Analysis of least squares finite element methods for a parameter-dependent first-order system *

We consider polynomials orthagonal with respect to a measure μ with an absolutely continuous component and a finite discrete part. We prove that subject to certatin integrability conditions, the polynomials satisfy a second order differential equation. The zeroes of such polynomials determine the equilibrium position of movable n unit charges in an external field determined by the measure μ. We also evaluate the discriminant of such orthagonal polynomials and use it to compute the total energy of the system at equilibrium in terms of the recursion coefficients of the orthonormal polynomials. We also investigate several explicit models, the Koornwinder polynomials, the Ginzburg-Landau potential and the generalized Jacobi weights.