On the approximation of singular source terms in differential equations

We study differential equations with singular source terms. For such equations classical convergence results do not apply, as these rely on the regularity of the solution and the source terms. We study some elliptic and parabolic problems numerically and theoretically, and show that, with the right approximation of the singular source terms, full convergence order can be achieved away from the singularities, whereas the convergence will be poor in a vicinity of these.