Monte Carlo methods are generally known for solving field problems one point at a time, unlike other numerical methods such as the finite difference and finite element methods which provide the solution at all the grid nodes simultaneously. This paper provides a Monte Carlo technique for obtaining the solution everywhere at once. The technique uses absorbing Markov chains to obtain the transition probabilities for all of the grid nodes at once. The procedure is illustrated with some examples for homogeneous and inhomogeneous, rectangular and axisymmetric solution regions, and is found to be accurate. © 1997 John Wiley & Sons, Ltd.

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The method of lines, a semianalytical procedure, has become one of the standard tools for solving practical, complex electromagnetic field problems. In this paper, the method of lines is used in solving axisymmetrical problems involving Laplace's equation. One numerical example is used to verify the procedure. The results obtained compare well with analytical solution.