The Numerical Solution of Laplace's Equation on a Wedge

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Abstract
Laplace's equation is considered on regions in the plane, with the boundary having corners; and the double-layer potential is used to derive a solution. The essential difficulties, both theoretically and numerically, are reduced to the case in which the boundary is a simple open wedge. The theoretical behaviour of the double layer integral equation is studied explicitly, and then piecewise linear and piecewise quadratic collocation methods are applied to the numerical solution of the equation. The major question of interest is the stability of the inverses of the approximating equations. The behaviour of the numerical methods is somewhat surprising, and it is much better than past analyses would have led one to expect.

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