In this article we develop a posteriori error estimates for second order
linear elliptic problems with point sources in two- and three-dimensional domains. We
prove a global upper bound and a local lower bound for the error measured in a weighted
Sobolev space. The weight considered is a (positive) power of the distance to the support
of the Dirac delta source term, and belongs to the Muckenhoupt’s class A2. The theory
hinges on local approximation properties of either Clément or Scott–Zhang interpolation
operators, without need of modifications, and makes use of weighted estimates for
fractional integrals and maximal functions. Numerical experiments with an adaptive
algorithm yield optimal meshes and very good effectivity indices.