NUMERICAL. ANALYSIS OF BOUNDARY VALUE PROBLEM OF ELLIPTIC TYPE BY MEANS PENALTY AND THE FINITE DIFFERENCE AND ITS APPLICATION TO FREE BOUNDARY PROBLEM

Abstract : The authors study a numerical method for solving free boundary problems of elliptic type. Usually these problems are prescribed with two boundary conditions on the free boundary. One of them is the Dirichlet condition and the other is the Neumann condition. Their method is to transform the original problem to an optimization problem. The state equation is approximated by an equation with a penalty term in which the Dirichlet condition on the free boundary is approximately satisfied. The outward normal derivative included in the Neumann condition through the free boundary is calculated by using the asymptotic behavior of the solution of the penalized state equation. Presented is a method to solve this penalized optimization problem. Also the error estimate of the discretized state equation by the finite difference method is given. (Author)