Discrete Solutions of Dirichlet Problems, Finite Volumes, and Circle Packings

Discrete & Computational Geometry, Jul 1999

Abstract. 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.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1007%2FPL00009447.pdf

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 , B-1050 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 1-norm (Theorem 3.5) and the L2-norm (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 volume-triangulations. 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 volume-triangulation of T in R2 (i.e., a 2-cell 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 volume-triangulation 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 volume-triangulations 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 multi-index (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 real-valued 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 sup-norms 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 1-seminorms 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 real-valued 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 à Z zi j 2Tn1 jzi j j ¡ z¤ i j 0 X jzi¤j j à 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 L2-convergence 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 volume-triangulation 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 L2-convergence of discrete solutions to the classical one. Theorem 3.6. Let Ä be a Jordan domain with C 2-boundary, ' 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 volume-triangulations, 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 2-boundary. 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 à 2 C 1.@ Ä/ such that jà ¡ 'j1;Ä < ". Write uQ for the solution of .¤/ with boundary condition à . 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 right-hand side of the above inequality. The smoothness assumptions on @ Ä and à 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. L1-Convergence 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 sup-norm on compact subsets. .2/ .3/ .4/ .5/ Let fTng and fTn¤g be regular families of triangulations and volume-triangulations. Throughout this section we assume in addition that fTng is quasi-uniform, 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 volume-triangulations, fTng is quasi-uniform, 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 quasi-uniform condition on fTng. We hope to resolve this issue in a sequel. 2. There are related results that address the convergence in sup-norm 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 2-domains 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 2-domain, 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 quasi-uniform 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 ° , where ° is an arc in @Ä and ° : @Ä ! f0; 1g is the characteristic function of ° (i.e., ° .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 ° .z@ /, z 2 @ Tn0, where z 7! z@ is the projection of @ Tn0 to @Ä defined earlier for C 2-domains. 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 '"¡ · ° · '"C, and the linear measure of sets fz 2 @Ä : j'"C.z/ ¡ ° .z/j C j'"¡.z/ ¡ ° .z/j > 0g goes to 0 as " ! 0. In other words, '"C and '"¡ are two continuous “step” functions on @Ä that approximate ° 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, volume-triangulations, 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 2-complex 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 volume-triangulation 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. CP-Regularity. 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 CP-regularity. 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 quasi-uniformity 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, quasi-uniform) if degrees (quasi-uniformity 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 (quasi-uniform). 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 quasi-uniform 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 complex-variable 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 quasi-uniform 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 real-valued 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 quasi-uniform 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 quasi-uniform 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 quasi-uniformity 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]. [Ad] R. A. Adams , Sobolev Spaces, Academic Press, New York, 1975 . [An] E. M. Andreev , Convex polyhedra in Lobacevskii space (in English) Math. USSR-Sb . 