Absorbing Markov Chain solution for Possion's equation

Abstract

The Markov chain method for solving Laplace's equation with Dirichlet boundary condition has been discussed in a few papers. This paper presents an absorbing Markov chain method to solve Possion's equation with Dirichlet boundary condition. In the Markov chain, the fundamental matrix N defines the transient relationships for a randomly walking particle from state s/sub j/ passing through state s/sub i/ before it reaches the absorbing state. From the fundamental matrix N and the probability matrix R from non-absorbing states to absorbing states, the contributions of boundary points and interior points to the potential of internal points are defined. The absorbing Markov chain method overcomes a major disadvantage of classic Monte Carlo methods that they are only capable of calculating the potential at a single point at a time unlike other numerical methods such as finite difference and finite element methods which provide simultaneously the solution at all the grid nodes. This paper presents an example to show the accuracy of the absorbing Markov chains solution.