On a Method to Subtract off a Singularity at a Corner for the Dirichlet or Neumann Problem

Let D be a plane domain partly bounded by two line segments which meet at the origin and form there an interior angle va > 0. Let U(x, y) be a solution in D of Poisson's equation such that either U or dll/dn (the normal derivative) takes prescribed values on the boundary segments. Let U(x, y) be sufficiently smooth away from the corner and bounded at the corner. Then for each positive integer N there exists a function Vn(x, y) which satisfies a related Poisson equation and which satisfies related boundary conditions such that U — Vn is AZ-times continuously differentiable at the corner. If 1/a is an integer Vn may be found explicitly in terms of the data of the problem for U. | In solving an elliptic partial differential equation by numerical methods the results proved about convergence of the numerical approximation to the actual solution frequently depend on differentiability properties of the (unknown) solution. In the work of Gerschgorin [2] and other papers written since, it is assumed that the solution of the partial differential equation has derivatives of order four which are continuous up to the boundary. If the boundary and all the data are sufficiently smooth there is, of course, no problem. In many cases, however, the boundary possesses a finite number of singularities, usually (in the two-dimensional case) in the form of corners; occasionally too, the boundary data may have jumps. Laasonen [3] has proved that convergence of the discrete solution to the actual solution holds for the Dirichlet problem, but that the convergence is slow in a neighborhood of the corner. In this paper we will consider a method to subtract off the singularity. The method is quite old (see Fox [1]), but includes results on the asymptotic behavior of solutions near a corner. In this light see the works of Lewy [4], Lehman [5], Wasow [6], and the author [7]. We consider a problem for which the solution is not known to be smooth. We then find, explicitly in terms of the boundary data, a solution to a related problem; then the difference between these two solutions is a solution to a third problem, and is sufficiently well-behaved to insure convergence of difference schemes. Finally, the sought solution can be found by adding the explicitly given one to the numerically-solved one. Let D be a plane domain partly bounded by two open line segments i\ and Ti, which share the origin as a common endpoint and form there an interior angle ira > 0. We assume that Ti is a subset of the positive x-axis and r2 makes an angle 7ra > 0 with the positive z-axis. Let F(x, y) be given in D and $,(x, y) (respectively ^i(x, y)) be given on r, (i = 1, 2). Let U(x, y) be a solution of AU = F in D, and have boundary values U(x, y) = $i(x, y) on T¿ (respectively dU/dn = ^¿; d/dn is Received January 16, 1967, revised July 18, 1968. * The author is indebted to the referee for many suggestions used in the preparation of this?. paper. 395 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use