Approximation of Solutions of Mixed Boundary Value Problems for Poisson's Equation by Finite Differences

Tile region R :is a bounded connected open set in the (x, y) plane whose boundary C consists of the two parts C,, and C2. The symbol k is the Laplace operator _~ (O~/Ox 2) + (02/0y2), and 0/0n denotes differentiation with respect to the outward-directed normal oa C~. The functions f, g and H are defined to be sufficierttly smooth functions on R, C, and C2 respectively. The function a is required to satisfy the following conditions on C~ : (a) piecewise continuity with a finite nmnber of discontinuities, (b) piecewise differentiability, (c) at all points of continuity, either a = 0 (the set C~ ')) or a => am > 0, where a,~ is a constaIlt (the set C~2)). We restrict our considerations to the cases in which either the set C2 or C[ ~) conl rains a nonempty open subset of C. In these eases (1.1) has a unique solution i provided the data and boundary are sufficiently smooth. The case in which C2 is i all of C is just the Diriehlet problem. Results for this special case are contained in [8]. We are interested in fornmlating a finite-difference analogue of (1.1) which has the following properties: (a) The boundary approximations involve at most three interior points (and one boundary point) (b) The matrix of the system is