The Rigidity Of Circle Packings And Discrete Boundary Value Problems

The theory of circle packings, which are configurations of circles hav-ing specified patterns of tangency, is a fast developing field of the border ofcomplex analysis and discrete di?erential geometry. Recent progress in thisarea was initiated by Thurston's idea about the approximation of the Rie-mann mapping by hexagonal circle packings in 1985. Shortly, B. Rodin andD. Sullivan proved the convergence of Thurston's scheme, which gave a newdiscrete geometry view of the Riemann mapping. After then, much researchon circle packings and their applications followed. In this thesis, our mainwork is as follows. First, we investigate the rigidity of infinite circle packingswith unbounded degree on Riemann sphere. Using the methods of windingnumber and index, combining with the finite covering theorem, we show thatany two infinite circle packings of unbounded degree, which have the samecombinatoric and almost fill the whole sphere, are M¨obius equivalent. Next,we study the discrete Dirichlet problems for di?usion equations. Based onthe regular quadrangulations of domain, we use the finite volume methodto define the discrete solutions of Dirichlet problems for di?usion equations.The estimation of the error between the discrete solutions and the classicalsolutions is obtained in terms of discrete semi-norms. It is proved that thesediscrete solutions converges in L2-space to the exact solutions.