Least-squares approximation method for solution of Laplace's equation

Least-squares approximation methods are described and analyzed for the solution of various boundary-value problems for Laplace's equation, in two- and three-dimensional regions. The Dirichlet problem is discussed on both interior and exterior regions. The main advantages of the method are: (a) The solution is reached without iteration. (b) It is not necessary to solve a large number of simultaneous equation as in finite-difference or finite- elements methods. The method is restricted to piecewise smooth and simply connected regions. In addition to that, in the three-dimensional case the region must be axisymmetrical and the boundary condition must be prescribed in such a way that it is symmetrical with respect to the symmetry axis, when spherical coordinates are used. A number of systematic problems for elliptical boundaries are carried out analytically (where possible) or numerically. The rates of convergence are observed, and it is found that the method is feasible, especially when the eccentricities of the ellipses, in the two-dimensional cases, are not too close to unity. The procedure is also formulated in general terms