10 ( 1970 ), 413 - 440 . [BR] R. E. Bank and D. J. Rose , Some error estimates for the box method , SIAM J. Numer. Anal . 24 ( 1987 ), 777 - 787 . [BeS1] A. F. Beardon and K. Stephenson , The uniformization theorem for circle packings , Indiana Univ. Math. J. 39 ( 1990 ), 1383 - 1425 . [BeS2] A. F. Beardon and K. Stephenson , The Schwarz-Pick lemma for circle packings , Illinois J. Math. [Ca] Z. Cai , On the finite volume element method, Numer . Math. 58 ( 1991 ), 713 - 735 . [CMM] Z. Cai , J. Mandel , and S. McCormick , The finite volume element method for diffusion equations on general triangulations , SIAM J. Numer. Anal . 28 ( 1991 ), 392 - 402 . [Ci] P. G. Ciarlet , The Finite Element Method for Elliptic Problems , North-Holland, Amsterdam, 1978 . [CdV] Y. Colin de Verdie`re, Un principe variationnel pour les empilements de cercles , Invent. Math . 104 ( 1991 ), 655 - 669 . [CdVM] Y. Colin de Verdie`re and F. Mathe´us, Empilements de cercles et approximations conformes, Actes de la Table Ronde de Ge´ome´trie Riemannienne en l'honneur de Marcel Berger, A. L. Besse (e´diteur), Collection SMF Se´minaires et Congre`s , vol. 1 , 1996 , pp. 253 - 272 . [Co] H. S. M. Coxeter , Introduction to Geometry, Wiley, New York, 1969 . [DS] P. Doyle and J. Snell , Random Walks and Electric Networks , Carus Mathematical Monographs , vol. 22 , MAA, Washington, DC, 1984 . [Du1] T. Dubejko , Recurrent random walks, Liouville's theorem, and circle packing , Math. Proc. Cambridge Philos. Soc . 121 ( 1997 ), 531 - 546 . [Du2] T. Dubejko , Approximation of analytic functions with prescribed boundary conditions by circlepacking maps, Discrete Comput . Geom . 17 ( 1997 ), 67 - 77 . [Du3] T. Dubejko , Random walks on circle packings, Contemp . Math. 211 ( 1997 ), 169 - 182 . [GT] D. Gilbarg and N. S. Trudinger , Elliptic Partial Differential Equations of Second Order, 2nd edn ., Springer-Verlag, New York, 1983 . [Ha] W. Hackbusch , On first and second order box schemes , Computing 41 ( 1989 ), 277 - 296 . [HR] Z.-X. He and B. Rodin , Convergence of circle packings of finite valence to Riemann mappings , Comm. Anal. Geom . 1 ( 1993 ), 31 - 41 . [Hn] B. Heinrich , Finite Difference Method s on Irregular Networks, Birkha¨user- Verlag , Basel, 1987 . [KS] I. Karatzas and S. E. Shreve , Brownian Motion and Stochastic Calculus , Springer-Verlag, New York, 1988 . [KSK] J. Kemeny , J. Snell , and A. Knapp , Denumerable Markov Chains, Springer-Verlag, New York, 1976 . [LV] O. Lehto and K. I. Virtanen , Quasiconformal Mapping in the Plane, 2nd edn ., Springer-Verlag, New York, 1973 . [OR] A. Oganesyan and L. A. Rukhovets , Variational-Difference Methods for Solving Elliptic Equations (in Russian) , Izdatielstvo Akademii Nauk Armyan SSR, Erevan , 1979 . [RS] B. Rodin and D. Sullivan , The convergence of circle packings to the Riemann mapping , J. Differential Geom . 26 ( 1987 ), 349 - 360 . [So] P. M. Soardi , Potential Theory on Infinite Networks, Lecture Notes in Mathematics , vol. 1590 , Springer-Verlag, Berlin, 1994 . [St1] K. Stephenson , Circle packings in the approximation of conformal mappings , Bull. Amer. Math. Soc. (Research Announcements) 23 ( 2 ) ( 1990 ), 407 - 415 . [St2] K. Stephenson , A probabilistic proof of Thurston's conjecture on circle packings, Rend . Sem. Mat. [St3] K. Stephenson , CirclePack (software) , http://www.math.utk.edu/»kens. [Th1] W. P. Thurston , The Geometry and Topology of 3-Manifolds , Princeton University Notes, Princeton University Press, Princeton, NJ, 1980 . [Th2] W. P. Thurston , The finite Riemann mapping theorem, Invited talk, an International Symposium at Purdue University on the occasion of the proof of the Bieberbach conjecture , March 1985 . Amer. Math. Soc . 69 ( 1950 ), 335 - 356 . [Wo] W. Woess , Random walks on infinite graphs and groups-a survey on selected topics , Bull. London Math. Soc . 26 ( 1994 ), 1 - 60 . [Ya] I. M. Yaglom , Complex Numbers in Geometry, Academic Press, New York, 1968 .


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FPL00009447.pdf

T. Dubejko. Discrete Solutions of Dirichlet Problems, Finite Volumes, and Circle Packings, Discrete & Computational Geometry, 1999, 19-39, DOI: 10.1007/PL00009447