, by
keeping angles of overlaps away from ¼=2, a bound on the degree will imply regularity
for packings and corresponding volumes. Also, the issue of quasiuniformity extends
without any changes. By allowing for overlaps in packings we add more flexibility to
the construction of triangulations and the volumes associated with them.
3. The reader may also be interested in the results [CdVM], [Du2], and [Ma]. As
was shown in Section 4(
2
) of [Du2], the ratio maps for hexagonal triangulations given
by solutions of a Dirichlet problem for radius functions of circle packings converge
uniformly on compacta to the classical solution of the Dirichlet problem.
Acknowledgments
The author would like to thank Prof. Francis Buekenhout for his hospitality. Figures 3
and 5 were created using CirclePack [St3].
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