An application of the integrated penalty method to free boundary problems of laplace equation

Free boundary problems of Laplace equation defined in the annulus in are numerically solved. The original problem is transformed to an optimization problem. The state equation is approximated by an equation with a penalty term which approximates one of boundary conditions on the free boundary. The flux through the free boundary is calculated by the integration of the penalty term introduced above ( Integrated Penalty Method ). This penalized optimization problem is numerically solved by finite difference method. Incomplete Cholesky decomposition combined with the conjugate gradient method is used to solve systems of linear equations. Some numerical examples are given.