The finite element-Galerkin method for singular self-adjoint differential equations

We investigate the finite element-Galerkin method for singular self-adjoint second-order differential expressions. The weak formulation of the problem involves integration by parts, which allows the use of the usual piecewise linear functions. Our analysis shows that the method produces the solution corresponding to a particular self-adjoint realization of the differential expression. We also propose two algorithms to approximate the solution of any self-adjoint realization. Numerical examples are given to illustrate the analysis as well as the proposed algorithms.