The third boundary value problem for elliptic equations

Abstract

In [I], the author discussed the numerical solution of the heat conduction equation in an open rectangular region, under boundary conditions involving a linear combination of the function and its normal derivative. It was shown that the instability which was observed in the difference methods examined, could be traced to the existence of solutions of the differential equation which grew, asymptotically, with time. This numerical instability was important in the solution of the heat equation only for large values of the time, and so did not affect calculations which were carried out over a few time steps. But, in the numerical solution of elliptic equations by iteration, such asymptotic instability will prevent convergence of the iterative process to the original system of equations. It is the purpose of this note to demonstrate this fact, and also to discuss the solution of Laplace's and Poisson's equations, when the Laplacian operator is singular, in a sense to be defined. It wilt be shown that, for certain boundary conditions, the numerical solution of Laplace's equation is best obtained by direct methods, rather than by iterative methods. In addition, a condition for the existence of a solution of the singular problem is obtained. 2. The Difference Equations Let D be any closed region in (x, y) space, with boundary aD. The outward normal to D on aD is denoted by n. Consider the equation ~: + -~- = ¢ (x, y)