Discrete solutions of Dirichlet problems, finite volumes, and circle packings
Convergence results are obtained for discrete solutions of Dirichlet problems for Poisson equations in the plane. The solutions are constructed for triangular grids, using the finite volume method where the volumes have sides perpendicular to, but not necessarily bisecting corresponding edges in underlying triangulations. It is shown that the discrete solutions of Poisson equations satisfy the maximum principle. Convergence of discrete solutions to the corresponding classical solutions is proved in the \(H^1\) norm and the \(L^2\) norms, under mild restrictions on the triangulations and volumes. With further restriction on the domain and the triangulations, uniform convergence on compacta is established. It is then shown that triangulations and volumes induced by circle packings have the desired properties, if the packings satisfy some natural combinatorial restrictions. The packings are then used via the discrete Riemann mapping theorem to pull the Dirichlet problem back to a problem in the disk. Finally, the connection between random walks induced by circle packings and Brownian motion in the plane is discussed.
scientific article; zbMATH DE number 1309665