Discrete solutions of Dirichlet problems by circle patterns and finite volumes

Abstract

Given a simply connected domain , the cellular decompositions for approximation, whose faces are not necessarily triangles, are produced by a family of Delaunay circle patterns. The regularities of cellular decompositions and associated volume cellular decompositions are derived by the properties of circle patterns. The discrete Dirichlet problems for Poisson equations with these cellular decompositions are defined by means of finite volume techniques. Based on the regularities of decompositions, the error estimate between the discrete solutions in these cellular decompositions and the classical one in is obtained in terms of discrete seminorms and it is proved that these discrete solutions converge in and to the exact solution, respectively.

Keywords:

• circle pattern
• finite volume
• Poisson equation
• discrete solution

AMS Subject Classifications:

• 65N10
• 65N30
• 52C15
• 30G62

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research is partially supported by NSF of China [grant number 11161004], [grant number 11171354] and NSF of Guangxi [grant number 2013GXNSFAA019015].