On the method of nets for a boundary problem with an oblique and a normal derivative

Abstract

In this article we construct error bounds for the solution of Poisson's equation by the method of nets, where the oblique or normal derivative is given on the boundary of the region. We shall be considering a system of difference equations, equal in number to the number of nodes of the net lying inside the region less one. The value of the function at one node is fixed and a corresponding equation “rejected”, and the system of difference equations thus reduced has a unique solution which can be calculated by an iterative method. An arbitrary constant is added to the resulting approximate solution. The book [1] contains a practical guide to the application of this method with a very simple approximation to the derivative on the boundary. However, the error bounds for the method in the solution of the problem with a given oblique or normal derivative on the boundary of the region have not been discussed before. The solution of the problem with an oblique derivative can be reduced to the solution of Hilbert's problem using the method of nets in [2], but since it is assumed that six derivatives of the required solution exist, the estimates obtained there are rough, of the order of O( h). The method of nets for Neumann's problem is discussed in [3]–[6]. In [3], in particular, a system of self-conjugate difference equations is constructed for the multidimensional case, its solution converging weakly to the generalized solution of Neumann's problem in the space W 2 (1) as h → 0. The difference scheme for the two-dimensional case is considered by reducing Neumann's problem for the Laplace equation to the Dirichlet problem, and in [5] the corresponding error bounds, of order O( h), are derived, on the assumption that the third derivatives of the required solution in the closed region are continuous and satisfy the Hölder condition, but that the fourth derivatives may have singularities at a finite number of points on the boundary. In the three-dimensional case, the method of orthogonal projections is used in [3] and [4]. In the solution of Neumann's problem, the local approximation errors of this method at internal nodes of the net are of the order of O( h 2). The error bounds of the difference ratios are evaluated in [5], and in [6] the smoothness of the difference solution inside the region is discussed. Finally, we must mention [7], which gives the error bounds of the method of nets for the third boundary problem and for mixed boundary conditions, when the normal derivative is given on part of the boundary. However, these bounds do not apply to Neumann's problem or to the problem with an oblique derivative on the whole boundary for Laplace and Poisson's equations, which we consider here. To approximate to the oblique and normal derivative we shall use the difference operators of high accuracy, which were suggested in [8]–[11], and, assuming that the fourth derivatives of the required solution are bounded, we derive error bounds of order O(h 2−2 ¦in h¦ In 2 h) , where h is the step of the net for the two-dimensional case. If first difference relations are used to define the derivative on the boundary of the region (see [12], [1]). the error estimate has the order O(h 1−1 ¦in h¦ in 2 h) , This is done for the two cases of “starred” boundary conditions (§ 2) and of an oblique derivative whose direction does not coincide with that of the tangent to the boundary (§ 3). In the concluding paragraph (§ 4) we consider the application of boundary difference operators of othe types [3], make estimates for the multidimensional case and give a new method for constructing the error bounds in the solution of Dirichlet's problem for the Laplace equation by the method of nets.